Central European Journal of Mathematics

CEJM 2(5) 2004 615–623 On almost hyperHermitian structures on Riemannian manifolds and tangent bundles Serge A. Bogdano...

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CEJM 2(5) 2004 615–623

On almost hyperHermitian structures on Riemannian manifolds and tangent bundles Serge A. Bogdanovich∗, Alexander A. Ermolitski† Cathedra of Mathematics, Belorussian State Pedagogical University, st. Sovietskaya 18, Minsk, 220050, Belarus

Received 15 December 2003; accepted 15 February 2004 Abstract: Some results concerning almost hyperHermitian structures are considered, using the notions of the canonical connection and the second fundamental tensor field h of a structure on a Riemannian manifold which were introduced by the second author. e on an almost Hermitian manifold M an With the help of any metric connection ∇ almost hyperHermitian structure can be constructed in the defined way on the tangent bundle TM. A similar construction was considered in [6], [7]. This structure includes two basic anticommutative almost Hermitian structures for which the second fundamental tensor fields h1 and h2 are computed. It allows us to consider various classes of almost hyperHermitian structures on TM. In particular, there exists an infinite-dimensional set of almost hyperHermitian structures on TTM where M is any Riemannian manifold. c Central European Science Journals. All rights reserved.

Keywords: Riemannian manifolds, almost hyperHermitian structures, tangent bundle MSC (2000): 53C15, 53C26

1

Some remarks on almost hyperHermitian structures

10 . Let (M, J, g) be an almost Hermitian manifold i.e. J 2 = −I and g(JX, JY ) = g(X, Y ) for X, Y ∈χ(M), where g is a fixed Riemannian metric on M. For any Riemannian metric g˜ on M such a metric g is defined by the formula g(X, Y ) = 12 (˜ g (X,Y ) + g˜(JX, JY )), X, Y ∈χ(M). Let ∇ be the Riemannian connection of the metric g. Then one can define a connection ¯ ∇ on M by ∗ †

E-mail: [email protected] E-mail: [email protected]

616 S.A. Bogdanovich, A.A. Ermolitski / Central European Journal of Mathematics 2(5) 2004 615–623

¯ X Y = 1 (∇X Y − J∇X JY ) = ∇X Y + 1 ∇X (J)JY, X, Y ∈ χ(M). ∇ (1) 2 2 ¯ is called the canonical connection of the pair (J, g), or more The connection ∇ ¯ precisely of the corresponding G-structure, where G = U(n), [2]. In particular, ∇g= 0, ¯ ∇J= 0. The tensor field h is called the second fundamental tensor field of the pair (J, g) [2], where ¯ X Y = − 1 ∇X (J)JY = 1 (∇X Y + J∇X JY ), X, Y ∈ χ(M); (2) hX Y = ∇X Y − ∇ 2 2 hXY Z = g(hX Y, Z) = −hXZY . (3) In particular, the classification given in [3] can be rewritten in terms of the tensor field h, [2]. Let dim M ≥ 6 and 2β(X) = δΦ(JX ), where Φ(X, Y ) = g(JX, Y ). Then we have Class

Defining condition h= 0

K U1 =NK

hX X= 0

U2 =AK

σhXY Z = 0

U3 =SK∩H U4

hXY Z – hJXJY Z = β(Z) = 0 1 hXY Z = 2(n−1) [<X, Y >β(Z) – <X, Z>β(Y ) – <X, JY >β(JZ ) + + <X, JZ >β(JY )]

U1 ⊕U2 =QK

hXY JZ = hJXY Z

U3 ⊕U4 =H

N (J) = 0 or hXY JZ = – hJXY Z

U1 ⊕U3

hXXY – hJXJXY = β(Z) = 0

U2 ⊕U4 U1 ⊕U4 U2 ⊕U3

σ[hXY JZ – hXXY = –

1 2(n−1) [<X,

1 (n−1) <JX,

Y >β(X) – ||X||2 β(Y ) – <X, JY >β(JX )]

σ[hXY JZ + hJXY Z ] = β(Z) = 0

U1 ⊕U2 ⊕U3 =SK U1 ⊕U2 ⊕U4

Y >β(Z)] = 0

β =0 hXY JZ – hJXY Z =

1 (n−1) [<X,

Y >β(JZ ) – <X, Z>β(JY ) + +<X, JY >β(Z) - <X, JZ >β(Y )]

U1 ⊕U3 ⊕U4

hXJXY + hJXXY = 0

U2 ⊕U3 ⊕U4

σ[hXY JZ + hJXY Z ] = 0 No condition

U Table 1

20 . We consider an almost hyperHermitian structure (ahHs) on a manifold M consisting of (J1 , J2 , J3 , g), where Ji2 = −I, J1 J2 = −J2 J1 = J3 , g(JiX, Ji Y ) = g(X, Y ), i= 1, 2, 3.

S.A. Bogdanovich, A.A. Ermolitski / Central European Journal of Mathematics 2(5) 2004 615–623 617

For any Riemannian metric g˜ such a metric g can be defined by the formula g (X,Y ) + g˜(J1 X, J1 Y ) + g˜(J2 X, J2 Y ) + g˜(J3 X, J3 Y )), X, Y ∈χ(M). g(X, Y ) = 41 (˜ ¯ If ∇ is the Riemannian connection of the metric g, then the canonical connection ∇ in the sense of [2] of the ahHs has the following form ¯ X Y = 1 (∇X Y − J1 ∇X J1 Y − J2 ∇X J2 Y − J3 ∇X J3 Y ), X, Y ∈ χ(M). ∇ 4 ¯ ¯ i = 0, i= 1, 2, 3. In particular, ∇g= 0, ∇J

(4)

Proposition 1.1. Let (M, J1 , g) be a Kaehlerian manifold i.e. ∇J1 = 0 on M. Then the connection given by (4) coincides with those defined by (1) for (M, J2 , g) and (M, J3 , g). In particular, the second fundamental tensor fields of (M, J2 , g) and (M, J3 , g) are the same. Proof. ¯ X Y = 1 (∇X Y − J 2 ∇X Y − J2 ∇X J2 Y − J1 J2 ∇X J1 J2 Y ) = ∇ 1 4 1 = (2∇X Y − J2 ∇X J2 Y − J1 J2 J1 ∇X J2 Y ) = 4 1 = (∇X Y − J2 ∇X J2 Y ) = 2 1 = (∇X Y − J3 J1 ∇X J3 J1 Y ) = 2 1 = (∇X Y − J3 ∇X J3 Y ). 2 To illustrate the situation described in proposition 1.1 we consider the following example. Example 1.2. Let (M, J, g) be an almost Hermitian manifold and let (J, g) belong to one of the classes in Table 1, dimM= 4n. Further, define orthonormal vector fields X1 , ..., X2n , JX 1 , ..., JX 2n on some open neighborhood Uof a point p ∈ M. Assuming J1 = J on U, we get an almost Hermitian manifold (U, J1 , g) which also belongs to the corresponding class. Then, we can define J2 by the following equalities J2 X1 = Xn+1 , J2 X2 = Xn+2 , . . . , J2 Xn = X2n ; J2 Xn+1 = −X1 , J2 Xn+2 = −X2 , . . . , J2 X2n = −Xn ; J2 (J1 X1 ) = −J1 Xn+1 , J2 (J1 X2 ) = −J1 Xn+2 , . . . , J2 (J1 Xn ) = −J1 X2n ; J2 (J1 Xn+1 ) = J1 X1 , J2 (J1 Xn+2 ) = J1 X2 , . . . , J2 (J1 X2n ) = J1 Xn . It is clear that J1 J2 = −J2 J1 , J12 = J22 = −I and g(JiX, Ji Y ) = g(X, Y ), i= 1, 2, 3 on U, where J3 = J1 J2 , X, Y ∈χ(U). If (M, J, g) is a Kaehlerian manifold (class K ) and dimM= 4n then we obtain the situation given in proposition 1.1 on the almost hyperHermitian manifold (U, J1 , J2 , J3 , g).


One can find examples of the structures from Table 1 in [2], [3]. To get an almost Hermitian manifold of dimension 4n we can take a Riemannian product M × M. Problem 1.3. Let (U α , U β ) be any pair of the classes from Table 1 (α, β = 0, ..., 15). Can one construct such examples of ahHs (J1 , J2 , J3 , g) on manifolds such that (J1 , g) belongs to the class U α and (J2 , g) belongs to the class U β ? The cases (K , K ) and (U , U ) are easily illustrated. 30 . Definition 1.4. [5] A connected Riemannian manifold (M, g) with a family of local isometries {sx : x ∈ M} is called a locally k-symmetric Riemannian space (k − s. l. R. s.) if the following axioms are fulfilled : (a) sx (x) = x and x is the isolated fixed point of the local symmetry s x ; (b) the tensor field S : Sx = (sx )∗x is smooth and invariant under any local isometry s x ; (c) S k = I and k is the least of such positive integers. e can be defined by the If M is a k-s. l. R. s. then the unique canonical connection ∇ following formula (see [2]) k−1

k−1

X 1X j e X Y = ∇X Y − 1 ∇ ∇X (S j )S k−j Y = S ∇X S k−j Y , X, Y ∈ χ(M). k j=1 k j=0

(5)

e =∇ eR ˜ = ∇h e = ∇S e = 0, S(R) ˜ = R, ˜ S(h) = h, S(g) = g, where Further, we have ∇g e and R ˜ is the curvature tensor field of ∇. e h=∇−∇ Let M be such a k-s. l. R. s. that Sx = (sx )∗x has only complex eigenvalues a1 ± b1 i, ..., ar ± br i. We define distributions Di , i= 1, ..., r by Di = ker(S 2 − 2ai S + I). Every X ∈χ(M) has the unique decomposition X = X1 + ... + Xr , where Xi ∈ Di , i= 1, ..., r. An almost complex structure J on M is defined by r X 1 (S − ai I)Xi . JX = b i=1 i

(6)

¯ It is proved in [2] that (J, g) is an almost Hermitian structure and the connections ∇ e defined by (1) and (5) respectively coincide i.e. ∇ e =∇ ¯ on M. and ∇ Proposition 1.5. Let (J1 , J2 , J3 , g) be such an ahHs on a k-s. l. R. s. (M, g) that J1 = J, where J is defined by (6). In this case J 2 and J 3 are not invariant with respect to the family {sx : x ∈ M}. Proof. Otherwise we have (sx )∗x · J2 X = J2 · (sx )∗x X i.e. S · J2 X = J2 ·SX. Using (6) we get r r X X 1 1 J2 J1 X = (J2 S − ai J2 Xi ) = (S − ai I)J2 Xi , b b i i i=1 i=1


J1 J2 X =

r X 1 (S − ai I)(J2 X)i , b i i=1

(S 2 − 2ai S + I)J2 Xi = J2 (S 2 − 2ai S + I)Xi = 0, therefore J2 Xi = (J2 X)i and J1 J2 X= J2 J1 X for any X ∈χ(M). But we have J1 J2 X = − J2 J1 X, hence J3 X = J1 J2 X= 0 i.e. J3 = 0. We obtained the contradiction because J3 is nonsingular. By similar arguments J3 can not be invariant with respect to the family {sx : x ∈ M}.

2

HyperHermitian structures on tangent bundles

00 . Let (M, g) be a Riemannian manifold and TM be its tangent bundle. For a metric e (∇g e = 0) we consider the connection map K ˜ of ∇ e [1], [4], defined by the connection ∇ formula e X Z = KZ ˜ ∗ X, ∇ (7) where Z is considered as a map from M into TM and the right side means a vector field ˜ ∗ Xp ∈ Mp . on M assigning to p ∈ M the vector KZ ˜ |T M and this n-dimensional subspace of If U ∈TM, we denote by HU the kernel of K U TM U is called the horizontal subspace of TM U . Let π denote the natural projection of TM onto M, then π∗ is a C ∞ -map of TTM onto TM. If U ∈TM, we denote by VU the kernel of π∗|T MU and this n-dimensional subspace of TM U is called the vertical subspace of TM U (dimTM U = 2dimM= 2n). The following maps are isomorphisms of corresponding vector spaces (p= π (U)) π∗|T MU : HU → Mp ,

˜ |T M : VU → Mp K U

and we have T MU = HU ⊕ VU . If X ∈χ(M), then there exists exactly one vector field on TM called the “horizontal lift” ¯ h (X ¯ v ), such that for all U ∈TM : (resp. “vertical lift”) of X and denoted by X ¯ Uh = Xπ(U ) , π∗ X

˜X ¯ Uh = 0π(U ) , K

(8)

¯ v = 0π(U ) , π∗ X U

˜X ¯ v = Xπ(U ) . K U

(9)

˜ be the curvature tensor field of ∇, e then following [1] we write Let R ¯ v , Y¯ v ] = 0, [X ¯Y v , ¯ h , Y¯ v ] = ∇ ˜X [X

(10)

¯ h , Y¯ h ]U ) = [X, Y ], π∗ ([X

(12)

˜ X ¯ h , Y¯ h ]U ) = R(X, ˜ K([ Y )U.

(13)

(11)


¯ =X ¯h ⊕ X ¯ v and Y¯ = Y¯ h ⊕ Y¯ v on TM the natural Riemannian For vector fields X metric < , > is defined on TM by the formula ¯ Y¯ >= g(π∗ X, ¯ π∗ Y¯ ) + g(K ˜ X, ¯ K ˜ Y¯ ). < X,

(14)

It is clear that the subspaces HU and VU are orthogonal with respect to < , >. ¯ 1h , X ¯ 2h , . . . , X ¯ nh , X ¯ 1v , X ¯ 2v , . . . , X ¯ nv are orthonormal vector fields It is easy to verify that X on TM if X1 , X2 , . . ., Xn are those on M i.e. g(Xi , Xj ) = δji . 10 . We define a tensor field J1 on TM by the equalities ¯h = X ¯ v , J1 X ¯ v = −X ¯ h , X ∈ χ(M). J1 X

(15)

For X ∈χ(M) we get ¯ = J1 (J1 (X ¯h ⊕ X ¯ v )) = J1 (−X ¯h ⊕ X ¯ v ) = −(X ¯h ⊕ X ¯ v ) = −I X ¯ J12 X and J12 = −I. For X, Y ∈χ(M) we obtain ¯ J1 Y¯ >=< −X ¯h ⊕ X ¯ v , −Y¯ h ⊕ Y¯ v >=< −X ¯ h , −Y¯ h > + < X ¯ v , Y¯ v >, < J1 X, ¯ Y¯ >=< X ¯h ⊕ X ¯ v , Y¯ h ⊕ Y¯ v >=< X ¯ h , Y¯ h > + < X ¯ v , Y¯ v > < X, ¯ J1 Y¯ > = <X, ¯ Y¯ >, (TM, J1 , < , >) is an almost Hermitian and it follows that <J1 X, manifold. Further, we want to analyze the second fundamental tensor field h1 of the pair (J1 , < , >). ˆ of the metric < , > on TM is defined by the formula The Riemannian connection ∇ (see [4]) ˆ X¯ Y¯ , Z¯ > = 1 (X ¯ < Y¯ , Z¯ > +Y¯ < Z, ¯ X ¯ > −Z¯ < X, ¯ Y¯ > + < Z, ¯ [X, ¯ Y¯ ] > + + < X, ¯ [Z, ¯ Y¯ ] >), X, Y, Z ∈ χ(M). + < Y¯ , [Z, (16) ¯ Y¯ , Z¯ on TM we obtain Using (2), (3) for orthonormal vector fields X, 1 ˆ X¯ Y¯ + J1 ∇ ˆ X¯ J1 Y¯ , Z¯ >= h1X¯ Y¯ Z¯ = < h1X¯ Y¯ , Z¯ >= < ∇ 2 1 ˆ X¯ Y¯ , Z¯ > − < ∇ ˆ X¯ J1 Y¯ , J1 Z¯ >) = = (< ∇ 2 1 ¯ Y¯ ], Z¯ > + < [Z, ¯ X], ¯ Y¯ > + < [Z, ¯ Y¯ ], X ¯ >− = (< [X, 4 ¯ J1 Y¯ ], J1 Z¯ > − < [J1 Z, ¯ X], ¯ J1 Y¯ > − < [J1 Z, ¯ J1 Y¯ ], X ¯ >). − < [X,

(17)

Using (10)–(13) and (17) we consider the following cases for the tensor field h1 assuming all the vector fields to be orthonormal.


h1X¯ h Y¯ h Z¯ h =

1 (< [X¯h , Y¯h ], Z¯h > + < [Z¯h , X¯h ], Y¯h > + < [Z¯h , Y¯h ], X¯h > − 4 − < [X¯h , J1 Y¯h ], J1 Z¯h > − < [J1 Z¯h , X¯h ], J1 Y¯h > − 1 − < [J1 Z¯h , J1 Y¯h ], X¯h >) = (g([X, Y ], Z) + g([Z, X], Y ) + 4 ¯ h v ¯ +g([Z, Y ], X)− < [X , Y ], Z¯v > − < [Z¯v , X¯h ], Y¯v > − 1 1 e X Y, Z) − − < [Z¯v , Y¯v ], X¯h >) = g(∇X Y, Z) − (g(∇ 2 4 e X Z, Y )) = 1 (g(∇X Y, Z) − g(∇ e X Y, Z)). −g(∇ (1.10 ) 2

1 ¯ h , Y¯ h ], Z¯ v > + < [Z¯ v , X¯h ], Y¯h > + < [Z¯v , Y¯h ], X¯h > − (< [X 4 ¯ h , J1 Y¯ h ], J1 Z¯ v > − < [J1 Z¯ v , X ¯ h ], J1 Y¯ h > − − < [X ¯ h >) = 1 (g(R(X, ˜ − < [J1 Z¯ v , J1 Y¯ h ], X Y )U, Z) + 4 ¯ h ], Y¯ v >) = 1 (g(R(X, ˜ ˜ + < [Z¯ h , X Y )U, Z) + g(R(Z, X)U, Y )) = 4 1 ˜ ˜ = − (g(R(X, Y )Z, U) + g(R(Z, X)Y, U)). (2.10 ) 4

h1X¯h Y¯h Z¯v =

By similar arguments we obtain 1 ˜ ˜ h1X¯ h Y¯ v Z¯h = − (g(R(Z, X)Y, U) + g(R(X, Y )Z, U)). 4 1 ˜ Y )X, U)). h1X¯ v Y¯ h Z¯ h = − (g(R(Z, 4 1 ˜ h1X¯ v Y¯ v Z¯v = (g(R(Z, Y )X, U)). 4 h1X¯ v Y¯ v Z¯h = 0. h1X¯ v Y¯ h Z¯ v = 0.

(3.10 ) (4.10 ) (5.10 ) (6.10 ) (7.10 )

1 ˜ X Y, Z) − g(∇X Y, Z)). h1X¯ h Y¯ v Z¯ v = (g(∇ (8.10 ) 2 20 . Now assume additionally that we have an almost Hermitian structure J on (M, g). We define a tensor field J2 on TM by the equalities ¯ h = JX J2 X

h

¯ v = − JX , J2 X

v

, X ∈ χ(M).

For X ∈ χ(M) we have ¯ = J2 (J2 (X ¯h ⊕ X ¯ v )) = J2 ( JX J22 X

h

v ¯h ⊕ X ¯ v ) = −I X ¯ ⊕ − JX ) = −(X

and J22 = −I.

(18)


For X, Y ∈ χ(M) we get ¯ J2 Y¯ > = < JX h ⊕ − JX v , JY h ⊕ − JY v >=< JX h , JY h > + < J2 X, v v + < JX , JY >= g(JX, JY ) + g(JX, JY ) = ¯ h , Y¯ h > + < X ¯ v , Y¯ v > = g(X, Y ) + g(X, Y ) =< X ¯h ⊕ X ¯ v , Y¯ h ⊕ Y¯ v >=< X, ¯ Y¯ > . =<X Further, we obtain ¯ = J1 ( JX J1 (J2 X)

h

v h v ⊕ − JX ) = JX ⊕ JX ,

¯ = J2 (−X ¯h ⊕ X ¯ v ) = − JX J2 (J1 X)

h

⊕ − JX

v

.

Thus, we get J1 J2 = −J2 J1 = J3 and ahHs (J1 , J2 , J3 , < , >) on TM has been constructed. ¯ Y¯ , Z¯ on TM we obtain Using (2), (3) for orthonormal vector fields X, 1 ˆ X¯ Y¯ + J2 ∇ ˆ X¯ J2 Y¯ , Z¯ >= h2X¯ Y¯ Z¯ = < h2X¯ Y¯ , Z¯ >= < ∇ 2 1 ˆ X¯ Y¯ , Z¯ > − < ∇ ˆ X¯ J2 Y¯ , J2 Z¯ >) = = (< ∇ 2 1 ¯ Y¯ ], Z¯ > + < [Z, ¯ X], ¯ Y¯ > + < [Z, ¯ Y¯ ], X ¯ >− = (< [X, 4 ¯ J2 Y¯ ], J2 Z¯ > − < [J2 Z, ¯ X], ¯ J2 Y¯ > − < [J2 Z, ¯ J2 Y¯ ], X ¯ >). − < [X,

(19)

Using (10)–(13) and (19) we consider the following cases for the tensor field h2 assuming all the vector fields to be orthonormal. h2X¯ h Y¯ h Z¯ h =

1 ¯ h , Y¯ h ], Z¯ h > + < [Z¯ h , X ¯ h ], Y¯ h > + < [Z¯ h , Y¯ h ], X ¯h > − (< [X 4 ¯ h , J2 Y¯ h ], J2 Z¯ h > − < [J2 Z¯h , X¯h ], J2 Y¯h > − − < [X 1 − < [J2 Z¯h , J2 Y¯h ], X¯h >) = (g([X, Y ], Z) + g([Z, X], Y ) + 4 +g([Z, Y ], X) − g([X, JY ], JZ) − g([JZ, X], JY ) − 1 −g([JZ, JY ], X)) = (g(∇X Y, Z) − g(∇X JY , JZ)) = hXY Z . (1.2 0 ) 2

1 (< [X¯h , Y¯ h ], Z¯v > + < [Z¯v , X¯h ], Y¯h > + < [Z¯v , Y¯ h ], X¯h > − 4 − < [X¯h , J2 Y¯h ], J2 Z¯v > − < [J2 Z¯v , X¯h ], J2 Y¯h > − 1 ˜ ˜ − < [J2 Z¯v , J2 Y¯h ], X¯h >) = (g(R(X, Y )U, Z) + g(R(X, JY )U, JZ)) = 4 1 ˜ ˜ = − (g(R(X, Y )Z, U) + g(R(X, JY )JZ, U)). (2.2 0 ) 4

h2X¯ h Y¯ h Z¯ v =

By similar arguments we obtain 1 ˜ ˜ h2X¯ h Y¯ v Z¯h = (g(R(X, Z)Y, U) + g(R(X, JZ)JY , U)). 4

(3.2 0 )


1 ˜ ˜ h2X¯ v Y¯ h Z¯h = − (g(R(Z, Y )X, U) − g(R(JZ, JY )X, U)). 4 h2X¯ v Y¯ v Z¯ v = 0.

(4.2 0 ) (5.2 0 )

h2X¯ v Y¯ v Z¯h = 0.

(6.2 0 )

h2X¯ v Y¯ h Z¯ v = 0.

(7.2 0 )

1 e X Y, Z) − g(∇ e X JY , JZ)). h2X¯ h Y¯ v Z¯v = (g(∇ (8.2 0 ) 2 It is clear that the construction of the ahHs on TM strongly depends on the connection e ∇ and we can obtain in this way an infinite dimensional set of ahHs. Theorem 2.1. Let (M, g, J) be an almost Hermitian manifold. Then there exists an infinite family of ahHs on TM (in particular, such structures can be constructed by the method above). Corollary 2.2. Let (M, g) be a Riemannian manifold. Then there exists an infinite set of ahHs on TTM.

References [1] P. Dombrowski: “On the Geometry of the Tangent Bundle”, J. Reine und Angew. Math., Vol. 210, (1962), pp. 73–88. [2] A.A. Ermolitski: Riemannian manifolds with geometric structures, BSPU, Minsk, 1998 (in Russian). [3] A. Gray and L.M. Herwella: “The sixteen classes of almost Hermitian manifolds and their linear invariants”, Ann. Mat. pura appl., Vol. 123, (1980), pp. 35–58. [4] D. Gromoll, W. Klingenberg and W. Meyer: Riemannsche geometrie im groβen, Springer, Berlin, 1968 (in German). [5] O. Kowalski: Generalized symmetric space, Lecture Notes in Math, Vol. 805, Springer-Verlag, 1980. [6] F. Tricerri: “Sulle varieta dotate di due strutture quusi complesse linearmente indipendenti”, Riv. Mat. Univ. Parma, Vol. 3, (1974), pp. 349–358 (in Italian). [7] F. Tricerri: “Conessioni lineari e metriche Hermitiene sopra varieta dotate di due strutture quasi complesse”, Riv. Mat. Univ. Parma, Vol. 4, (1975), pp. 177–186 (in Italian).

CEJM 2(5) 2004 624–662

Review article

Lie groupoids as generalized atlases Jean Pradines∗ Université Paul Sabatier, 26, rue Alexandre Ducos, F31500 Toulouse, France

Received 15 December 2003; accepted 1 April 2004 Abstract: Starting with some motivating examples (classsical atlases for a manifold, space of leaves of a foliation, group orbits), we propose to view a Lie groupoid as a generalized atlas for the “virtual structure” of its orbit space, the equivalence between atlases being here the smooth Morita equivalence. This “structure” keeps memory of the isotropy groups and of the smoothness as well. To take the smoothness into account, we claim that we can go very far by retaining just a few formal properties of embeddings and surmersions, yielding a very polymorphous unifying theory. We suggest further developments. c Central European Science Journals. All rights reserved.

Keywords: Lie groupoids, spaces of leaves, orbit spaces MSC (2000): 58H05

1

Introduction

The aim of the present lecture is, rather than to present new results, to sketch some unifying concepts and general methods wished to be in Charles Ehresmann’s spirit. As usual I am expecting that geometers will think these sorts of concepts are too general and too abstract for being useful, while categoricists will estimate they are too special and too concrete for being interesting. However let us go. In the following, I shall be concerned with a certain structure B (basically thought as a manifold) endowed with a certain equivalence relation denoted by ∼ or R, and I would like to describe what kind of smoothness or structure is inherited from B by the quotient q set Q = B/R. The canonical projection will be denoted by B → Q. The relation R will ∗

E-mail: [email protected]

J. Pradines / Central European Journal of Mathematics 2(5) 2004 624–662

625

be identified with its graph, defined by the following pullback square, in which β = pr1 , R α

∨ B

β

>B

pb

q

∨ > Q q

τ

R α = pr2 . We also denote by R → B × B the canonical injection, with τR = (β, α). In case when the structure B is just a topology, the well known answer is given by the so-called quotient or identification topology on Q, which owns the good expected universal property in the category Top. However we notice that, when given other similar data q′ B ′ → Q′ , one has not in general, in spite of a famous error (in Bourbaki’s first edition), a homeomorphism between the product Q × Q′ and the quotient space of B × B ′ by the product of the two equivalence relations, though this is true in two important cases, when q and q ′ are both open or proper, since q × q ′ has the same property. On the opposite, when B is a manifold, it is well known that there is no such satisfactory answer when staying inside the category D = Dif of (smooth maps between) smooth manifolds, i.e. there is no suitable manifold structure for Q. Now for facing this situation there may be two opposite, or better complementary, styles of approaches. b The first one consists in “completing” D, i.e. embedding D in a larger category D b has better categorical properties, i.e. has by adding new objects in such a way than D enough limits for allowing to define a good universal quotient. For instance one can wish b to be a topos. D Various interesting solutions do exist, the study of which is out of our present scope. We just mention, besides Ehresmann’s approaches, two dual ways (considered, under various aspects, by several lecturers at the present Conference) of defining generalized smooth structures on Q, one (first stressed by Frölicher) consisting in defining the smooth curves, while the second method (emphasized by Souriau with his diffeologies) considers the smooth functions on Q. Alain Connes’ “non-commutative” approach is also related. We follow here an opposite path, avoiding to add too many (necessarily pathological) new objects, and trying to stay within D. We do not attempt to define a generalized smooth structure (in the set-theoretical sense) on the most general quotients, and limit ourself to objects which are sufficiently close to manifolds in the sense that they can be described by means of equivalence classes of some simple types of diagrams in D ; we do not try to introduce the limits of such diagrams in the categorical sense.

Indeed we think that the classical categorical concept of limit involves in general a certain loss of the information encapsulated in the concept of a suitable equivalence class of diagrams, but we shall not attempt to develop more formally such a general concept here, though we think it a very promising way, being content with illustrating this point of view by the important special case sketched presently.

626

2


Some motivating examples

Before going to abstract general definitions, I start by giving some elementary examples (to be made more precise later) of the kinds of objects I have in mind.

2.1 Regular equivalences The ideal situation is of course that of the so-called regular equivalences. This means q that there exists on Q a (necessarily unique) manifold structure such that B ◮ Q is a surmersion (= surjective submersion). (Here we start anticipating some pieces of notation for arrows to be systematized later within a more general setting). Godement’s theorem gives a characterization of those equivalences by properties of the graph R summarized by the following notations :

α

R β

◮ B and R ◮τR > B × B ◮

where again the black triangle head for an arrow stands for “surmersion”, while the black triangle tail means “embedding” (in the sense of Bourbaki), or “proper embedding” when dealing with Hausdorff manifolds. These conditions express that R, regarded (in a seemingly pedantic way) as a subgroupoid of the (banal) groupoid B × B , is indeed a smooth (or Lie) groupoid in the sense introduced by Ehresmann, embedded in B × B, and the manifold Q may be viewed as the orbit space of this groupoid. A Lie groupoid R satisfying the framed conditions will be called a principal or Godement groupoid. We shall see in the next example why it is convenient to consider

α

R β

◮B ◮

q

◮Q

as a “generalized (non étale) atlas” for Q. α ′ ◮ B ′ q ◮ Q of the same manifold Q, we can take If we have another “atlas” R′ ◮ β ′ the fibred product of q and q , and we get a commutative diagram


627

S f

′

f

∼

◭

◮

HH E

R

R′

p

′

p

βR

∼

αS

βS

αR

β R′

◭

◮

HH B

HH B′

q

q′

r

αR′

◭

◮

H Q

expressing the “compatibility” of these “atlases”, which means that they define the same (manifold) structure on Q. More precisely the graph S (which in turn may be viewed as a groupoid) can be obtained in the following way by means of the commutative cube below, the bottom face f

S H

f′

◮ ◮ Q H

∼

τR′

τR

∨ ◮B×B

p×p

r ×r

q q ′ ×q ′

◮ ∨ ◮Q×Q

◮

◮ ∨ B′ × B′

τQ

q×

p′

◮

g′

′

R H

p′ ×

∼

∼

◮

τS

g

h

∼

∨ E ×E

◮R H

∼

of which is the pullback of q × q and q ′ × q ′ , and which is constructed step by step by pulling back along the vertical arrows, starting with τQ . The last one is just the diagonal of Q, and may be considered as the anchor map of the “null ” groupoid Q (consisting of just units). The upper face is then also a pullback, as well as all the six faces, and also the vertical diagonal square with three dashed edges. The ∼ symbols emphasize (very special instances of) “surmersive equivalences” between Lie groupoids. (One can observe on this diagram the general property of “parallel transfer by pulling back” for the embeddings and surmersions). Thus we see that the “compatibility” of the two (generalized) atlases α

R β

◮B ◮

q

◮ Q and R′

α β

◮ B′ ◮

q′

◮Q

628


for Q is expressed by the existence of a common “refinement” (pictured above with dashed α ◮ E r ◮ Q. arrows) : S ◮ β One might prove directly a converse, which indeed follows from more general considerations.

2.2 Classical atlases An important special case of the previous one (which explains the terminology) will be given by the following diagrammatic description of atlases and covers of a manifold Q. Let (ϕi : Vi → Ui )(i∈I) be an atlas of the manifold Q, where (Vi )(i∈I) is a cover of Q and the codomains Ui ’s of the charts ϕi ’s are open sets in some model space (which may be Rn or a Banach space). Let Vij be Vi ∩ Vj and Uij be the image of Vij in Ui by the restriction of ϕi . ` Set V = i∈I Vi , with its canonical projection r : V → Q (whose datum is equivalent ` ` to the datum of the covering), U = i∈I Ui (a trivial manifold), R = (i,j)∈I×J Uij , and ` S = (i,j)∈I×J Vij . The charts ϕi ’s define a bijection ϕ : V → U as well as a bijection φ : S → R. Note that S, together with its canonical projection onto Q, defines the intersection covering, while, with its two canonical projections onto V , it can be viewed also as the graph of the equivalence relation associated to the surjection r. Using the bijections ϕ, φ, we have analogous considerations for R and U, but moreover the latter are (trivial) manifolds, and the equivalence is regular, so that we recover a (very) special instance of the situation in the first example. Here the projections αR , βR are not only surmersions but moreover étale maps (of a special type, which might be called trivial); here they will be pictured by arrows of type ⊲ . More precisely their restrictions to the components of the coproduct R define homeomorphisms onto the open sets Uij ’s, and the datum of the smooth groupoid R with base U is precisely equivalent to the datum of the pseudogroup of changes of charts. The situation is summed up by the following diagram (with q = r · ϕ−1 ), which φ

S ............ >R .. .. ≈ .. .. βS ... ... αS βR αR ∨.. ∨.. ϕ ▽ ▽ V ............ >U .. ≈ .. r ... q ▽ ∨.. Q ===== Q α

⊲ U q ⊲ Q associated to a classical atlas (whence ⊲ β the terminology). The dotted arrows in the diagram are to remind that the left column lies in Set, while the right column lies in Dif.

describes the generalized atlas R


629 q

Now if we define a refinement of the previous atlas, denoted by U ⊲ Q for brevity, r p 1 as an atlas W ⊲ Q such that r admits a surjective factorization W ⊲ U, it is easy ′ q ′ q to see that the compatibility of two atlases U ⊲ Q and U ⊲ Q may be expressed by r the existence of a common refinement W ⊲ Q, and we get a special case of the notion of compatibility introduced in the previous subsection, where the general surmersions are replaced by (very special) surjective étale maps. The compatibility diagram now reads as below, and this explains the terminology of Z αZ

∼

∼

f

′

f

βZ

⊳

⊲

▽▽

R

W p

′

p

βR

R′

αR

β R′

αR′ ▽▽

⊳

⊲

▽▽

U

U′

q

q′

r

⊳

⊲

▽

Q

the previous section.

2.3 Group and groupoid actions β

An action of a Lie group G on a manifold E, defined by the (smooth) map : G × E → E, (g, x) 7→ g · x = β(g, x), can be described by the graph of the map β. Modifying the ι order in the products, this graph defines an embedding H = G × E ◮ > G × (E × E), (g, x) 7→ (g, (g · x, x)). Regarding G × (E × E) as a (Lie) groupoid with base E (product of the group G by the “banal” groupoid E × E), the associativity property of the action law may be expressed by the fact that H is an (embedded) subgroupoid of G × (E × E). f

Composing ι with pr1 yields a (smooth) functor H → G which owns the property that the commutative diagram generated by the source projections is a pullback. α ◮ B acting on This construction extends for the action of a (smooth) groupoid G ◮ β p

a manifold over B : E ◮ B, replacing the product G × (E × E) by the fibred product of the anchor map τG and p × p, and one gets a pullback square : H αH

H E

f

pb

f (0)

◮G αG

H ◮B

630


where f (0) = p (induced by the functor f on the bases E = H (0) , B = G(0) ). The action law may be recovered by β = βH , using the isomorphism of H with the fibred product of G and E over B, which results from the pullback property. p Note that the pullback property remains meaningful even when E → B is not a surmersion, since αG still is, (though the fibred product of τG by p × p may then fail to exist), and βH still defines an action of G on p. f For that reason we call a functor H act > G owning the previous pullback property an actor, which is emphasized by the framed label of the arrow. Such functors received various unfortunate names in the categorical literature, among which “discrete opfibrations” and “foncteurs d’hypermorphisme” (Ehresmann), and, better, “star-bijective” (Ronnie Brown), but note that the present concept encapsulates a smoothness information, included in the pullback property, and not only the purely set-theoretic or algebraic conditions (see below for a more general setting). There is an equivalence of categories between the category of equivariant maps between action laws and the category admitting the actors as objects and commutative squares of functors as arrows. Note than in the literature H is currently called the action groupoid, but it is only the whole datum of the actor f : H → G which fully describes the action law, whence our terminology. Here we let Q be the (set-theoretic) quotient of the manifold E by the action of the Lie group(oid) G. By the previous construction it appears too as the orbit space of the Lie groupoid H, so we have again for Q = E/G = E/H a generalized atlas α ◮ E ......◮ Q, the dotted arrow meaning here that we have now just a set-theoretic H ◮ β surjection (we have here to go out of Dif, since Q is no more a manifold).

2.4 Foliations on B Here we need a more restrictive notion for our surmersions, called retroconnected (it is in a certain sense precisely the opposite of being étale, which might be called as well “retrodiscrete”), and, in the present subsection, unlike the previous one, a notation such p as E ⊲ B will indicate that the surmersion p is retroconnected. This means that the inverse image of any x ∈ B is connected, or, equivalently, that the inverse image of any connected subset of B is connected. In the following, when we have to use simultaneously étale and retroconnected surmersions, we shall distinguish them by means of circled labels : A ⊲ B or A c ⊲ B . ´ e There is an obvious notion of foliation induced by pulling back a foliation along such a retroconnected surmersion, and the induced foliation keeps the same set-theoretical space of leaves. Then the notion of F-equivalence (in the sense introduced by P. Molino in the 70’s) between two foliations (B, F ), (B, F ′ ) can be expressed by the existence of a commutative


631

⊲

⊳

′

(B, F )

(B, F ′)

⊳ ... .... q ′ .... .... .

.. ⊲ q .... .. .... ....

(E, G) .. .. .. .. .. . r ... .. .. .. .. . ▽ Q

p

p

diagram as below, which means that F and F ′ induce the same foliation G on the manifold

E. Now we note that again Q = B/F may be viewed as the orbit space of a Lie groupoid (here with connected source fibres), to know the Ehresmann holonomy groupoid α ⊲ B (sometimes renamed much later as the graph of the foliation), and Molino H ⊲ β equivalence may alternatively be described by the commutative diagram below (with all K

∼

∼

f

′

f

⊳

⊲

αH ′

βH ′ ▽▽

B′

.... ⊲ .... .... .... .... .... q .... .... ....

⊳ ... .... .... .... .... q ′ .... .... .... .... .

f

⊳

B

H′

⊲

▽▽

0) ′(

(0 )

E .. .. .. .. .. .. .. .. .. .. .. .. r .. .. .. .. .. .. .. .. .. . ▽ Q

f

αH

αK ▽▽

H

βH

βK

maps retroconnected), where f (0) = p, f ′(0) = p′ , and the symbols ∼ mean (surmersive retroconnected) equivalences of Lie groupoids, in a sense to be made more precise later. As we shall see also later, such a diagram defines a (smooth in a very precise sense) Morita equivalence between H and H ′ . Remark 2.1. Any surmersion A

f

A

◮ Q admits of an essentially unique2 factorization q′ c

⊲ Q′

e ´ e

⊲Q

with q ′ retroconnected and e étale. Though this might be proved directly, it is better to apply the general theory of Lie groupoids, in the special case of principal (or Godement) groupoids (see 2.1 above):

632


if R is the graph of the regular equivalence on A defined by q, its neutral (or αconnected) component Rc is an open (possibly non closed3 , but automatically invariant) subgroupoid of R, hence it is still a Godement groupoid, with now connected α-fibres, and it defines the regular foliation admitting Q′ 4 for its space of leaves. Then Q′ >Q is a surmersion with discrete fibres, i.e. étale, and the graph of the equivalence on Q′ thus defined is isomorphic to the two-sided quotient groupoid R//Rc (cf. [P2]).

3

Diptychs

I am now enough motivated for introducing more dogmatically some general abstract definitions modelling and unifying the previous situations, as well as myriads of others.

3.1 Definition of diptychs The notions presented in this section have a much wider range than it would be strictly necessary for the sequel, if one wants to stay in Dif, but give to it a much wider scope, even when aiming only at applications in Dif, as illustrated by some of the previous examples. We introduced them a long time ago, in [P1], and think they deserve being better known and used. In the presentation of the examples of the previous section, we emphasized the role played by embeddings/surmersions (these are special mono/epimorphisms of Dif, but not the most general ones, which would indeed be pathological), with possibly some more restrictive conditions added. Our claim is that an incredible amount of various constructions can be performed without using the specificity of these conditions, but just a few very simple and apparently mild stability properties (of categorical nature) fulfilled in a surprisingly wide range of situations encountered by the “working mathematicians”. The power of these properties comes from their conjunction. Then the leading idea (illustrated beforehand in the previous section) will be to describe the set-theoretical constructions by means of diagrams, emphasizing injections/surjections, and then rereading these diagrams in the category involved, using the distinguished given mono/epi’s. 3.1.1 Diptychs data A “diptych” D = (D; Di, Ds ) (which may be sometimes denoted loosely by D alone) is defined by the following data : • D is a category which comes equipped with finite non void5 products. The subgroupoid of invertible arrows (called isomorphisms) is denoted by D∗ . • Di /Ds is a subcategory of D, the arrows of which are mono/epi-morphisms (by axiom (iii) below), called good mono/epi ’s and denoted generally by arrows with a triangular


tail/head such as ⊲ > / resp.).

⊲ , or ◮ > /

633

◮ , and so on (here / is written loosely for

The arrows belonging to Dr = Di Ds (i.e. composed of a good mono and a good epi) are called regular. In general Dr will not be a subcategory. When Dr = D, the diptych may be called regular. 3.1.2 Diptychs axioms These data have to satisfy the following axioms (which look nearly self dual, but not fully, and that has to be noticed6 ) : (i) Di ∩ Ds = D∗ ; (ii) Di and Ds are stable by products ; (iii) (a)/(b) the arrows of Di /Ds are monos/strict 7 epis ; (iv) (“strong/weak source/range -stability” of Di /Ds ) : (a) 8 (h = gf ∈ Di ) ⇒ (f ∈ Di ) ; (b) ((h = gf ∈ Ds ) and (f ∈ Ds )) ⇒ (g ∈ Ds ) ; (v) (“transversality”, denoted by Ds ⋔ Di ) : (a) (“parallel transfer ”) : s i given A ◮ B and B ′ ◮ > B (which means : s ∈ Ds , i ∈ Di ), there exists a pullback with moreover s′ ∈ Ds , i′ ∈ Di (the question marks frame the objects ¿A′ ? ◮

¿i′ ?

>A

¿ pb ?

¿s′ ?

H B′ ◮

s

H >B

i

or properties, as well as the dashed arrows, which appear in the conclusions, as consequences of the data). (b) (conversely : “descent”, or “reverse transfer ”) : given a pullback square as below (with i′ ∈ Di , s,s′ ∈ Ds , i ∈ D), one has i ∈ Di i′

A′ ◮ s′

H B′

>A

pb

¿◮

?

s

i

H >B

(conclusion pictured by the question marks around the dashed triangular tail). 3.1.3 Full subdiptychs Remark 3.1. Let be given a subclass C of the class of objects of D, which is stable by products and let D ′ , Di′ , Ds′ be the full subcategories of D, Di , Ds thus generated.

634


In order they define a new dyptych D′ (full subdiptych), one of the following conditions is sufficient : (1) given a good epi A ◮ B, the condition B ∈ C implies A ∈ C ; (2) given a good mono A ◮ > B, the condition B ∈ C implies A ∈ C ;

3.2 Some variants for data and axioms 3.2.1 Prediptychs When dealing with diagrams, we shall need a weaker notion : Definition 3.2. A prediptych is a triple T = (T ; Ti , Ts ) where one just demands Ti ,Ts to be subcategories of T , such that : Ti ∩ Ts = T∗ (isomorphisms). Most of the prediptychs we shall consider are regular, i.e. T = Tr = Ti Ts . 3.2.2 Alternatives for axioms f

i

pr

2 > B ′ , we have the “graph factorization” : B ◮ > B × B ′ > B′, For any arrow B with i = (1B , f ) ∈ Di by axioms (iv) (a) and (i) (as a section of pr1 ), whence it is readily deduced that one has Ds ⋔ D ; this means : • (v)(a) remains valid when omitting Di both in assumptions and conclusions.

We have also Ds ⋔ Ds . In particular the pullback square generated by two good epis p, q, has its four edges in Ds . Such pullback squares will be called perfect squares. The case p = q : B ◮ Q covers the general situation embracing various examples above. It can be shown, using composition of pullback squares, that the axiom (iii) (b), in presence of the other ones, may be rephrased in the following equivalent two ways (using the notations of (2.1)) : α ◮ B, where • (iii) (b’) any good epi q : B ◮ Q is the coequalizer of the pair R ◮ β R is the fibred product of q by q ; • (iii) (b”) any perfect square is a push out too (this last property is very important and remarkable). 3.2.3 Terminal object The existence of the void product (i.e. of a terminal object), is not always required, in view of important examples ; when it does exist, it will be denoted by a plain dot • (though its support has not to be a singleton). Though, in many examples, not only there exists a terminal object, but moreover the canonical arrows A → • are in Ds , it may be useful however not to require this property in general. Then those objects owning this property will be called s-condensed (see examples below).


635

If A ◮ B is in Ds , then A is s-condensed iff B is. Observe that, if A is s-condensed, then, for any object Z, the canonical projection pr2 : A × Z ◮ Z will be in Ds (which is always true whenever all objects are required to be s-condensed). If D ′ , Di′ , Ds′ are the full subcategories generated by the s-condensed objects it can be checked that they define a (full) subdiptych D′ = (D ′ ; Di′ , Ds′ ).

3.3 A few examples of basic large diptychs In fact most of the categories used by the “working mathematicians” own one or several natural diptych structures, and checking the axioms may sometimes be a more or less substantial (not always so well known) and often non trivial part of their theory, which is in this way encapsulated in the (powerful) statement that one gets a diptych structure.9 This is all the more remarkable since a general category (with finite products) bears no canonical non trivial (i.e. with Ds 6= D∗ ) diptych structure, the crucial point being that in general the product of two epimorphisms fails to be an epimorphism. 3.3.1 Sets The category E = Set of (applications between) sets owns a canonical diptych structure E = Set, which is regular, by taking for Ei /Es the subcategories of injections/surjections (here these are exactly all the mono/epi -morphisms). The same is true for the dual category (exchanging injections and surjections), but the dual diptych E∗ is not isomorphic to E .10 3.3.2 Two general examples There are two remarkable and important cases when one gets a (canonical) diptych structure, which is moreover regular, by taking as good monos/epis all the mono/epi -morphisms, to know : the abelian categories and the toposes 11 . Of course these two very general examples embrace in turn a huge lot of special cases in Algebra and Topology. As a consequence all the constructions we carry out by using diptychs are working for general toposes, but the converse is false, since the most interesting and useful diptychs are not toposes. 3.3.3 Topological spaces In Top (resp. Haus12 ), we can take as good monos the (resp. proper13 ) topological embeddings, and as good epis the surjective open maps. All objects are then s-condensed. This diptych is not regular. These canonical diptychs will be denoted by Top and Haus. We may alternatively take as good epis in Top the étale/retroconnected (or, in Haus, proper) surjective maps. Then the s-condensed objects are the discrete/connected (compact) spaces. This is illustrated by examples above.

636


3.3.4 Banach spaces In Ban, the category of (continuous linear maps between) Banach spaces, one can take as good monos/epis the left/right invertible arrows. All objects are s-condensed. This diptych is not regular (save for the full subdiptych of finite dimensional spaces). It will be denoted by Ban. 3.3.5 Manifolds In Dif, the category of (smooth maps between) possibly Banach and possibly non Hausdorff manifolds (in the sense of Bourbaki [B]), the basic diptych structure, denoted by Dif, is defined by taking as good monos the embeddings and as good epis the surmersions. But ones gets a very large number of very useful variants and of full subdiptychs, as in Top, when suitably adding and combining extra conditions for objects or arrows such as being Hausdorff, proper, étale, retroconnected, and also various countability conditions, either on the manifolds (e.g. existence of a countable dense subset) or on the maps (for instance finiteness or countability of the fibres : retro-finiteness, retro-countability). This basic diptych is not regular 14 . 3.3.6 Vector bundles Let VecB denote the category of (morphisms between) vector bundles (for instance in Dif) ; the arrows are commutative squares (see (4.1): E′ p′

H B′

f

>E p

H >B

f0

There are several useful diptych structures. The basic one, denoted by VecB, takes for good monos the squares with f, f0 ∈ Di , and for good epis those with f, f0 ∈ Ds , and which moreover are s-full in the sense to be defined below (4.1). (One can also use pullback squares, which means E is induced by B along f0 ).

3.4 (Pre)diptych structures on simplicial (and related) categories Small diptychs (and especially prediptychs) may be also of interest : 3.4.1 Finite cardinals The category Nc of all maps between finite cardinals (or integers) defines a regular diptych Nc by taking for (Nc )i / (Nc )s all the injections/surjections. The set of objects is N. The s-condensed objects are those which are 6= 0. The dual or opposite category, denoted by Nc∗ , defines also a regular diptych N∗c with (Nc∗ )i = ((Nc )s )∗ , (Nc∗ )s = ((Nc )i )∗ . The product in (Nc )∗ is the sum in Nc . The terminal object is now 0, and all the objects are now s-condensed.


637

+∗ 15 When dropping 0 one gets still diptychs denoted by N+ , the objects of c and Nc 16 which will be denoted by ·n = n + 1 (n ∈ N) . 17 The diptych N+∗ c will be basic for our diagrammatic description of groupoids .

3.4.2 Finite ordinals Considering now the integers as finite ordinals, we get the subcategories of monotone maps which will be denoted here by No , No+ , (denoted by ∆, ∆+ in [McL]) (simplicial categories), as well as their duals, but these have no (cartesian) products and therefore ∗ +∗ cannot define diptychs. They define only prediptychs denoted here by : No , N+ o , No , No . Remark 3.3. Though No is by no means isomorphic to its dual N∗o , however there are two canonical isomorphisms Φ, Ψ = (Φ∗ )−1 (defined only on the privileged subcategories ! ), which can be defined using the canonical generators as denoted in [McL]: Φ : (No )i → (No∗ )i = ((No )s )∗ , n 7→ ·n, δjn 7→ (σj·n )∗ , Ψ : (No )s → (No∗ )s = ((No )i )∗ , ·n 7→ n, σj·n 7→ (δjn )∗ , where a star bearing on an arrow means that this very arrow is regarded as belonging to the dual category (with source and target exchanged). 3.4.3 Canonical prediptychs To each (small or not) category T , we can associate three canonical prediptychs (the last two being regular): T(∗) = (T ; T∗ , T∗ ) , T(ι) = (T ; T , T∗ ) and T(σ) = (T ; T∗ , T ) .

3.4.4 Silly prediptychs Let I (or sometimes, more pictorially, ↓ or →) denote the (seemingly silly) category with ε two objects 0, 1, and one non unit arrow 0 → 1 (it owns products and sums) 18 . It is canonically isomorphic with its dual I∗ , by exchanging the two objects. All the arrows are both mono- and epi-morpisms, but ε is not strict, and cannot be accepted as a good epi. As we shall see, it will be convenient to endow I with one of its canonical prediptych structures (3.4.3): I(∗) ,I(ι) ,I(σ) , according to what is needed.

4

Commutative squares in a diptych D = (D; Di , Ds )

4.1 Three basic types of squares Let D 19 denote the category of commutative squares in D with the horizontal composition, which can be regarded (pedantically) as the category of natural transformations between functors from I to D, with the (unfortunately so-called !) vertical composition. Its arrows might alternatively be described as functors from I × I to D.

638


It turns out that the following properties of a commutative square (K) play a basic A′ u

∨ A

f′

> B′

(K)

v

∨ > B f

role (the terminology will be explained by the application to functors). Definition 4.1. the commutative square (K) is said to be : 4.1.0.1

(u,f ′ )

◮>

a) i-faithful if A′ ◮ > A × B ′ is a good mono ; notation : f′ f′ A′ A′ > B′ > B′ · v or u fid v u (K) ∨ ∨ ∨ ∨ >B >B A A f f

4.1.0.2 b) a good pullback 20 if it is a pullback square (in the usual categorical sense) and moreover i-faithful 21 ; one writes f ⋔ v (f and v are “weakly transversal ”) to mean that the pair (f, v) can be completed in such a square ; notation : f′ f′ A′ A′ > B′ > B′ u

(K)

∨ A

f

v

or

∨ >B

u

∨ A

pb

f

v

∨ >B ◮ A×B

◮

4.1.0.3 c) s-full 22 if one has f ⋔ v 23 and if moreover the canonical arrow A′ B ′ (which is then defined) is a good epi ; notation as below. f′ f′ A′ > B′ A′ > B′ · ful u (K) v or u v ∨ ∨ ∨ ∨ A > B A > B f f

4.2 Basic diptych structures on D The three previous kinds of squares have remarkable composition stability properties resulting from the axioms, which we shall not state here (cf Prop. A 2 of [P3], with a different terminology). The (purely diagrammatical) proofs are never very hard, but may be lengthy. A substantial part of these properties is expressed by the following important (non exhaustive) statements, which deserve to be considered as theorems, as they bring together a very large number of various properties, which acquire much power by being gathered.


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Let be given a diptych D = (D; Di, Ds ) . 4.2.1 Canonical structure On D, we get a first diptych structure , called canonical by setting : D = ( D; (D, Di), (D, Ds )) , which is pictured by :

· >· ·◮ >· · ◮· ∨· >·∨ ; ∨·◮ >·∨ , ∨· ◮·∨ . and also, using remark (3.1.3), the full subdiptych (still defined for the diptychs of the following paragraph): i

D = ( (Di , D); (Di , Di), (Di, Ds )) ,

pictured by :

· >· · ◮·H H H ; H·◮ >·H , H ∨· >·∨ ∨·◮ >·∨ ∨· ◮·∨ . Defining an analogous full subdiptych with Ds replacing Di requires a more restrictive choice for the squares taken as good epis. (These full subdiptychs will be essential to get fibred products of diagrams, hence of groupoids). 4.2.2 Full and pullback epis We have the two basic diptychs: (i,ful) D = ( D; (D, Di), ful (D, Ds )) (i,pb) D = ( D; (D, Di ), pb (D, Ds )) which can be pictured by :

·

·◮

>·

·

,

>·∨ ∨·◮ >· ·◮

◮·

◮

; ∨· ·

·

>· >·∨ >·

∨· ·

;

◮·∨ ◮·

,

.

∨· >·∨ ∨·◮ >·∨ ∨· ◮·∨ and in this way, thanks to a parallel transfer property, we get the expected basic full subdiptychs :

pictured by :

s (i,ful)

D = ( (Ds , D); (Ds , Di), ful (Ds , Ds ))

s (i,pb)

D = ( (Ds , D); (Ds , Di ), pb (Ds , Ds )) ·

·◮

>·

>·H >·

; H ·

>·H

·

,

H H·◮ >· >· ·◮

◮·

◮

; H · ·

·

>·

H · ·

H ◮· ◮·

, H ·◮

>·H

. H·

H ◮·

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One also defines two basic subdiptychs by taking : pb

s ful

D = ( pb D; pb (D, Di ), pb (D, Ds ))

D = ( ful (Ds , D); ful (Ds , Di ), ful (Ds , Ds ))

pictured by :

· ∨· ·

· ◮ >·

>· ;

◮·

∨· ◮·∨ · ◮· ful ful ful ; , . H H H H H H · >· ·◮ >· · ◮· >·∨ >·

∨·◮ >·∨ ·◮ >·

· ,

4.2.3 Iteration One immediately notices than this theorem allows an iteration of the construction of commutative squares giving rise to new diptych structures for commutative cubes, and so on, which would be very difficult to handle directly. This is especially interesting when dealing with commutative cubes since such a diagram gives rise to three commutative squares in D, each edge of which (which is actually a square of D) belonging to two different squares of D, and this gives a powerful method for deducing properties of certain faces (for instance being a pullback), or edges from properties of the others, using diptych properties of parallel transfer. But we cannot develop more here.

5

Diagrams of type T in a diptych

We are now going to replace the silly category I of 3.4.4 by a more general notion generalizing the construction of commutative squares, and allowing to perform various settheoretic constructions in a general diptych D. Let T be a small prediptych 24 (3.2.1), and let D = (D; Di , Ds ) be a diptych.

5.1 Definitions : objects and morphisms We denote by T D : = D T

the category having :

– as objects the elements of I T D : = Hom(T, D)

, called diagrams of D of type T,

which means those functors F from T to D such that one has : (1)

F (Ti ) ⊂ Di and F (Ts ) ⊂ Ds

(these last conditions are void when T = T(∗) (3.4.3)) 25 ; – as arrows the morphisms (i.e. the natural transformations) between such functors26 . These morphisms may be described :


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- either as functors Φ from T × I to D such that (denoting by (:) the set of the two units of I): Φ(Ti × (:)) ⊂ Di and Φ(Ts × (:)) ⊂ Ds - or as functors Ψ from T to vert D 27 such that, with the notations of (4.2) : Ψ(Ti ) ⊂ (Di , D) = i D and Ψ(Ts ) ⊂ (Ds , D) = s D . In words this means that such a morphism may be viewed : - either as a diagram in D of type T × I(∗) (cf. 3.4.3 and 3.4.4) - or as a diagram of the same type T in vert D, regarded with its canonical (vertical) diptych structure (4.2). However be careful that, in such a description, though the previous definition of the morphisms uses the vertical composition of squares, the composition of these morphisms (called vertical in [McL]!!) involves the horizontal composition of squares.

5.2 Diptych structures on the category of diagrams 5.2.1 Definitions Using the latter interpretation, and taking now into account the horizontal composition, it is clear that any prediptych structure on D determines a prediptych structure on T D. For instance, using on T its trivial prediptych structure, we can consider the canonical prediptych structure T D, but this not in general a dyptich structure (one needs parallel transfer properties for good epis in order to ensure conditions (1) of (5.1)). Using the parallel transfer properties of s-full and pullback squares, one can get three useful diptych structures denoted by: ((i,ful) )T D , ((i,pb) )T D and

pb T

D .

Of special interest, as we shall see, will be those diagrams which preserve certain pullbacks, since groupoids may be described by diagrams of this type. 5.2.2 The silly case The case of commutative squares is just the special case when one takes for T the silly category I , with one of its prediptych structures (3.4.4), since one has: I I(∗) D = |D| , I I(ι) D = |Di | , I I(σ) D = |Ds | , 28 I(∗) D = D , I(ι) D = i D ; I(σ) D = s D ; more precisely : I(∗) D = D , I(ι) D = i D ((i,ful) )I(∗) D = (i,ful) D , ((i,pb) )I(∗) D = (i,pb) D . and also : (s (i,ful) )I(σ) D = s (i,ful) D , (s (i,pb) )I(σ) D = s (i,pb) D .

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5.3 The exponential law for diagrams Given another similar datum S, we have the exponential law, which, with our notations, reads (at least when S and T are trivial diptychs) : S (T D) = S×T D (with a lot of possible variants), and the canonical isomorphism : S × T ≈ T × S yields a canonical isomorphism : S (T D) ≈ T (S D) . Particularly one has:

n

( T)n = T

and, specializing for T = I(∗) :

, n

()n = (I(∗) )

.

Remarks about notations The reader may have observed that the symbols and I used above are intendedly ambiguous and protean, since this allows to memorize and visualize a lot of various properties : – the symbols I or I(•) , where • stands for ∗, ι, or σ : - denote the silly category, possibly with various prediptych structures on it; - denote the bifunctor Hom (?,?), with the first argument treated as an exponent); - often behaves formally as a 1; - may sometimes suggest the functor T 7→ T × I(•) ; – the symbol with possibly labels in various positions ⋄ ? ⊤ (♯,♮) : - may picture or suggest the product I ×I or more precisely various instances of I(△) ×I(▽) ; - may create the commutative squares of a category or of a diptych with possible extra conditions : ∗ on the left, they bear on the vertical edges ; ∗ on the right, they describe the subcategories for a (pre)diptych structure ; ∗ inside, they describe global properties of the squares ; - may become the bifunctor Hom, with possible restrictions on the squares involved.

6

Groupoids as diagrams in E = Set

We shall be concerned only with small groupoids, viewed as generalizing both groups and graphs of equivalence relations, as well as specializing (small) categories. According to our program, we need a diagrammatic description of both groupoid data and axioms, which will be transferred from the diptych E = Set to a general diptych D. This can be achieved in two complementary ways : - either using finite diagrams (called sketches in Ehresmann’s terminology) ; - or using the simplicial description by the nerve. Though seemingly more abstract, the latter turns out to be often the most convenient for theoretical purposes, while the former is adapted to practical handling. We refer to [McL] and also to (3.4) for general properties and notations concerning


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simplicial categories and simplicial objects, which are just functors from No+∗ to E, and more precisely diagrams of type N+∗ o in E. + 6.1 Some remarks about N+ c and No

(see 3.4) 6.1.1 The arrows as functors For each n ∈ N, we shall denote by : → ∗− ·n the ordinal ·n = n + 1 viewed as a small category (defining the order) 29 ; ˙ the subcategory consisting of arrows of this order category with source 030 ; ∗ ∧n → = (·n)2 the banal groupoid (·n) × (·n) , which ignores the ordering 31 . ∗← ·n With these structures on the objects, the arrows of No+ and Nc+ may be regarded (though this looks pedantic) as functors (or morphisms) between small categories. It may be sometimes suggestive to write : ← → ← → ← → ← → ← → ← → ← → ← → ˙ = ∧ , ·2 = 3 = △ , ·3 = 4 = ⊠ . ·0 = 1 = · , ·1 = 2 = ↔ , ∧2 6.1.2 Pullbacks in Nc∗ Pullbacks in Nc∗ come from pushouts in Nc . One can check that all the commutative squares describing the relations between the canonical generators of No (see [McL]) become pushouts when written in Nc , and they generate, by composition, all the pushouts of Nc . More precisely (with the notations in [McL]) the squares involving the injections δj ’s alone or mixing both δj ’s and σk ’s are pullbacks too, but not those with the surjections σj ’s alone 32 . This is still valid when dropping 0 (but one looses the squares describing the sums as pushouts).

6.2 Characterization of the nerve of a groupoid The special properties of a groupoid among categories yield very special and remarkable properties as well as alternative descriptions for its nerve. 6.2.1 Three descriptions for the nerve Given a (small) groupoid G, denoted loosely by G or G ⇉ B, we can associate to it three canonically isomorphic simplicial objects (among which the first one is the nerve of G regarded as a category). First we define and denote the three images of the generic object ·n = n + 1 33 : → (1) ↓ G(n) = hom(− ·n, G) = {paths of G of length n} ; ∧ (n) ˙ G) = {n -uples of arrows of G with the same source}; (2) G = hom(∧n, ← → l (n) (3) G = hom( ·n , G) = {commutative diagrams of G with ·n vertices}34 .

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Identifying these three sets, we can write loosely: G(n) = ↓ G(n) = ∧ G(n) = l G(n) . We shall also sometimes feel free to write loosely and suggestively : G(0) = B , G(1) = ↓ G = l G = G , G(2) = ∧G = △G , G(3) = ⊠ G . The images of the canonical generators of Nc+ are then especially easy to define with the interpretation (3), since they consist in repeating or forgetting one vertex. But, with the descriptions (1) and (3), one can immediately interpret the required contravariant functors from No+ to E as being just special instances of the classical contravariant hom-functors hom(?, G), which consist in letting the arrows of No+ act by right morphism composition (in the category of morphisms or functors between small categories) with the elements of ↓ G(n) or l G(n) . 6.2.2 Properties of the nerve of a groupoid 6.2.2.1 Extension of the nerve With the interpretation (3), we get a bonus, since it is now obvious that this contravariant functor extends to Nc+ , and so defines a diagram in G of type N+∗ . (We remind that, in this interpretation, integers are interpreted as c small banal groupoids, and the arrows of Nc+ as morphisms.) 6.2.2.2 Exactness properties Moreover this extended functor, which is now defined on a diptych, is “exact”, there meaning that it preserves pullbacks 35 . It would be enough to check this for the generating pullbacks of Nc∗ (6.1.2). 6.2.2.3 Conversely one might check that the datum of an exact diptych morphism G = (G(n) )(n∈N) 36 from N+∗ c to E = Set determines a groupoid, the nerve of which is the restriction of this morphism to N+∗ o . 6.2.2.4 Sketch of a groupoid 37 Actually one might check that the groupoid data are fully determined by the restriction of G to the (truncated) full subcategory [·2] of Nc+∗ generated by ·0, ·1, ·2, while the groupoid axioms are expressed by its restriction to [·3] (and the conditions that the images of the previous generating squares be pullbacks) 38 . 6.2.3 Concrete description Actually the previous data are somewhat redundant, and it is convenient for our purpose to observe that G may be fully described by the data (G, B, ωG , αG , δG ) (satisfying axioms which we shall not make explicit), where : • B (“base”) and G (“set of arrows”) are objects of E ; • ωG : B ◮ > G is an injection (“unit law”) ; • αG : G ◮ B is a surjection (source map) ; • δG : ∧G = △G ◮ G is a surjection (“division map”39 ), with : ∧G = G ×B G (fibred product of α by α).


645

One often writes α for ωα, and we set : • τG = (βG , αG ) : G → B × B (“anchor map” 40 , or “transitor ”). From these data, it is not difficult to get the range map βG , the inverse law ιG , and the composition law γG (defined on the fibred product of αG and βG ). With notations from Mac Lane: ωG , αG , βG , δG would be the respective images of : 1 ∗ (σ0 ) , (δ11 )∗ , (δ01 )∗ , (δ02 )∗ ( in the dual N∗c ).

7

D-groupoids

7.1 Definition of D-groupoids 7.1.1 Groupoids in a diptych The diagrammatic description of a set-theoretical groupoid given above leads to define a D-groupoid as an exact diptych morhism (6.2.2) , i.e. preserving good monos, good epis, and good pullbacks: G : N+∗ c → D. As above, the exactness properties allow to characterize G by restriction to various subcategories, and to recover in this way the simplicial description as well as the skecth ones and the (G, B, ωG , αG , δG ) presentation. We shall also use as previously the relaxed notations (G(n) )n∈N (omitting the effect α ◮ B and its of G on the arrows), or briefly G (meaning G(1) ), B for G(0) , and G ◮ β variants. We shall denote by Gpd(D) the class of D-groupoids. 7.1.2 Examples This very general notion may be specialized using for instance the various examples given in (3.3). Note that the Top -groupoids are those for which the source map has to be open, a condition which can hardly be avoided for getting a useful theory. Lie groupoids 41 are of course the Dif -groupoids, with the usual diptych structure on Dif , but, using some of the above-mentioned variants, the theory will include, for instance, among others, étale or α -connected groupoids as well. VecB-groupoids were used in [P6]. 7.1.3 Null groupoids For any object B of D, the constant simplicial object B is a D-groupoid, called “null ” 42 , and denoted by ◦B. All arrows are units, and the map ωG is an isomorphism.

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7.2 Principal and Godement D-groupoids 7.2.1 The transitor τG From a purely set-theoretical point of view, the transitor τG : G → B × B measures the (in)transitivity 43 of the groupoid G. It turns out that, in the diptych setting, its properties encapsulate a very rich “structured” information. We just mention, without developing here, that, for instance in Dif, one can fully characterize, just by very simple properties of τG , not only the graphs of regular equivalences, but (among others) the gauge groupoids of principal bundles, the Poincaré groupoids of Galois coverings, the holonomy groupoids of foliations, the Barre Q-manifolds, the Satake V-manifolds (or orbifolds). 7.2.2 Principal D-groupoids The notion of graph of regular equivalence (2.1) may be carried over in any diptych as follows. q Given a good epi B ◮ Q of D, we can construct the iterated fibred product of ·n (n) q, denoted by R = ×Q B (n ∈ N) and check that (n 7→ R(n) )(n∈N) allows to define a D-simplicial object which is a groupoid. We shall say that R = (R(n) )n∈N is the principal groupoid associated to q (with base B). Moreover we have an“augmentation”, which means an extended “exact”(6.2.2) diptych morphism (6.2.2) from N∗c to D. This augmentation carries 0 44 to R(−1) = Q, and the q added generator η ∗ = (δ00 )∗ 45 to B ◮ Q. This situation gives rise to a perfect square (3.2.2): R αR

H B

βR pb po

q

◮B q

H ◮Q

The pushout property of this square shows that the good epi q is uniquely determined by the knowledge of the D-groupoid R. αG Applied to G ◮ B, this construction allows considering ∧G as a principal groupoid (see below (8.2.2)). 7.2.3 Banal groupoids This construction applies in particular when B is an s-condensed object (3.2.3). Then one has R(n) = B ·n (·n times iterated product). It is called the “banal groupoid ” 46 associated to B ; it is principal. It is denoted by B × B or B 2 . More generally such a banal groupoid (possibly non principal in the absence of a terminal object) is associated to an object B whenever the canonical projections pri : B × B → B are good epis 47 .


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One might call “proper ” those objects B for which the banal groupoid B 2 is defined. When such is the case, the transitor τG may be viewed as a D-functor. Given an integer n , we can attach to it, besides (for n 6= 0) the banal groupoid ← → n = n2 (which uses the product in Nc ), a banal cogroupoid 2 n or 2n, using the product ×∗ of Nc∗ (which is the coproduct + of Nc ). 7.2.4 Godement groupoids and Godement diptychs Definition 7.1. A D-groupoid G is called a Godement groupoid if τG : G ◮ > B × B is a good mono. Every principal groupoid is a Godement groupoid. • We say the Godement axiom is fulfilled, and D is a Godement diptych if conversely every Godement D-groupoid is principal. The fact that the diptych Dif (3.3.5) is Godement is the content of the so-called Godement theorem, proved in Serre [LALG] 48 . But it is highly remarkable that nearly all of the examples of diptychs given above are indeed Godement diptychs, as well as most of the diagram diptychs we constructed above, provided one starts with a Godement diptych. Such a statement includes a long list of theorems, which are not always classical. • From now on, we shall assume D is Godement whenever this is useful.

7.3 Regular groupoids More generally, a D-groupoid G is called regular if τG is regular (3.1.1). Then we have the following factorization of τG : G

π

τR

◮R◮ >B×B,

where R is a Godement D-groupoid, hence principal, so that we can construct the perfect square (7.2.2) ; then the orbit space Q exists as an object of D, and is also the pushout of αG and βG . π However the object Q inherits an “extra structure” from the arrow G ◮ R, a good epi which measures how much the s-full pushout square GBBQ, fails to be a pullback. • In the general case, when G is neither principal nor even regular, the aim of the present paper is to define a kind of “virtual augmentation” (as a substitute for the failing one), which is the D-Morita equivalence class of G, and which has to be considered intuitively as defining the “ virtual structure” of the orbit space. 7.3.1 Plurigroups An important special case of regular D-groupoid is when R = ◦B (7.1.3) (this is indeed equivalent to αG = βG ). When such is the case, we shall say G is a “D-plurigroup” 49.

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7.3.2 s-transitive D-groupoids The opposite degeneracy is when τG is a good epi: we shall say G is s-transitive. When D = Set, this just means that the orbit space is reduced to a singleton, but in Dif, for instance, this has very strong implications, since this means essentially that G may be viewed as the gauge groupoid of a principal bundle 50 . This can be proved in a purely diagrammatic way, which allow to extend these concepts to all (Godement) diptychs. In fact the orbit space has to be thought as “a singleton structured by a group”. One of the basic reason of the strength of the notion of D-groupoid is that it unifies and gathers in a single theory all these various degeneracies.

8

The category Gpd(D)

8.1 D-functors. 8.1.1 D-functors as natural transformations D-functors (or morphisms) between D -groupoids are of course special cases of diagram morphisms (5) and, as such, are defined as natural transformations between the D groupoids viewed as functors. But the pullback property of the generating squares of the diagrams defining groupoids (6.2.2) has very strong implications (arising from the preliminary study of commutative squares in a diptych) which we cannot develop here, referring to [P4] for more details. We just mention a few basic facts. A D -functor f : H → G is fully determined by f (1) : H (1) → G(1) and hence often denoted loosely by f : H → G ; we shall write : H (0) = E, G(0) = B. It is called principal if H is principal. We say f is a i/s-functor when f lies in Di /Ds ; as a consequence, one can check this is still valid for all the f (n) . We get in this way the category Gpd(D) , with Gpd(D) as its base, of (D-functors or morphisms between) D-groupoids, and two subcategories Gpdi (D) and Gpds (D), but, as announced earlier, the second one will not be the right candidate for good epis in Gpd(D) (see below). An arrow of Gpd(D) is said to be split if it is right invertible, in other words if it admits of a section. 8.1.2 D-functors as (vert D)-groupoids Following (5.1), a D-functor may be viewed as a (vert D)-groupoid 51.

8.2 Actors It turns out that the basic (algebraic) 52 properties of a D-functor f : H → G are encapsulated into two fundamental squares, written below, which immediately acquire a


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“structured” (and hence more precise) meaning when written in a diptych. It is enough to write these properties at the lowest level (f (0) , f (1) ), since it turns out that the special pullback properties of the nerve, as a diagram, allow to carry them over to all levels. 8.2.1 The activity indicator A(f ) : in/ex/-actors We consider first the commutative square generated by the source maps : f H >G A(f ) αG . H f (0) H E >B We shall say f is : an actor 53 , an inactor, an exactor 54 , depending on whether the square A(f ) is a pullback, i-faithful, or s-full (4.1). One can show that any exactor f : H → G, one can define its kernel K ◮ > H, which is null when f is an actor. f A principal actor is an actor R → G with R principal. For instance , taking for R the graph associated to a covering, as described in (2.2), we recover the notion of G-cocycle, including cocycles defining a principal 55 fibration (when G is a Lie group) and Haefliger cocycles defining a foliation (when G is a pseudogroup). αH

8.2.2 The canonical actor δG The map δG : ∧G = △G ◮ G may be viewed as a functor, and indeed a principal s-actor, associated to the right action of G on itself. This will be enlightened by the functorial considerations to be developed below.

8.3 D-equivalences 8.3.1 The full/faithfulness indicator T(f ) The second basic square is built with the transitors (anchor maps) . f H >G τH

∨ E ×E

T(f )

τG

∨ >B×B

.

f (0) ×f (0)

8.3.2 Equivalences and extensors We shall say f is an inductor/i-faithful/s-full, depending on whether the square T(f ) is a pullback/i-faithful/s-full 56 (4.1). When f is s-full/a D-inductor, and moreover f (0) lies in Ds , then f is an s-functor, and we say f is an s-equivalence/an s-extensor 57 . While the concept of D-inductor derives from the diagrammatic description of “full and faithful”, the general concept of D-equivalence demands to add a diagrammatic description

650


of the “essential (or generic) surjectivity”, which uses the A(f ) square and will not be given here (we refer to [P4]). This general notion allows to speak of i-equivalences too. Parallel to the notion of canonical actor, we have those of canonical equivalences. These will be defined below, when we have given a diagrammatic construction of G and of the canonical morphisms: G → G ⇉ G.

8.4 Diptych structures on Gpd(D) The following result , which we can just mention here, is of basic importance for a unified study of structured groupoids. Using the preliminary study of diagrams in a diptych, one can define several useful (Godement) diptych structures on Gpd(D). We stress the fact that the s-functors are not the right candidates for good epis 58 . Among various possibilities, one can take: • for good monos: · either the i-functors · or the i-actors; • for good epis those s-functors which moreover belong to one of the following types: · s-exactors · s-actors · s-equivalences.

8.5 The category Gpd(N+∗ c ) D∗ -groupoids may be called D-cogroupoids. Some constructions for D-groupoids may be better understood from a study of N+∗ c + groupoids or Nc -cogroupoids (which are not Set-groupoids) (3.4.1). Some pieces of notations are needed to avoid confusions arising from duality. 8.5.1 Notations for End(Nc+ ) and End(Nc+∗ ) In contrast to sections (4) and (5), we shall, for a while, stick to Mac Lane’s terminology concerning horizontal and vertical composition of natural transformations (also called functorial morphisms), in order to allow free use of [McL] as reference. This means that groupoids (viewed as diagrams or functors) have here to be thought as written horizontally (instead of vertically as above), hence the groupoid functors or morphisms (i.e. the arrows of Gpd(D)) as written vertically, though this is somewhat uncomfortable. The natural transformations between endofunctors make up categories denoted by End(Nc+ ) and End(Nc+∗), which are indeed double categories (and even more precisely 2-categories [McL]) when considering both horizontal and vertical composition. The identity maps define canonically:


651

• a contravariant functor: Nc+ → Nc+∗ , λ 7→ λ∗ , where λ∗ is λ with source and target exchanged ; denoting by ×∗ the product in Nc+∗ (i.e. the sum + in Nc+ ), we can write (for any pair of arrows λ, µ): (λ + µ)∗ = λ∗ ×∗ µ∗ ; • a covariant functor: ∗

End(Nc+ ) → End(Nc+∗ ) , Φ 7→ ∗ Φ∗ = Φ , ∗

with Φ (λ∗ ) = (Φ(λ))∗ ; • a bijection ∗

·

·

∗

End(Nc+ ) 7→ End(Nc+∗) , (ϕ : Φ → Φ′ ) 7→ (ϕ∗ : Φ ← Φ′ ) , with ϕ∗ (n) = (ϕ(n))∗ , which is: · covariant with respect to horizontal composition laws ; · contravariant with respect to vertical composition laws. 8.5.2 Description of Gpd(N+∗ c ) By the previous bijection: + • N+∗ c -groupoids derive covariantly from the endofunctors of Nc preserving surjections, injections and pushouts ; • morphisms of N+∗ c -groupoids derive contravariantly from morphisms between endofunctors of the previous type. Moreover we know from the general descriptions of D-groupoids (6.2.3) that: ∗

+∗ • the endofunctors Γ = Φ of N+∗ c defining Nc -groupoids are uniquely determined by the data: (Γ(0) , Γ(1) , ωΓ , αΓ , δΓ ),

hence (resulting from the previous study) by the data in N+ c : (n0 = Φ(·0), n1 = Φ(·1), n2 = Φ(·2), ω : n0 ◭

n1 , α : n1 < ◭ n0 , δ : n2 < ◭ n1 ).

8.5.3 Some basic examples of N+∗ c -groupoids and morphisms As just explained, such a groupoid morphism γ : Γ ← Γ′ derives from a functorial · morpism ϕ : Φ → Φ′ where Φ, Φ′ preserve surjections, injections and pushouts. These define a 2-subcategory of End(Nc+ ) denoted by ̥. Those Φ ’s which preserve sums too are of type : p× : ·n 7→ p × ·n, λ 7→ p × λ (p times iterated sum in N+ c ). The associated groupoids, denoted by p ×∗ are the banal (but not principal 59 ) N+∗ c 60 groupoids, in the sense of (7.2.3) .

652


For p = 0, we get the “null groupoid ”, denoted by 0, associated to the constant functor 0 : ·n 7→ ·0, and, for p = 1, the “unit groupoid ”, denoted by 1, associated to the identity functor. For p = 2, one has the “square groupoid ” 61 , denoted by ∗ = 2 ×∗ , associated to: 2× : ·n 7→ 2(·n) = ·n + ·n, λ 7→ 2λ = λ + λ . Among the non principal ones (7.2.2), the simplest is denoted by △ ∗ : it is associated to the “shift functor ”: ·0+ : ·n 7→ ·0 + ·n, λ 7→ ·0 + λ where the last ·0 is understood as the identity of the object ·0 = 1. We have also a N+∗ c -groupoid morphism: (1)

δ0∗ : △ ∗ −→ 1

associated, with the notations of (7.1.1) and of [McL], to the the “shift morphism”, defined by the family (δ0·n )n∈N (injections skipping the 0 in ·1 + ·n). As to ∗ , associated to 2×, we have in N+ c the coproduct morphisms (n ∈ N) : ι2 codiag (·n) + (·n) < (·n) < < ι (·n) 1 which define morphisms of N+∗ c -groupoids denoted suggestively by: 1

ω

̟2

> ∗ ̟1

>1 . >

(2)

62 By iteration we can even get a canonical groupoid ( ∗(n) )n∈N in Gpd(N+∗ c ) .

The “symmetry groupoid ” Σ∗ proceeds from the (involutive) “symmetry map” Σ : → −); → Nc+ , derived by reversing the order on the integers viewed as ordinals (− n 7→ ← n it is defined by the identity on the objects, and, on the generators (with Mac Lane’s notations), by (n ∈ N+ ):

Nc+

n n Σ : δjn 7→ δn−j , σjn 7→ σn−j ,

and the family of maps : (ςn : n → n, j 7→ n − j)(n∈N+ ) defines a natural transformation from identity towards Σ, hence also a groupoid morphism: ς : Σ∗ −→ 1 .

9

(3)

Double functoriality of the definition of D-groupoids

9.1 Bivariance of D-groupoids and D-functors The definition of a D-groupoid as a diptych morphism G : N+∗ c → D shows immediately Γ

N+∗ c

↓γ Γ′

> +∗ N > c

H ↓f G

> D >

T ↓t T′

> ′ D >

(look at the above diagram, where we go on sticking to Mac Lane’s conventions) that this definition is:


653

• functorial with respect to composition, on the left (target side), with an exact diptych morphism T : D → D′ , as well as with respect to right (horizontal) composition with · a natural transformation t : T → T′ ; this means that any D-functor f : H → G gives rise to a D′ -functor T◦f : T◦H→T◦G 63

and to a natural transformation: ·

t ◦ f : T ◦ f → T′ ◦ f ; the latter gives rise to the following commutative square of D′ -groupoids 64 : Tf TH > TG t(H)

t(G)

;

∨ ∨ T ′f ′ TH > T ′G actually Gpd behaves here like a functor and we can define: · Gpd(T) : Gpd(D) → Gpd(D′ ), Gpd(t) : Gpd(T) → Gpd(T′ ) ; • functorial with respect to horizontal composition on the right (source side), with ′ a N+∗ c -groupoid Γ, as well as with a groupoid morphism γ : Γ → Γ viewed as a natural transformation between endofunctors of N+∗ c ; this means that any D-functor f : H → G gives rise to a D-functor: Γ• f : Γ• H → Γ• G, where Γ• H = H ◦ Γ, Γ• G = G ◦ Γ, Γ• f = f ◦ Γ , and to a natural transformation γ • f = f ◦ γ : Γ• G → Γ′• G ; this defines a (covariant) functor: Γ• : Gpd(D) → Gpd(D) , hence a canonical (vertical) representation: Gpd(N+∗ c ) → Gpd(D) , but this representation depends in a contravariant way upon Γ, with respect to the horizontal composition of N+∗ c -groupoids defined above, since • ′• Γ ◦ Γ = (Γ′ ◦ Γ)• , γ • ◦ γ ′• = (γ ′ ◦ γ)• . Note that, when going back to the generating endofunctors of N+ c , we get a doubly contravariant canonical representation of ̥ (notation of 8.5.3) into Gpd(D). We give a few examples.

9.2 Examples for the left functoriality We can either “forget” the structures on objects of D, or “enrich” them. We give examples of these opposite directions.

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9.2.1 Concrete diptychs Thinking to the basic examples of Top and Dif, we shall say D is “concrete” if it comes equipped with an adjunction from E = Set to D defined by the following adjoint pair [McL] of functors 65 : ˙ (discrete, forgetful) = ( ˙ , | |) : E < > D . || Then we can speak of the underlying E-groupoid, and make use of set-theoretical descriptions. 9.2.2 The tangent functor Thinking now to the case when D = Dif, we can consider (see 3.3.6) the tangent functor: T : Dif −→ VectB , which is equipped with two natural transformations: 0

·

>T o

· t

> 0, .

Once one has checked it defines an exact diptych morphism, we can immediately transfer to the tangent groupoids all the general constructions valid for general D-groupoids (for instance constructions of fibred products, and so on). 9.2.3 Double groupoids The idea of defining and studying the notion of double groupoids as groupoids in the category of groupoids is due to Ehresmann, who proved the equivalence with the alternative description by means of two category composition laws satisfying the “exchange law” 66 . In the diptych setting we get several notions depending on the choice for the diptych structure on Gpd(D) (see 8.4). We stress the point that, even in the purely set-theoretical setting, the choice we made of exactors for good epis implies adding a certain surjectivity condition 67 which does not appear in Ehresmann’s definition, but was encountered by several authors, mainly Ronnie Brown (filling condition), and seems useful to develop the theory beyond just definitions. Applying the general theory of D-diptychs, for instance for getting fibred products, or quotients, or Morita equivalences, gives results which it would be very hard to get by a direct study (which has never been done), even in the purely algebraic setting.

9.3 Examples for the right functoriality As announced,the coherence of various notations introduced above will appear just below (to be precise, we find : △ ∗ • = △, ∗ • = ). The examples given in 8.5.3, allow to transfer to D-groupoids some classical set-theoretic constructions, announced above. Taking for γ : Γ → Γ′ :


655

• formula (1) of 8.5.3, we recover the morphism: δG : △G −→ G ; • formula (2), we get the canonical equivalences: G → G ⇉ G; • formula (3), we get the dual groupoid G∗ = ΣG, and the inverse law ςG = ιG , which defines an involutive isomorphism: ςG : G → G∗ .

9.4 D−natural transformations, holomorphisms

p( f)

>

Once G is defined, one can also define D-natural transformations between functors from H to G as D-functors H → G. The canonical i-equivalence G → G defines the identical transformation of the identity functor. Since G is a groupoid, such natural transformations are necessarily functorial isomorphisms. An isomorphy class of D-functors will be called a “holomorphism” (an alternative terminology might be “exomorphism”, since this notion generalizes the outer automorphisms of groups). Since the horizontal composition of natural transformations commutes with the vertical one, it defines a composition between holomorphisms, and this yields a quotient category of Gpd(D), which we shall denote by Hol(D). Any D-functor f : H → G generates the following commutative diagram, in which q(f ) G N K q(f ) ∼

H H

π2 ∼

> G pb

f

π1 ∼

H >G

is an s-equivalence, and p(f ) an exactor. The pair (p(f ), q(f )) is called the “holograph” of f . Moreover the s-equivalence q(f ) is split (8.1.1) since π1 is.

10

The butterfly diagram

This section illustrates, in the case of orbital structures presently described, the use of diagrams in a diptych for transferring constructions in Set to constructions in Dif or other various categories, as well as the use of the various kinds of D-functors introduced above. We can be only very sketchy. More precise descriptions and results may be found in [P4], where they are stated for the differentiable case, but written to be easily transferable to general diptychs. More details about the diagrams used for proofs will be given elsewhere.

656


10.1 Generalized “structure” of the orbit space 10.1.1 Algebraic “structure” First, from a purely set-theoretic point of view, any equivalence between two groupoids, in the general categorical sense [McL] 68 , preserves 69 the set-theoretic orbit space Q, but indeed it preserves much more, since, to each orbit (or transitive component), there is a (well defined only up to isomorphism) attached isotropy group, and this endows Q with a kind of an algebraic “structure”, in a (non set-theoretical) generalized sense. For instance, the leaves of a foliation are marked by their holonomy groups, the orbits of a group(oid) action are marked by their fixators. Such a “structure” is sometimes called a group stack. 10.1.2 D -“structure” of the orbit space Then, in the D-framework, replacing algebraic equivalences by D-equivalences will moreover encapsulate in this generalized structure the memory of the D-structure as well. It turns out (though this is by no means a priori obvious) that it is enough to make use of s-equivalences (8.3.2). Finally we are led to the following: Definition 10.1. Two D-groupoids H, G, are said to be D-equivalent if they are linked q p by a pair of s-equivalences: H ◭ ∼ K ∼ ◮ G . The fact that this is indeed an equivalence relation is an easy consequence of the results stated in 8.4, taking s-equivalences as good epis in Gpd(D), and using fibred products of good epis. A D-equivalence class of D-groupoids may be called an orbital structure. Any representative of this equivalence class is called an atlas of the orbital structure.

10.2 Inverting equivalences 10.2.1 Meromorphisms Note than in general orbital structures cannot be taken as objects of a new category. However one can define [P4] a new category which shall be denoted here by Mero(D), with the same objects as Gpd(D), and arrows called “meromorphisms”, in which the s-equivalences (and indeed all the D-equivalences) become invertible, in other words are turned into isomorphisms. In the topological case, these isomorphisms may be identified with the Morita equivalences. In this new category the orbital structures now become isomorphy classes of D-groupoids, though the objects still remain D-groupoids and not isomorphy classes, so that the orbital “structures” are not carried by actual sets, and, as such, remain “virtual”. This means that Mero(D), is the universal solution for the problem of fractions consisting in formally inverting the s-equivalences, and indeed all the D-equivalences. This kind of problem always admits a general solution [G-Z]: the arrows are given by equivalence classes of diagrams, the description of which is simpler when the conditions


657

for right calculus of fractions are satisfied. It is worth noticing that these assumptions are not fulfilled here, while our construction is in a certain sense much simpler, since, as is the case in Arithmetic, each fraction will admit of a simplified or irreducible canonical representative (p, q). In the topological case these representatives may be identified with the “generalized homomorphisms” described by A. Haefliger in [Ast 116] and attributed to G. Skandalis, or the “K-oriented morphisms” of M. Hilsum and G. Skandalis. Φ We can say more. The canonical functor Gpd(D) → Mero(D) admits of the following Φ Φ2 factorization: Gpd(D) →1 Hol(D) → Mero(D) (see 9.4) with Φ1 full (i.e. here surjective) and Φ2 faithful (or injective). It turns out that Hol(D) is the solution of the problem of fractions for split (8.1.1) D-equivalences. It is embedded in Mero(D) by means of the holograph (9.4). 10.2.2 Description of fractions Let (p, q) denote a pair of exactors with the same source K: p : K → G, q : K → H. We set R = Ker q, S = Ker p (see 8.2). Let (p′ , q ′ ) another pair p′ : K ′ → G, q ′ : K ′ → H with the same G and H. Letting for a while G and H fixed, we start considering arrows k : (p′ , q ′ ) → (p, q) defined as D-functors k : K ′ → K such that the whole diagram commutes. Let p/q denote the isomorphy class of (p, q), and call it a “fraction”. On the other hand we say two pairs (pi , qi )(i = 1, 2) are equivalent if there exist two s-equivalences ki : (p, q) → (pi , qi ). This is indeed an equivalence relation, and the class of (p, q) will be denoted by pq −1 . We consider now those pairs (p, q) satisfying the subsequent extra conditions, which turn out to be preserved by the previous equivalence: (1) q is an s-equivalence; (2) p and q are “cotransversal ”. The former condition implies that the kernel R = Ker q is principal (7.2.2). The latter condition will be expressed by means of the following (commutative) “butterfly diagram” gathering the previous data: S◮

j

i

> v

∼

K
∨ H ◭ G. Then the condition of cotransversality means that u or (this is indeed equivalent) v is an exactor (then v will be an s-exactor). When u (and v) are actors, p and q are said to be “transverse”, and the fraction p/q is called “irreducible” (or simplified). One can show (using the theory of extensors) that the class pq −1 owns a unique irreducible representative p/q.

658


Our meromorphisms (from H to G) are then defined as the classes pq −1 or their irreducible representatives p/q. A “Morita equivalence”is the special case when p is an s-equivalence too. The butterfly diagram is then perfectly symmetric. We say that (u, v) is a pair of “conjugate principal actors”. Each one determines the other one up to isomorphism. Using irreducible representatives and forgetting the D-structures, one then recovers easily the set-theoretical part of the description of Skandalis-Haefliger homomorphisms (two commuting actions, one being principal). Now, in the differentiable case, the local triviality conditions are automatically encapsulated in the surmersion conditions (imposed to the good epis) by means of the Godement theorem. One of the immense advantages of this presentation (apart from being defined in many various frameworks), is that the use of non irreducible representatives allows a very natural definition of the composition of meromorphisms (note that in [Ast 116] this composition is defined by A. Haefliger but in very special cases, when one arrow is a Morita equivalence). This composition is defined by means of the following diagram (using the diptych properties of Gpd(D)) (8.4): L

>

◭

∼

a ex

K′

∼

◭

∼

◭

> G′

mero

>

>

mero

K a ex

a ex

G′′

pb

> G.

(Of course there are many things to check to justify all our claims).

10.3 Example Let us come back to the example of the space of leaves of a (regular) foliation (2.4). We invite the reader to look at what happens when we take as good epis: (1) all the surmersions; (2) the retroconnected ones; (3) the retrodiscrete (or étale) ones; (4) the proper ones. In the first case, we are allowed to take as an atlas the holonomy groupoid and the transverse holonomy pseudogroups associated to various tranversals as well, which all belong to the same Morita class. The second choice is adapted to the search for invariants of the Molino equivalence class of a foliation: the Morita class of the holonomy groupoid is such an invariant. The third choice is adapted to the use of holonomy pseudogroups and of van Est S-atlases [Ast 116], and to the study of the effect of coverings on foliations. The fourth choice is adapted to the study of compact leaves of foliations and of properties related to Reeb stability theorem, as well as to the study of orbifolds (or Satake manifolds).


659

In fact it is useful to use various choices simultaneously.

11

Epilogue

• We have not given here statements concerning diptych structures on the category Mero(D), since our present results are still partial and demand some further checks to ensure them completely, especially concerning the Godement property. It is clear for us that such types of statements would be very useful, since, for instance, groupoids in such diptychs would be fascinating objects. Anyway it seems clear for us that this category has to be explored more deeply. • We know that our formal construction for the previous category of fractions seems to work perfectly as well when replacing the category of s-equivalences by that of sextensors or by various subcategories of the latter (adding for instance conditions of connectedness on the isotropy groups). We are convinced that such categories, which are much less known (not to say totally unknown) than the previous one, are basic for ˇ an f oliations), the understanding of holonomy of foliations with singularities (Stef and that these enlarged Morita classes certainly encapsulate some deep and hidden properties of the orbit spaces.

Acknowledgment (1) I would like to thank warmly the organizers for their invitation to this Conference at Krynica. I was happy with the friendly and stimulating atmosphere which was reigning throughout this session. (2) I am indebted to Paul Taylor for the (certainly very awkward) use I made of his package “diagrams”, allowing various styles of arrows, which are very useful as condensed visual mnemonics for memorizing properties of maps and functors.

Notes 1

This surjectivity is not implied by the usual definition of a refinement of a covering, but one can always impose it by the following slight modification of the definition, which changes nothing for the common use made of it : one demands in addition that a refinement of a covering contains this latter covering, which can always be achieved by taking their union. 2 i.e. up to isomorphisms. 3 As in [B], we have to deal with possibly non Hausdorff manifolds. 4 Possibly non Hausdorff. 5 See below. 6 It is said that the same thing happened at the very instant of the big bang, with analogous consequences. 7 This means that they are coequalizers [McL]. See below for an alternative formulation. 8 We just mention that it may be sometimes useful to work with only the “weak source-stability” condition, dual of (b). It is then possible to define a suitable full subcategory of D in which the strong axiom is satisfied. The objects of this subcategory own (in particular) the property that their diagonal

660


maps are in Di , which is formally a Hausdorff (or separation) type property. These objects may be called i-scattered. 9 It might be also advisable to look for a way of adapting the axioms without weakening their power in order to include some noteworthy exceptions such as measurable spaces or Riemannian or Poisson manifolds. 10 Owing to the lack of symmetry for the axioms, one cannot in general define the dual of a diptych. 11 See for instance, for a good part, but not the whole, of the properties involved in this statement, the textbooks by Mac Lane and Peter Johnstone. 12 The full subcategory of Hausdorff spaces. 13 If we had taken these ones for Top, we would have been in the situation alluded to in footnote 8 and the Hausdorff spaces might be constructed as the i-scattered objects. 14 This is indeed an important source of difficulty, but also of richness for the theory. 15 The latter with a terminal object, the former without such. 16 The notation suggests that the elements of such an object have to be numbered from 0 to n, the dot symbolizing the added 0. 17 We stress again that it is not isomorphic to Nc . 18 This is the category denoted by 2 in [McL], since it represents the order of the ordinal 2. 19 Ehresmann’s notation. 20 In Dif, there are plenty of useful pullback squares generated by pairs (f, v) which are not transversal in the classical sense of [B] (for instance two curves intersecting neatly in a high dimensional manifold), but there are also (actually pathological) pullback squares existing without (f, v) being weakly transversal. We think the weak transversality is the most useful notion. 21 Observe that perfect squares are good pullbacks, as well as those arising from axiom (v), or more generally from (3.2.2), but it may happen that more general ones are needed. 22 Such a square owns the parallel transfer property : (v ∈ Ds ) ⇒ (u ∈ Ds ). 23 This was not demanded in a). 24 It may be sometimes useful to extend the following definitions when T is just a graph with two given subgraphs. 25 Forgetting the prediptych structure of T and dropping conditions (1) would oversimplify the theory, but deprive it of all its strength. 26 Again with the “vertical composition”, which we prefer here to write horizontally, drawing the diagrams vertically, and the morphisms horizontally. Of course one can exchange everywhere simultaneously “vertical” and “horizontal”, since the distinction is purely psychological and notational, and since vert D and hor D are canonically isomorphic. 27 This notation means that D has to be considered here with its vertical composition law. 28 In the second members of this line, the signs | | denote the forgetful functor forgetting the composition laws, since they are not defined by the first members. → − → 29 − ·1 = 2 is just what we called in (3.4.4) the silly category I = ↓ = → . 30 And of course the units. 31 Though the numbering of the base is kept! 32 This induces a strong dissymmetry between the dual diptychs Nc and N∗c . 33 Though these images may be regarded as instances of diagrams of finite type in G, the notations used below differ slightly from those used in the previous section for the general case. 34 With two-sided edges and numbered vertices. 35 But not products (and not the pushouts which don’t arise from perfect squares). When some risk of confusion might arise, it would be more correct to say something like “diptych-exact” or “p.b.-exact”, since here the term “exact” has to be understood in a much weaker sense than the general meaning for functors between categories : it has not to preserve all (co)limits.


36

661

Relaxed notation, using only the images of the objects. This notion was introduced by Ehresmann for general categories. 38 We cannot give more details here. 39 This means the map : (y, x) 7→ yx−1 . 40 Mackenzie’s terminology. 41 Introduced by C. Ehresmann under the name of differentiable groupoids. 42 We cannot accept the traditional terminology “discrete”, which has another topological meaning, and which moreover does not agree with the group terminology, unlike ours. 43 Whence the terminology adopted here, preferably to “anchor map”. 44 Which might be written 0 = ·(−1) . 45 Dual arrow, from Mac Lane’s notations. 46 We cannot accept the term “coarse”, often used in the literature for the same reason as for “discrete”. 47 We gave above examples of diptychs in which such is not always the case. 48 Where the formal aspect of this theorem is clearly visible, and inspired our Godement axiom. 49 We keep the term D-group for the case when moreover B is a terminal object. On the other hand, the possible term “multigroup” would create confusion with the multiple categories, which have nothing to do with the present case. 50 The term “Lie groupoid” was first reserved to that special case (see for instance the first textbook by K. Mackenzie), till A. Weinstein and P. Dazord changed the terminology, with my full agreement. 51 One has to check the exactness property. 52 Most of these properties own various names in the literature, depending on the authors, and equally unfortunate for our purpose, since these purely algebraic properties received often names issued from Topology, which cannot be kept when working in Top or Dif. 53 See (2.3) for the terminology. 54 The underlying algebraic notion is known in the categorical literature under the name of “fibering functors” (or “star-surjective” functors for Ronnie Brown), which cannot be kept when working in a topological setting. 55 Whence the terminology. 56 As announced, this explains the terminology used for the squares. 57 The terminology derives from the following fact : it turns out that such functors are the exact generalizations of Lie group extensions (save for the fact that one has to use two-sided cosets, which, in general, don’t coincide with right or left cosets.). The smooth case is treated in [P2], which is written in order to be read possibly in any diptych without any change. 58 This problem doesn’t arise in the set-theoretic case. Many authors seem to believe that pullbacks along s-functors always exist in the Dif case , but actually the delicate point, often forgotten, is to prove the surmersion condition for the source map 59 Since the terminal object 0 has been dropped. 60 While the banal (and principal) N+ c -groupoids are p × p. 61 The terminology will become clear below in 9.3. 62 More precisely, with a suitable diptych structure (see 8.4). This is indeed a double groupoid, or better a “groupoid-cogroupoid ”. 63 As in [McL], the same notation is used for a functor and the identity natural transformation associated to this functor, here T 64 Written in loose notations, i.e. identifying groupoids and functors with their 1-level part. 65 Assumed moreover to be faithful, to preserve products, and to define exact diptych morphisms. An object B of D is viewed as a “structure” on the “underlying set” |B|, and an arrow A → B of D is fully ˙ described by the triple (A, |f |, B) . Any set E may be endowed with the “discrete structure” E. 66 In our framework this would result from 5.3. 37

662


67

If we consider the square made up by the sources and targets of both laws, this condition means that three of the four edges may be given arbitrarily. 68 i.e. a functor which is full, faithful, and essentially surjective, but possibly non-surjective. 69 More precisely, this means that it defines a bijection between the two orbit spaces.

References [1] [P1] J. Pradines: “Building categories in which a Godement theorem is available”, Cahiers Top. et Géom. Diff., Vol. XVI-3, (1975), pp. 301–306. [2] [P2] J. Pradines: “Quotients de groupo¨ıdes différentiables”, C.R.A.S., Paris, Vol. 303(16), (1986), pp. 817–820. [3] [P3] J. Pradines: “How to define the graph of a singular foliation”, Cahiers Top. et Géom. Diff., Vol. XVI-4, (1985), pp. 339–380. [4] [P4] J. Pradines: “Morphisms between spaces of leaves viewed as fractions”, Cahiers Top. et Géom. Diff., Vol. XXX-3, (1989), pp. 229–246. [5] [P5] J. Pradines: “Feuilletages: holonomie et graphes locaux”, C.R.A.S., Paris, Vol. 298(13), (1984), pp. 297–300. [6] [P6] J. Pradines: “Remarque sur le groupo¨ıdes cotangent de Weinstein-Dazord”, C.R.A.S, Paris, Vol. 306, (1988), pp. 557–560. [7] [B] N. Bourbaki: Variétés différentielles et analytiques, Hermann, Paris, 1971. [8] [McL] S. Mac Lane: Categories for the Working Mathematician, Springer-Verlag, New York, 1971. [9] [LALG] J.-P. Serre: Lie Algebras and Lie Groups, W.A. Benjamin Inc., New York, 1965. [10] [G-Z] P. Gabriel and M. Zisman: Calculus of fractions and homotopy theory, Ergebn. Math. 35, Springer, 1965. [11] [Ast. 116] Astérisque 116, Structure transverse des feuilletages, Société Mathématique de France, Paris, 1984. A. Haefliger: “Groupo¨ıdes d’holonomie et classifiants”, pp. 70-97. W.T. van Est: “Rapport sur les S-atlas”, pp. 235-292.

CEJM 2(5) 2004 663–707

Nondegenerate cohomology pairing for transitive Lie algebroids, characterization Jan Kubarski1∗, Alexandr Mishchenko2† 1

Institute of Mathematics, Technical University of L´ od´z, al. Politechniki 11, 90-924 L´ od´z, Poland 2 Department of Mathematics, Moscow State University, Leninskije Gory 119992, Moscow, Russia

Received 15 December 2003; accepted 15 April 2004 Abstract: The Evens-Lu-Weinstein representation (QA , D) for a Lie algebroid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as or well, this representation is slightly modified to (Qor A , D ) by tensoring by orientation flat line or = D⊗∂ or . It is shown that the induced cohomology pairing bundle, Qor A = QA ⊗or (M ) and D A or is nondegenerate and that the representation (Qor A , D ) is the unique (up to isomorphy) line representation for which the top group of compactly supported cohomology is nontrivial. In the case of trivial Lie algebroid A = T M the theorem reduce to the following: the orientation flat bundle (or (M ) , ∂ or ) is the unique (up to isomorphy) flat line bundle (ξ, ∇) for which the twisted de Rham complex of compactly supported differential forms on M with values in ξ possesses the nontrivial cohomology group in the top dimension. Finally it is obtained the characterization of transitive Lie algebroids for which the Lie algebroid cohomology with trivial coefficients (or with coefficients in the orientation flat line bundle) gives Poincaré duality. In proofs of these theorems for Lie algebroids it is used the Hochschild-Serre spectral sequence and it is shown the general fact concerning pairings between graded filtered differential R-vector spaces: assuming that the second terms live in the finite rectangular, nondegeneration of the pairing for the second terms (which can be infinite dimensional) implies the same for cohomology spaces. c Central European Science Journals. All rights reserved.

Keywords: twisted cohomology, cohomology of Lie algebras, Poincaré duality, Lie algebroid, modular class, cohomology pairing, Evens-Lu-Weinstein pairing, Hochschild-Serre spectral sequence MSC (2000): 58 H 99, 17 B 56, 18 G 40, 55 R 20, 55 T 05, 57 R 19, 57 R 22, 58 A 10 ∗ †


664

1

J. Kubarski, A. Mishchenko / Central European Journal of Mathematics 2(5) 2004 663–707

Introduction

The cohomology pairing coming from Evens-Lu-Weinstein representation of a Lie algebroid [4] is very important in many applications of Lie algebroids (Poisson geometry, intrinsic characteristic classes). This pairing generalizes the well known pairings that give Poincaré duality for Lie algebra cohomology and de Rham cohomology of a manifold and real cohomology of transitive invariantly oriented Lie algebroids [14]. For a Poisson manifold, this pairing agree with the pairing on the Poisson homology. The authors of [4] give an example of a nontransitive Lie algebroid for which the pairing is not necessarily nondegenerate and post the problem of when it is nondegenerate. This paper gives the positive answer for the case of any transitive Lie algebroids and proves the property of this representation: it is the one (up to isomorphy) for which the top group of compactly supported cohomology is nontrivial. Finally, we prove that for the nonregular transformation Lie algebroid corresponding d , there is no line representation to the action γ : R → X (R) , γ (t) = t· X where X = xN dx for which the cohomological pairing is nondegenerate. More detailed, this paper is devoted to prove two cycles of theorems, mutually overcoming. FIRST CYCLE concerns nondegenerate cohomology pairings for manifolds (Theorem 2.3), Lie algebras (Theorem 3.4) and Lie algebroids (Theorem 7.3). — Assume that M is a connected m-dimensional manifold (oriented or not) and ξ1 , ξ2 are two flat vector bundles with flat covariant derivatives ∇1 and ∇2 respectively. Denote by or (M) the orientation bundle with canonical flat structure ∂ or . Let F : ξ1 × ξ2 → or (M) be a pairing (i.e. 2-linear homomorphism) of vector bundles compatible with the flat structures (∇1 , ∇2 , ∂ or ), nondegenerate at least at one point (therefore, at every). From such a pairing one obtains a pairing on differential forms and the induced pairing in cohomology j H∇ 1

(M, ξ1 ) ×

m−j H∇ 2 ,c

F#

(M, ξ2 ) −→

H∂mor ,c

R or,#

M (M, or (M)) −→ R

is nondegenerate in the sense that ∼ =

j m−j H∇ (M, ξ1 ) → H∇ (M, ξ2 ) 1 2 ,c

∗

.

The index ”c” means that the compactly supported cohomology are considered. This theorem generalizes the classical Poincaré duality as well as the one for dω -cohomologies [6]. — Assume that g is an arbitrary n-dimensional Lie algebra and ∇1 , ∇2 : g → LR are two representations of g in R. Denote by ∇trad : g → LR the trace-representation n (∇trad )a = tr (ada ) · id. Then the top group of cohomology Htrad (g) of g with respect ∼ = n to ∇trad is nonzero, Htrad (g) → R, and if the multiplication of reals is compatible with respect to (∇1 , ∇2 , ∇trad ) then the exterior multiplication ∧ : Λr g∗ × Λn−r g∗ → Λn g∗ ∼ =R yields the induced nondegenerate pairing in cohomology n−i n i H∇ (g) × H∇ (g) → Htrad (g) ∼ = R, 1 2


i.e.

∼ =

i n−i H∇ (g) → H∇ (g) 1 2

∗

665

.

In particular, for (∇1 , ∇2 , ∇trad ) = (0, ∇trad , ∇trad ) we obtain ∗ ∼ = n−i H i (g) → Htrad (g) . For unimodular Lie algebra g the usual Poincaré duality is obtained in this way. — Let A = (A, [[·, ·]], #A ) be a Lie algebroid on M and QA = Λtop A ⊗ Λtop T ∗ M the line vector bundle with canonical Evens-Lu-Weinstein representation [4], Dγ (Y ⊗ ϕ) = Lγ (Y ) ⊗ ϕ + Y ⊗ L#A (γ) (ϕ) . To consider non-oriented manifolds we modify it into Qor A = QA ⊗ or (M) and D or = D ⊗ ∂Aor tensoring by the orientation bundle and its flat structure ((∂Aor )γ σ = (∂ or )#A (γ) σ, σ ∈ Γ (or (M)), #A : A → T M is the anchor of A). For transitive Lie algebroids with nor dimensional isotropy Lie algebras and multiplications by reals (M × R) ⊗ Qor A → QA the induced pairing in cohomology m+n−j m+n or H j (A) × HD (A, Qor or ,c A ) → HD or ,c (A, QA ) → R m+n or ∼ is nondegenerate, i.e. HD or ,c (A, QA ) = R and m+n−j H j (A) ∼ = HDor ,c (A, Qor A)

∗

.

SECOND CYCLE shows the uniqueness of the line representation for which the top group of compactly supported cohomology is not zero (Theorems 2.10, 3.5, 7.10). m — H∇,c (M, ξ) 6= 0 if and only if (ξ, ∇) is, up to isomorphy, the orientation flat line m bundle (or (M) , ∂ or ) . In particular, for oriented manifold, H∇,c (M, ξ) 6= 0 if and only if (ξ, ∇) is, up to isomorphy, the trivial flat line bundle (M × R, ∂) .

— For an n-dimensional Lie algebra g the trace-representation ∇trad is the unique line n representation ∇ for which H∇ (g) 6= 0. — Let A be a transitive Lie algebroid and ∇ a representation of A in a line vector m+n bundle ξ. Then H∇,c (A, ξ) 6= 0 if and only if (ξ, ∇) is, up to isomorphy, the E-L-Wor or representation (QA , D ) . In conclusion we obtain a full classification of transitive Lie algebroids for which the algebra of real cohomologies with trivial coefficients satisfies the Poincaré duality.

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— If A is a transitive Lie algebroid then the following conditions are equivalent: • Hcm+n (A) 6= 0, ∗ • Hcm+n (A) ∼ = R and H j (A) ∼ = (Hcm+n−j (A)) , • A is orientable vector bundle and the modular class of A is zero, θA = 0. In particular, — For an orientable manifold M we have: Hcm+n (A) 6= 0 if and only if A is a TUIO-Lie algebroid [13], i.e. the adjoint Lie Algebra Bundle g = ker #A is oriented and there is a global nonsingular section ε ∈ Γ (Λng ) invariant with respect to the adjoint representaion. The above theorem for a compact oriented manifold M and 1-rank adjoint LAB g = M × R was proved earlier in [15]. To prove Theorem 7.10 we use Theorem 4.4 concerning a pairing · : 1A × 2A → 3A between graded filtered differential R-vector spaces and theirs spectral sequnces. Roughly (m+n) speaking, if the second terms rE2j,i live in the rectangular 0 ≤ j ≤ m, 0 ≤ i ≤ n, 3E2 = 3 m,n ∼ 1 (j) 2 (m+n−j) 3 m,n ∼ E2 = R and the multiplication of the second terms h·, ·i2 : E2 × E2 → E2 = 1 (j) ∼ 2 (m+n−j) ∗ R is nondegenerate in the sense that E2 = E2 , then the cohomology pairing for ∼ ∗ = cohomologies of spaces is nondegenerate as well, i.e. 3H m+n ∼ = R and 1H j −→ (2H m+n−j ) . We must stress that the spaces rE2j,i may be infinite dimensional.

2

Non-degenerate pairings for twisted cohomology of a manifold

Many of the facts from this section belong to ”the folklore”. We call 1-dimensional vector bundles line bundles.

2.1 Twisted cohomology, elementary properties Let M be an m-dimensional paracompact manifold and ξ a vector bundle of rank p and ∇X ν, X ∈ X (M) , ν ∈ Γ (ξ) , a flat covariant derivative on M in ξ. (•) The differerential equation ∇ν = 0 (with respect to the local section ν of ξ) is locally uniquelly integrable. The local section ν satisfying ∇ν = 0 is called ∇-constant (or sometimes ∇-invariant). To set a flat covariant derivative ∇ is equivalent to set local trivializations {(Uα , ϕα )} relative to which the transitive functions are locally constant which is, in turn, equivalent to set a homomorphism of Lie algebroids ∇ : T M → A (ξ) where A (ξ) is the Lie algebroid of ξ. The flat bundles (ξ, ∇) form a category with morphisms F : (ξ1 , ∇1 ) → (ξ2 , ∇2 ) being linear isomorphisms F : ξ1 → ξ2 compatible with flat covariant derivatives (∇1 , ∇2 ), i.e. for which F (∇1,X ν) = ∇2,X (F ν) . We write also F

(ξ1 , ∇1 ) ∼ (ξ2 , ∇2 ) F

or briefly ∇1 ∼ ∇2 .


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Two flat line bundles over a connected manifold are isomorphic if and only if they have the same holonomy homomorphism h : π1 (M, x) → GL (R, 1) . For a flat vector bundle (ξ, ∇) the differential operator d∇ of the degree 1 for ξ-valued differential forms Ω∗ (M, ξ) is defined by the standard formula d∇ (φ) (X0 , ..., Xq ) X X = (−1)i ∇Xi (φ (X0 , ...ˆı..., Xq )) + (−1)i+j φ ([Xi , Xj ] , X0 , ...ˆı....ˆ ..., Xq ) . i

i<j

Let σα1 , ..., σαp be local sections of ξ over Uα corresponding to the standard basis e1 , ..., ep ∈ Rp under the trivialization ϕα , σαi (x) = ϕα,x (ei ) . The local sections σαi are ∇-constant, P i ∇σαi = 0. Over Uα a ξ-valued q-form φ can be written as φ ⊗ σαi , φi ∈ Ωq (Uα ) P i P and we have d∇ ( φ ⊗ σαi ) = ddR (φi) ⊗ σαi . The flatness of ∇ implies that d∇ is a differential operator, d2∇ = 0, therefore Ω∗ (M, ξ) is a differential complex and the (twisted) cohomology ∗ H∇ (M, ξ) = H (Ω∗ (M, ξ) , d∇ ) makes sense. By the definition the 0-group of cohomology can be written as 0 H∇ (M, ξ) = {ν ∈ Γ (ξ) ; ∇X ν = 0 for all X ∈ X (M)} .

(1)

0 (••) If (ξ, ∇) is a line nontrivial flat vector bundle then according to (•) above H∇ (M, ξ) = 0. If ∇ is a flat covariant derivative in a vector bundle ξ and ω ∈ Ω1 (M) is a closed real 1-form, then ∇ωX ν = ∇X ν + ω (X) · ν (2)

is a flat covariant derivative as well. If ξ is a line bundle and ∇ and ∇1 are two flat covariant derivatives then there exists a closed 1 -form ω such that ∇1 = ∇ω . Each flat covariant derivative ∇ in the trivial vector bundle M ×R is of the form ∂ ω for some closed 1-form ω (∂ is the standard covariant derivative in the trivial vector bundle M × R defined by differentiation of functions ∂X (f ) = X (f ) ). Differential operator d∂ ω is given directly by d∂ ω (φ) = ddR φ + ω ∧ φ. The operator d∂ ω is in the literature denoted rather by dω than by d∂ ω [6], [8] and the cohomology space H∂ ω (M, M × R) is denoted by Hω (M). If ω = 0 the usual de Rham cohomology of M is obtained. It is easy to see that    0 ⇐⇒ ω is nonexact, 0 Hω (M) = (3)   R ⇐⇒ ω is exact. The space of ξ-valued q-forms with compact support Ω∗c (M, ξ) is a differential complex F ∗ as well and we have the compactly supported cohomology H∇,c (M, ξ) . If (ξ1 , ∇1 ) ∼ (ξ2 , ∇2 ) then F∗ : Ω∗ (M, ξ1 ) → Ω∗ (M, ξ2 ) commutes with the differential operators d∇1

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∗ ∗ and d∇2 and gives rise to an isomorphism in cohomology F# : H∇ (M, ξ1 ) → H∇ (M, ξ2 ). 1 2 ∗ Analogously, for compact supports, we have an isomorphism F#,c : H∇1 ,c (M, ξ1 ) → ∗ H∇ (M, ξ2 ) . 2 ,c For an open subset U ⊂ M we have the restricted flat covariant derivative ∇U on U in the vector bundle ξU and the twisted cohomology H∇ (U, ξ) and H∇,c (U, ξ) are defined. Similarly as in the case of real coefficients (see for example [1]) we can obtain the short exact Mayer-Vietoris sequences (U1 , U2 ⊂ M are open subset, U = U1 ∪ U2 , and U12 = U1 ∩ U2 ) β

α

0 → Ω∗ (U, ξ) → Ω∗ (U1 , ξ) ⊕ Ω∗ (U2 , ξ) → Ω∗ (U12 , ξ) → 0 and βc

α

0 ← Ω∗c (U, ξ) ←c Ω∗c (U1 , ξ) ⊕ Ω∗c (U2 , ξ) ← Ω∗c (U12 , ξ) ← 0. They give rise to long exact sequences in cohomology α#

β#

∂q

q q q q q+1 → H∇ (U, ξ) → H∇ (U1 , ξ) ⊕ H∇ (U2 , ξ) → H∇ (U12 , ξ) → H∇ (U, ξ) →

and αc#

βc#

∂q

q q q q c q+1 ← H∇,c (U, ξ) ← H∇,c (U1 , ξ) ⊕ H∇,c (U2 , ξ) ← H∇,c (U12 , ξ) ← H∇,c (U, ξ) ← ∗ ∗ ∗ (U, ξ) of H∇ (M, ξ) Remark 2.1. There is a natural isomorphism H∇ (M, ξ) ∼ = HI(∇) ∗ with HI(∇) (U, ξ) , the cohomology of M in the sheaf I (∇) of local ∇-constant sections of ∗ ξ. In other words, H∇ (M, ξ) are cohomology of M with local system of coefficients.

2.2 Orientation flat bundle and its characterization Let {(Uα , xa )} be a coordinate open cover for the manifold M, with transition functions gαβ = xα ◦ x−1 β . Take the orientation bundle or (M) , i.e. the line bundle on M with a distinguished system of local trivializations {ϕα } such that the transition functions are equal to sgn J (gαβ ) [1]. Let {eα } be a family of local sections corresponding to 1 under the trivializations {ϕα } , eα (x) = ϕα,x (1) . In the bundle or (M) there exists exactly one flat covariant derivative ∂ or such that eα are ∂ or -constant, ∂ or (eα ) = 0. The notation eα and ∂ or is valid in the whole paper. The flat orientation bundle (or (M) , ∂ or ) is characterized by the holonomy homomorphism s : π1 (M, x0 ) → Z2 ⊂ GL (R, 1) that can be identified with monodromy to the group of the local orientations in the fixed point x0 which also is Z2 . In the sequel it will be useful to give other characterization of the flat orientation bundle. Proposition 2.2. Let (ξ, ∇) be a flat line bundle. The following conditions are equivalent: (a) (ξ, ∇) ∼ = (or (M) , ∂ or ) ,


669

(b) there exists a collection of local sections {σα } of ξ such that σa are ∇-constant and the transition functions are equal to sgn J (gαβ ) , (c) (or (M) ⊗ ξ, ∂ or ⊗ ∇) ∼ = (M × R, ∂) , (d) there exists a global nonsingular section t ∈ Γ (or (M) ⊗ ξ) which is ∂ or ⊗∇-constant. Proof. Equivalences (a)⇔(b) and (c)⇔(d) are evident by definition. (b)⇒(c) The linear homomorphism F : or (M)⊗ξ → M ×R defined by F (eα ⊗ σα ) = 1 is a well defined linear isomorphism compatible with (∂ or ⊗ ∇, ∂) . (d)⇒(b) Locally t = eα ⊗ σα for some local nonsingular sections σα of ξ. Since 0 = ∂ or ⊗ ∇ (eα ⊗ σα ) = ∂ or ⊗ σα + eα ⊗ ∇σα = eα ⊗ ∇σα , it follows that σα are ∇-constant and have the same transition functions sgn J (gαβ ) . The or (M)-valued m-differential forms are called densities. There exists an operator Z or : Ωm c (M, or (M)) → R M

of the integration of densities and the Stoke’s Theorem for densities holds Z or d∂ or (ω) = 0 M

for ω ∈ Ωcm−1 (M, or (M)) [1]. Hence it produces a linear operator Z

or,#

: H∂mor ,c (M, or (M)) → R.

(4)

M

2.3 Pairings and cohomology, nondegeneracy Now let (ξ1 , ∇1 ) , (ξ2 , ∇2 ) , and (ξ3 , ∇3 ) be three flat vector bundles. We say that (ξ1 , ∇1 ) and (ξ2 , ∇2 ) are paired to (ξ3 , ∇3 ) if there is a bilinear homomorphism F : ξ1 × ξ2 → ξ3 compatible with flat covariant derivatives (∇1 , ∇2 , ∇3 ), i.e. such that, for every X ∈ X (M) , ∇3,X F (ν1 , ν2 ) = F (∇1,X ν1 , ν2 ) + F (ν1 , ∇2,X ν2 ) .. (5) Then we write F : (ξ1 , ∇1 ) × (ξ2 , ∇2 ) → (ξ3 , ∇3 ) . From such a pairing one obtains a pairing (φ, ψ) 7→ φ ∧ ψ := F∗ (φ, ψ) of Ωq (M, ξ1 ) and Ωr (M, ξ2 ) to Ωq+r (M, ξ3 ) fullfilling the equality d∇3 F∗ (φ, ψ) = F∗ (d∇1 φ, ψ) + (−1)deg φ F∗ (φ, d∇2 ψ) . Clearly, φ ∧ ψ := F∗ (φ, ψ) is the usual wedge product of differential forms with F multiplication of values, see [7, Vol.II]. The pairing of differential forms induces a pairing of cohomology classes ∗ ∗ ∗ F# : H∇ (M, ξ1 ) × H∇ (M, ξ2 ) → H∇ (M, ξ3 ) 1 2 3

670


as well as the pairing for compact supports ∗ ∗ ∗ F#,c : H∇ (M, ξ1 ) × H∇ (M, ξ2 ) → H∇ (M, ξ3 ) . 1 2 ,c 3 ,c

Consider two flat vector bundles (ξ1 , ∇1 ) , (ξ2 , ∇2 ) and a pairing F : (ξ1 , ∇1 ) × (ξ2 , ∇2 ) → (or (M) , ∂ or ) .

(6)

For an open subset U ⊂ M we define a pairing Z or,# R or,# F# q m−q m ◦F# : H∇1 (U, ξ1 ) × H∇2 ,c (U, ξ2) → H∂ or ,c (U, or (M)) U→ R, U

and the Poincaré linear homomorphism DUq

:

q H∇ 1

(U, ξ1 ) →

m−q H∇ 2 ,c

(U, ξ2 )

∗

,

DUq

([Φ]) ([Ψ]) =

Z

(Φ ∧ Ψ) .

U

Similarly as in the case of real coefficients we check that the family of Poincaré homomorphisms {DUq } induces a map from the long exact sequences in cohomology to the long exact sequences in compactly supported cohomology (the symbols of vector bundles ξ1 and ξ2 in the diagram below are ommitted and the sign ± is equal precisely to (−1)q+1 ) q H∇ (U)  D y U

α#

−−−→

β#

q q H∇ (U1 ) ⊕ H∇ (U2 )  D ⊕D y U1 U2

q −−−→ H∇ (U12 )  D y U12

∂q

q+1 −−−→ H∇ (U)  D y U

(7)

∗ ∂cq

∗ ±( ) α∗c# βc# q q q q q+1 H∇,c (U, ξ)∗ −−−→ H∇,c (U1 )∗ ⊕ H∇,c (U2 )∗ −−−→ H∇,c (U12 )∗ −−−−→ H∇,c (U)∗ ` For an infinite disjoint open subsets U = Ui we deduce that DU can be identifying Q with DUi .

Theorem 2.3. Assume that M is connected. If pairing (6) is nondegenerate at least one point then the cohomology pairing Z or,# R or,# F# q m−q m → R, ◦F# : H∇1 (M, ξ1 ) × H∇2 ,c (M, ξ2 ) → H∂ or ,c (M, or (M)) M M

is also nondegenerate in the sense that ∼ =

q q m−q DM : H∇ (M, ξ1 ) → H∇ (M, ξ2 ) 1 2 ,c

∗

is an isomorphism, q ∈ {0, 1, ..., m} . Proof. We can use the standard method from [7, Vol.I] (or a slightly modified method by using Riemannian structure and properties of geodesically convex neighbourhoods, [1], [18]). According to [7, Vol.I, Prop.II, p.16] and the commutativity of diagram (7) and remark on infinite disjoint open subsets we need only to prove the theorem for the manifold M = Rm .


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Each vector bundle ξ over Rm is trivial, each flat covariant derivative ∇ has trivial holonomy, so the differential equation ∇ν = 0 is globally integrable. Therefore for an arbitrary point x0 ∈ M there exists an isomorphism of flat vector bundles ϕ : (ξ, ∇) → (Rm × ξx0 , ∂) where by ∂ is denoted the standard flat covariant derivative ∂X f = X (f ) . (Remark: for the line bundle ξ the isomorphism ϕ can be given directly as follows. For ξ = Rm × R any flat covariant derivative ∇ is of the form ∇X f = ∂X f + ∂X (α) · f for some function α. Then ϕ (f ) = e−α f is a required isomorphism.) The isomorphism ϕ gives rise to an isomorphism in cohomology ∼ =

ϕ# : H∇ (Rm , ξ) → HdR (Rm , ξx0 ) , especially for the zero level 0 H∇

ϕ0#

ρ

=

=

0 (R , ξ) → HdR (Rm , ξx0 ) → ξ . ∼ ∼ x0 m

On the other hand, the isomorphism ϕ also gives rise to an isomorphism in compactly ∼ = supported cohomology ϕ#,c : H∇,c (Rm , ξ) → HdR,c (Rm , ξx0 ) , especially for the top level ϕm #,c

:

m H∇,c

ϕm #,c

ρc

=

=

m HdR,c (Rm , ξx0 ) → ξ , (R , ξ) → ∼ ∼ x0 m

where ρc is defined by the formula Z X i X ρc f · ∆ ⊗ vi = i

i

Rm

f i · ei

where vi is a basis of ξx0 , ∆ is a determinant function on Rm and f i ∈ Cc∞ (Rm ) are functions with compact support. ρc is independent of the choice of the basis vi and fulfils R the equality ρc ([f · ∆ ⊗ v]) = Rm f · v, f ∈ Cc∞ (Rm ) , v ∈ ξx0 . Now take flat vector bundles (ξi, ∇i ) on Rm and linear isomorphisms ϕi : (ξi , ∇i ) → Rm × (ξi )x0 , ∂ . For any pairing F : (ξ1 , ∇1 ) × (ξ2 , ∇2 ) → (ξ3 , ∇3 ) we get easily the commutative diagram 0 m m F# : H∇ (Rm , ξ1 ) × H∇ (Rm , ξ2 ) −−−→ H∇ (Rm , ξ3 ) 1 2 ,c 3 ,c   ϕ0 ×ϕm ϕm y 1# 2#,c y 3#,c 0 m m F¯# : HdR Rm , (ξ1 )x0 × HdR,c Rm , (ξ2 )x0 −−−→ HdR,c Rm , (ξ3 )x0   ρ×ρ ρc y c y

Fx0 : (ξ1 )x0 × (ξ2 )x0

−−−→

(ξ3 )x0

where the middle pairing comes from the ”constant” pairing F¯ : Rm , (ξ1 )x0 × Rm , (ξ2 )x0 → Rm , (ξ3 )x0 , F¯x = Fx0 . To prove the theorem take (ξ3 , ∇3 ) = (or (M) , ∂ or ) and choose a point x0 such that Fx0 is nondegenerate.

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2.4 Applications of the nondegenerate cohomology pairing Now we give a number of applications of Theorem 2.3. Example 2.4. For a connected orientable manifold M and the trivial flat vector bundles (ξi , ∇i ) = (M × R, ∂) and the multiplication of reals · : R × R → R we obtain the classical Poincaré duality H j (M) × Hcm−j (M) → Hcm (M) → R. Especially Hcm (M) = R and ∗ H j (M) ∼ = (Hcm−j (M)) . Example 2.5. More generally, for arbitrary connected manifold M taking (ξ1 , ∇1 ) = (M × R, ∂) and (ξ2 , ∇2 ) = (or (M) , ∂ or ) and the multiplication by reals F : (M × R) × or (M) → or (M) we get the Poincaré duality also for nonorientable manifold [1]. Especially operator (4) is an isomorphism, H∂mor ,c (M, or (M)) ∼ = R. Example 2.6. [6], [8] Let M be an oriented connected manifold. The following conditions are equivalent: m (1) Hω,c (M) = 0, 1 (2) H (M) ∋ [ω] 6= 0. If [ω] = 0 then Hωm (M) = R. Indeed, consider multiplication by reals F : (M × R) × (M × R) → M × R. This pairing is nondegererate and compatible with (∂ −ω , ∂ ω , ∂) . By Theorem 2.3 we get the p m−p nondegenerate pairing H−ω (M) × Hω,c (M) → Hcm (M) ∼ = R. In particular, we get ∗ 0 m H−ω (M) = Hω,c (M) , so all follows from (3). Each flat covariant derivative in or (M) is of the form (∂ or )ω for a closed 1-form ω. Concider the multiplications by reals F : (M × R) × or (M) → or (M). Then we easily get: Example 2.7. For any connected manifold M (oriented or not) the following conditions are equivalent: m (1) H(∂ or )ω ,c (M, or (M)) = 0, 1 (2) H (M) ∋ [ω] 6= 0. m If [ω] = 0 then H(∂ or )ω ,c (M, or (M)) = R. The next applications are given in the following propositions. Proposition 2.8. If M is orientable and ξ is an arbitrary line nonorientable (i.e. nontrivial) vector bundle then for any flat covariant derivative ∇ in ξ m H∇,c (M, ξ) = 0.

Proof. Indeed, consider the natural nondegenerate pairing F : (ξ, ∇) × (ξ, ∇) → (ξ ⊗ ξ, ∇ ⊗ ∇) , (ν, µ) 7→ ν ⊗ µ,


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and any linear isomorphism ϕ : ξ ⊗ ξ → M × R. The latter transforms the flat covariant derivative ∇ ⊗ ∇ to the ∂ ω for some closed 1-form ω. We recall that (∇ ⊗ ∇)X (ν ⊗ µ) = ∇X ν ⊗ µ + ν ⊗ ∇X µ. Then the pairing ϕ ◦ F : ξ × ξ → M × R is compatible with (∇, ∇, ∂ ω ) and, in consequence, with (∇−ω , ∇, ∂) (for ∇−ω see (2)). By Theorem 2.3 we have the nondegenerate pairing R

M 0 m m H∇ −ω (M, ξ) × H∇,c (M, ξ) → Hc (M) → R.

In consequence we obtain by the nontriviality of ξ and observation (•) from section 2.1 ∗ 0 m 0 = H∇ −ω (M, ξ) = H∇,c (M, ξ) m which imply H∇,c (M, ξ) = 0.

Proposition 2.9. If ξ is a line bundle not isomorphic to or (M) then for arbitrary flat covariant derivative ∇ in ξ we have m H∇,c (M, ξ) = 0.

Proof. Indeed, fix a linear isomorphism ϕ : ξ ⊗ ξ → M × R. Such isomorphism ϕ exists since ξ ⊗ ξ is orientable line vector bundle, therefore, trivial. F Let ∇ ⊗ ∇ ∼ ∂ ω for a closed 1-form ω. Take the multiplication by reals τ : or (M) ⊗ (M × R) → or (M) and notice that τ is compatible with (∂ or ⊗ ∂ ω , (∂ or )ω ) . Consider the canonical nondegenerate pairing F : (or (M) ⊗ ξ) × ξ → or (M) ⊗ ξ ⊗ ξ which is compatible with (∂ or ⊗ ∇, ∇, ∂ or ⊗ ∇ ⊗ ∇) . The composition F

id⊗ϕ

τ

F ′ : (or (M) ⊗ ξ) × ξ → or (M) ⊗ ξ ⊗ ξ −→ or (M) ⊗ (M × R) → or (M) , clearly, is nondegenerate and is compatible with (∂ or ⊗ ∇, ∇, (∂ or )ω ) . Therefore F ′ is compatible with (∂ or ⊗ ∇−ω , ∇, ∂ or ) . According to Theorem 2.3 applied to F ′ we get ∗ m H∂0or ⊗∇−ω (M, or (M) ⊗ ξ) ∼ (M, ξ) . = H∇,c Since ξ is not isomorphic to or (M) the vector bundle or (M) ⊗ ξ is not trivial (indeed, if or (M)⊗ξ ∼ = M ×R then or(M) ∼ = ξ∗ ∼ = ξ ) which produces H∂0or ⊗∇−ω (M, or (M) ⊗ ξ) = 0 m and further H∇,c (M, ξ) = 0. Finally we have the main application. Theorem 2.10. The following conditions are equivalent:

674


m a) H∇,c (M, ξ) 6= 0, m b) H∇,c (M, ξ) = R, c) (ξ, ∇) ∼ (or (M) , ∂ or ) ,

Proof. For c) =⇒ b) see Example 2.5 or [1]; b) =⇒ a) is evident. It remains to show that a) =⇒ c). Keep the notation ϕ and ω from the proof of the previous proposition. By the same reasoning we check H∂0or ⊗∇−ω (M, or (M) ⊗ ξ) 6= 0. It means that or (M) ⊗ ξ is trivial and there exists a nonsingular global cros-section ν ∈ Γ (or (M) ⊗ ξ) which is ∂ or ⊗∇−ω -constant. Express locally ν in the form ν = eα ⊗fα for some local sections fα of ξ, for eα see subsection 2.2. It is evident that {fα } has the transition function equal to sgn Jgαβ and that ∇−ω fα = 0, i.e. ∇X fα = ω (X) · fα . The ϕ formula f = ϕ (fα ⊗ fα ) determines correctly a nonsingular function f . Since ∇⊗∇ ∼ ∂ ω ω then ∂X f + ω (X) · f = ∂X f = 2 · ω (X) · f , one has ∂X f = ω (X) · f. The global cros-section ν ′ = f1 ν is ∂ or ⊗ ∇-constant. The proposition follows now from Proposition 2.2.

3

A generalization of the Chern-Hirzebruch-Serre Lemma and applications to cohomology of Lie algebras

We generalize Lemma 3 from [2] concerning Poincaré differentiation from algebras to pairings. The assumption on finite dimensionality is superfluous. L Lemma 3.1. Let As = ni=0 Ais , ds : As → As , s = 1, 2, 3, be three graded differential R-vector spaces such that (1) ds [Ais ] ⊂ Ai+1 s , 2 (2) ds = 0, (3) d3 A3n−1 = 0. (4) An3 ∼ = R, Ai3 = 0 for i > n. Let · : A1 × A2 → A3 be a (5) (6) (7)

pairing such that Ai1 · Aj2 ⊂ Ai+j 3 , d3 (x · y) = d1 x · y + (−1)deg x x · d2 y, · : Ar1 × A2n−r → An3 ∼ = R, r = 0, 1, ..., n are nondegenerate in the sense that the induced mappings ∗ ∼ = ir : Ar1 −→ A2n−r


675

are linear isomorphisms. Then the induced homomorphisms in cohomology · : H r (A1 , d1 ) × H n−r (A2 , d2) → H n (A3 , d3 ) ∼ =R are nondegenerate as well, i.e. the induced linear homomorphism i′r : H r (A1 , d1) → H n−r (A2 , d2 )

∗

are linear homomorphisms. Proof. The proof is identical with the original proof by Chern-Hirzebruch-Serre for an algebra and it is sufficient to check that (ir )#

∼ =

i′r : H r (A1 , d1 ) −→ H n−r (A∗2 , d∗2 ) −→ H n−r (A2 , d2) , where (A∗2 , d∗2 ) denotes the dual complex. Now we give some applications to the cohomology of Lie algebras with coefficients. Let g be a real Lie algebra of dimension n and let ∇ : g → LR = End R ∼ =R be an arbitrary representation in 1 dimensional vector space. We will distinguish two representations • ∇0 = 0, • (∇trad )a = tr (ada ) · id . We see that ∇0 = ∇trad if and only if g is unimodular. Denote the differential with respect to ∇trad by dtrad and the cohomology of g by Htrad (g) . Straightforward n−1 = 0. Therefore computations show that dtrad n Proposition 3.2. Htrad (g) = Λn g∗ ∼ = R for every Lie algebra.

Let us notice the following Remark 3.3. (1) Each representation ∇ : g → LR is equal to 0 on g2 and, conversely, each linear homomorphism ∇ : g → LR such that ∇|g2 = 0 is a representation. 0 (2) The zero group of cohomology H∇ (g) = 0 if and only if ∇ 6= 0. (3) The multiplication of reals · : R × R → R is compatible with (∇1 , ∇2 , ∇3 ) if and only if ∇3 = ∇1 + ∇2 . Point (1) from the remark above implies that any linear combination of representations is a representation. Take an arbitrary representation ∇ and put ∇′ = ∇trad − ∇.

676


Then the multiplication of reals is compatible with (∇′ , ∇, ∇trad ) by (3) from the remark. Therefore, for differential operators d∇′ , d∇ , dtrad the condition (6) from Lemma 3.1 holds. Since the exterior multiplication ∧ : Λr g∗ × Λn−r g∗ → Λn g∗ ∼ =R is nondegenerate then according to Lemma 3.1 the multiplication in cohomology r n−r n H∇ (g) → Htrad (g) ∼ =R ′ (g) × H∇

is nondegenerate as well, i.e. in particular ∗ 0 n ∼ H∇ ′ (g) = (H∇ (g)) .

Immediately from the above reasoning we obtain the following theorems. Theorem 3.4. The multiplication of reals is compatible with the representations (0, ∇trad , ∇trad ) and the induced cohomology pairing r n H n−r (g) × Htrad (g) → Htrad (g) ∼ = R,

is nondegenerate. In particular we obtain a noncanonical isomorphism r r Htrad (g) ∼ (g) = Htrad

∗

∼ = H n−r (g) .

n Theorem 3.5. ∇trad is the unique representation ∇ for which H∇ (g) 6= 0.

Proof. For any representation ∇ take ∇′ = ∇trad − ∇. By (2) from the remark above we have n 0 ⇐⇒ ∇′ = 0 ⇐⇒ ∇ = ∇trad . H∇ (g) ∼ = H∇ ′ (g) 6= 0

4

Pairings for graded filtered differential R-vector spaces and spectral sequences

The aim of this chapter is to prove that for any pairing of graded regularly filtered differential R-vector spaces, if the second terms of spectral sequences gives the nondegenerate pairing then the same holds for the cohomology algebras of the spaces. This holds without assumption that dim E2 is finite and generalizes the suitable theorem for graded filtered differential algebras [16]. Given three graded filtered differential R-vector spaces M r r i r r A= A , d, Aj , r = 1, 2, 3, (8) i≥0

denote for shortness r

H := H (rA, rd) .


677

Assume · : 1A × 2A → 3A preserves gradations and filtrations 1 s

A · 2At ⊂ 3As+t ,

(9)

Aj · 2Ak ⊂ 3Aj+k ,

(10)

1

and that the differentials rd satisfy the compatibility condition d (x · y) = 1dx · y + (−1)deg x x · 2dy.

3

(11)

Clearly, there exists a multiplication of cohomology classes · : 1H j × 2H k → 3H j+k ,

([x] , [y]) 7→ [x · y] ..

Let r

Esj,i , rds

be spectral sequences of graded filtered differential R-vector spaces (8). Lemma 4.1. (1) 1 j,i Zs

· 2Zsk,l ⊂ 3Zsj+k,i+l,

0 ≤ s ≤ ∞,

(2) j,k 1 j,i 2 k,l Zs · Ds−1 + 1Ds−1 · 2Zsi,l 1 j,i 2 k,l j,i 2 k,l Z∞ · D∞ + 1D∞ · Z∞

j+k+1,i+l−1 j+k,i+l ⊂ 3Zs−1 + 3Ds−1 , 0 ≤ s < ∞, j+k,i+l ⊂ 3D∞ ,

(s = ∞).

Proof. Straightforward calculations. Conclusion 4.2. There exists a multiplication of s-terms of spectral sequences 1 j,i Es

× 2Esk,l → 3Esj+k.i+l , ([x] , [y]) 7→ [x · y] ,

0 ≤ s ≤ ∞.

The differentials 1ds , 2ds , 3ds fulfils the compatibility condition with respect to the total gradation 3 ds (x · y) = 1ds x · y + (−1)total deg x x · 2ds y. There exists a multiplication of cohomology classes of s-terms H j,i 1Es , 1ds × H k,l 2Es , 2ds → H j+k,i+l 3Es , 3ds , ([˜ x] , [˜ y ]) 7→ [˜ x · y˜] . The linear isomorphisms of bigraded spaces r

σs : rEs+1 → H (rEs , rds ) ,

r

σ∞ : rE∞ → E0 (rH)

678


conserve the multiplications 3

σs ([x] · [y]) = 1σs [x] · 2σs [y] , 3

σ∞ (¯ x · y¯) = 1σ∞ (¯ x) · 2σ∞ (¯ y) .

Remark 4.3. For s ≥ i + 2 we consider the canonical epimorphisms j,i r j,i j,i r j+1,i−1 j,i j+1,i−1 j,i ∼ r j,i γs : rEsj,i ∼ / Z∞ + rDs−1 ։ rZ∞ / rZ∞ + rD∞ = rZ∞ = E∞ . For s ≥ i + l + 2 the canonical epimorphisms 1γsj,i, 2γsk,l , 3γsj+k,i+l are compatible with multiplications 3 j+k,i+l γs ([x] · [y]) = 1γsj,i [x] · 2γsk,l [y] . This implies that if spectral sequences (rEs , rds ) collapse at the mth ¯ term then the ca∼ = r r r nonical isomorphisms βm¯ : Em¯ → E∞ , see [7, Vol.III. §1.1.2], conserve bigradations and are compatible with multiplications. We recall the construction of rβm¯ . For arbitrary (j, i) we select arbitrary s ≥ max (m, ¯ i + 2) and put r j,i βm¯

rγ j,i s

rσ j,i

j,i m ¯ r j,i r j,i j,i : rEm Em+1 ←− ... ←− Es ։ rE∞ . ¯ ←− ¯ ∼ ∼ ∼ =

=

=

The following main result of this chapter generalizes Corollary 12 from [16]. Theorem 4.4. Given three graded filtered differential R-vector spaces (8) and a pairing · : 1A × 2A → 3A satisfying (9), (10), (11), assume that the filtrations are regular in the sense rA0 = rA and that the second terms rE2j,i live in the rectangular 0 ≤ j ≤ m, (m+n) 0 ≤ i ≤ n and that 3E2 = 3E2m,n ∼ = R. If the multiplication in the second terms (j)

(m+n−j)

h·, ·i2 : 1E2 × 2E2

→ 3E2m,n ∼ =R

is nondegenerate in the sense that 1 (j) E2

is a (a) (b) (c)

∼ =

(m+n−j) ∗

−→ 2E2

, x 7→ hx, ·i2,

linear isomorphism, then 3 m+n ∼ H = R, r t H = 0 for t > m + n, the multiplication in cohomology classes h·, ·iH : 1H j × 2H m+n−j → 3H m+n ∼ =R is nondegenerate as well, i.e. ∼ =

H j −→ 2H m+n−j

1

is a linear isomorphism.

∗

, [x] 7→ h[x] , ·iH ,

(12)


679

Proof. The terms rE3 , rE4 , ..., rE∞ live also in the same rectangular 0 ≤ j ≤ m, 0 ≤ i ≤ n. The bidegree argument of the second differential operator 3d2 implies (compare (m+n−1) with [16]) the condition 3d2 3E2 = 0. By the generalized Chern-Hirzebruch-Serre 3 (m+n) 3 m,n ∼ Lemma 3.1 we get E2 = E2 = R and nondegeneracy of the multiplication for third terms. Proceeding inductively we get the same for all finite terms. The bidegree argument for the further differential operators rds implies the colapsing of spectral sequ(m+n) m,n ∼ ences (rEs ,r ds ) , say at rm > max (m + 1, n + 2) places. Then 3E∞ = 3E∞ = R so (a) (m+n) 3 m+n ∼ 3 1 2 3 ∼ holds because H ¯ ≥ max ( m, m, m) the canonical = E∞ = R and next, for m r isomorphisms βm¯ (see Remark 4.3) are compatible with multiplications. In consequence, the multiplication in the infinite terms (j) (m+n−j) m,n ∼ · : 1E∞ × 2E∞ → 3E∞ =R

(13)

is nondegenerate as well. It remains to prove the nondegeneracy of the multiplication of cohomology classes (12). The spaces rH possess a natural graded filtration rH j,i, and thanks to the regularity of filtrations we have H t = rH 0,t ⊃ rH 1,t−1 ⊃ ... ⊃ rH t,0 ⊃ 0

r

(14)

and a noncanonical isomorphism Ht ∼ = (rH 0,t / rH 1,t−1 ) ⊕ (rH 1,t−1 / rH 2,t−2 ) ⊕ .... ⊕ rH t,0 =

r

M

E0j,i (rH) .

(15)

j+i=t

Analogously to the proof of Theorem 11 from [16] we assert that (m+n) r

E0

( H) = E0m,n (rH) = rH m,n ,

(16)

and r

H j,i = rH j+1,i−1 for j > m or i > n.

Therefore by (15) rH t = 0 for t > m + n which proves (b). As in [16] we check the rule: j,i j,i m−j,n−i m−j,n−i • if 1σ∞ (¯ x) = [x] for x¯ ∈ 1E∞ , x ∈ 1H j,i, and if 2σ∞ (¯ y ) = [y] for y¯ ∈ 2E∞ , 2 m−j,n−i y∈ H , then 3 m,n σ∞ (¯ x · y¯) = [x] · [y] = x · y. (17) m,n m,n We fix generators ξ∞ ∈ 3E∞ and ξH ∈ 3H m,n in such a way that 3σ∞ (ξ∞ ) = ξH . Consider the pairings, see (13), (j) (m+n−j) h·, ·i∞ : 1E∞ × 2E∞ → R, h¯ x, y¯i∞ · ξ∞ = x¯ · y¯,

h·, ·iH : 1H j × 2H m+n−j → R, hx, yiH · ξH = x · y. By (17) we have h¯ x, y¯i∞ = hx, yiH

(18)

680


j,i m−j,n−i where 1σ∞ (¯ x) = [x] and 2σ∞ (¯ y ) = [y]. According to (13) the pairing h·, ·i∞ is 1 (j) ∼ 2 (m+n−j) ∗ nondegenerate, that is E∞ = E∞ . Consider the induced linear mapping ∗ κ : 1H j → 2H m+n−j , x 7→ hx, ·iH .

Similarly to [16] we easily check the monomorphy of κ. It remains to check that κ is an epimorphism. Take a linear function 0 6= l : 2H m+n−j → R and consider the filtration (14) for r = 2 and t = m + n − j. Let V p ⊂ 2H p,m+n−j−p be a subspace complementary to 2 p+1,m+n−j−p−1 H , p = 0, 1, ..., m + n − j and ψ p : V p → E0p,m+n−j−p 2H , x 7→ [x] , the induced isomorphism. Put X M M p,m+n−j−p 2 ψ= ψ p : 2H m+n−j = Vp ∼ E0 H . = p

p

p

L p,m+n−j−p 2 ∗ The composition l ◦ψ −1 ∈ ( H) determines a family of linear functions p E0 ∗ p p,m+n−j−p 2 l0 ∈ E0 ( H) . Define Il = {p; l0p 6= 0} . For each p ∈ Il we define - through isomorphisms 2 p,m+n−j−p σ∞

∼ =

p,m+n−j−p : 2E∞ −→ E0p,m+n−j−p 2H

- a linear nonzero functions p l∞ ∈

2 p,m+n−j−p ∗ E∞

p p,m+n−j−p , l∞ = l0p ◦ 2σ∞ .

m−p,p+j−m p,m+n−j−p The nondegenerate pairing h·, ·i∞ : 1E∞ × 2E∞ → R determines an m−p 1 m−p,p+j−m m−p p p,m+n−j−p ∗ element 0 6= x¯ ∈ E∞ such that h¯ x , ·i∞ = l∞ ∈ (2E∞ ) . Let m−p,p+j−m 1 1 m−p,p+j−m m−p m−p σ∞ (¯ x ) = [x ] ∈ E0 ( H) , where

xm−p ∈ 1H m−p,p+j−m and xm−p ∈ / 1H m−p+1,p+j−m−1. Put x :=

X

xm−p .

p∈Il

We prove the equality κ (x) = hx, ·iH = l ∈ 2H m+n−j L

∗

.

p p p p Since 2H m+n−j = p V , we need only to prove κ (x) (y ) = hx, ·iH (y ) = l (y ) for y p ∈ V p ⊂ 2H p,m+n−j−p. If p ∈ / Il , i.e. l0p = 0, then l (y p) = 0 and for all p′ ∈ Il by (16) and (17) ′ ′ ′ ′ ′ hxm−p , y piH · ξH = xm−p · y p = xm−p · [y p ] ∈ E0m−p+p ,n+p−p 3H = 0.


681

If p ∈ Il and 0 6= y p ∈ V p then p p,m+n−j−p l (y p ) = (l|V p ) ◦ (ψ p )−1 [y] = l0p [y p] = l∞ ◦ 2σ∞ p,m+n−j−p −1 = h¯ xm−p , 2σ∞ ([y p ])i∞

−1

([y p ])

(18)

= hxm−p , y p iH

= hx, y piH . The last equation holds because for p′ 6= p, p′ ∈ Il , we have ′ ′ ′ 0 6= xm−p ∈ E0m−p ,p +j−m 1H and by (16) and (17) ′ ′ ′ ′ xm−p · y p = xm−p · [y p ] ∈ E0m−p ,p +j−m 3H = 0.

5

Hochschild-Serre filtration and the spectral sequence for transitive Lie algebroids

We fix a transitive Lie algebroid A = (A, [[·, ·]], #A ) with the Atiyah sequence 0 → g ֒→ #A

A → T M → 0 and a representation ∇ : A → A (ξ) of a Lie algebroid A on a vector bundle ξ. ∇ is a homomorphism of Lie algebroids, then ∇ induces a homomorphism of vector bundles ∇+ : g → End (ξ) ∇+

g −−−→ End (ξ)     y y ∇

A −−−→

A (ξ)

and ∇+ x : g x → End (ξx ) is a representation of the isotropy Lie algebra g x in the vector space ξx . We will consider the pair of R-Lie algebras (g, k) where g = Γ (A) , k = Γ (gg ) . Below, the elements of g will be denoted by γ, γ1 , γ2 , ... while elements of k by σ, σ1 , σ2 , .... Of course, k is an ideal of g (actually, k is C ∞ (M)-Lie algebra but it is not interesting here). The space Γ (ξ) is a g-modul with respect to the induced representation denoted by the same letter ∇ : Γ (A) → Γ(A (ξ)) ⊂ LΓ(ξ) . Following Hochschild-Serre [9] we can consider a graded cochain group of R-linear alternating functions M AR = Ai , Ai = C i (g, Γ (ξ)) , i≥0

with the R-differential operator of degree 1 d∇ : C i (g, Γ (ξ)) → C i+1 (g, Γ (ξ))

682


defined by the standard formula X X (d∇ f ) (γ0 , ..., γt ) = (−1)i ∇γi (f (γ0 , ...ˆı..., γt )) + (−1)i+j f ([[γi , γj ]], ...ˆı...ˆ ...) . i

i<j

For the trivial representation ∂A : A → A (M × R) , (∂A )γ (f ) = ∂#(γ) (f ), this operator is denoted by dA . Clearly, for a real alternating t-cocycle ϕ and σ ∈ Γ (ξ) we get d∇ (ϕ ⊗ σ) = dA ϕ ⊗ σ + (−1)t ϕ ∧ d∇ σ. L t In the space the Hochschild-Serre filtration Aj ⊂ AR as follows: t≥0 A we haveL Aj = AR for j ≤ 0. If j > 0, Aj = t≥j Atj , Atj = Aj ∩ At , where Atj consists of all those t-cochains f for which f (γ1 , ..., γt ) = 0 whenever t − j + 1 of the arguments γi belongs to k. In this way we have obtained a graded filtered differential R-vector space M AR = At , d∇ , Aj (19) t≥0

and we can use its spectral sequence Esj,i, ds .

(20)

Following K. Mackenzie [17] (see also V.Itskov, M.Karashev, and Y.Vorobjev [11]) we will consider the C ∞ (M)-submodule of C ∞ (M)-linear altarnating cochains with values in the vector bundle ξ (i.e. A-differential ξ-valued forms) Ωt (A, ξ) ⊂ C t (g, Γ (ξ)) and the induced filtration Ωj = Ωj (A, ξ) = Aj ∩ Ω (A, ξ) of C ∞ (M)-modules. The differential d∇ of a C ∞ (M)-cochain is a C ∞ (M)-cochain, so we get dA,∇ : Ω (A, ξ) → Ω (A, ξ) . We obtain in this way a graded filtered differential space M Ω (A, ξ) = Ωt (A, ξ) , dA,∇, Ωj (21) t

and its spectral sequence

j,i EA,s , dA,∇,s .

(22)

Now we consider as well a submodule of C ∞ (M)-linear altarnating cochains with compact support Ωtc (A, ξ) ⊂ Ωt (A, ξ) and the corresponding filtration Ωc,j = Ωj ∩ Ωc (A, ξ) of C ∞ (M)-modules. Since supp d∇ f ⊂ supp f then we obtain dAc ,∇ : Ωc (A, ξ) → Ωc (A, ξ)


and we get a graded filtered differential space with compact support M Ωc (A, ξ) = Ωtc (A, ξ) , dAc ,∇ , Ωc,j

683

(23)

t

and its spectral sequence

EAj,ic ,s , dAc ,∇,s .

(24)

Sometimes we can deduce directly properties of the last two spectral sequences (22), (24) from the suitable properties of (20), see [9], denoted further by AR , sometimes we must use some additional observations. Lemma 5.1. The homomorphisms ρ0 and ρc,0 in the sequence ρc,0

ρ0

j+i j+i j+i j,i j+i EAj,ic ,0 = Ωc,j /Ωc,j+1 ֌ Ωjj+i /Ωj+1 = EA,0 ֌ Ajj+i /Aj+1 = E0j,i

are monomorphisms. For differentials dAc ,∇ , dA,∇ , d0 the following diagram is commutative ρc,0

ρ0

ρc,0

ρ0

֌

֌

j,i EAj,ic ,0 −−−→ EA,0 −−−→ ֌ ֌   dj,i dj,i y Ac ,∇,0 y A,∇,0

E0j,i   j,i yd0

j,i+1 EAj,i+1 −−−→ EA,0 −−−→ E0j,i+1. c ,0

From • AR For R-cochains there exists an isomorphism aj,i : E0j,i → C j g/k, C i (k, Γ (ξ))

such that aj,i [f ] ([γ1 ] , ..., [γj ]) (σ1 , ..., σi ) = f (σ1 , ..., σi , γ1, ..., γj ) , .

(25)

we can easily obtain the following Conclusion 5.2. The homomorphisms j,i j i ∗ aj,i A : EA,0 → Ω M, Λ g ⊗ ξ j,i j i ∗ aj,i : E → Ω M, Λ g ⊗ ξ c Ac Ac ,0 defined by the formula aj,i A [f ] (X1 , ..., Xj ) (σ1 , ..., σi ) = f (σ1 , ..., σi , λX1 , ..., λXj ) , Xj ′ ∈ X (M) , σi′ ∈ k, (aj,i Ac defined by the identical formula) where λ : T M → A is an arbitrary connection, are correctly defined linear isomorphisms of C ∞ (M)-modules. j,i Proof. Monomorphy of aj,i A and aAc follows from the commutativity of the diagram

684


EAj,ic ,0  aj,i y Ac

ρc,0

−−−→ ֌

ρ¯c,0

j,i EA,0   j,i yaA

ρ0

−−−→ ֌

ρ¯

E0j,i   j,i ya

0 Ωjc (M, Λig ∗ ⊗ ξ) −−−→ Ωj (M, Λig ∗ ⊗ ξ) −−−→ C j (g/k, C i (k, Γ (ξ))) .

֒→

֒→

To prove that aj,i it is sufficient to check that if aj,i [f ] is a C ∞ (M)A is an epimorphism linear cochain, i.e. aj,i [f ] = ρ¯0 f¯ for some f¯ ∈ Ωj (M, Λig ∗ ⊗ ξ), i.e. f¯ (#A (γ1 ) , ..., #A (γj )) (σ1 , ..., σi ) = f (σ1 , ..., σi , γ1 , ..., γj ) , j,i then there exists a representative f ′ ∈ [f ] ∈ EA,0 which is C ∞ (M)-linear cochain such ′ ¯ that aj,i ω0 : A → g coresponding to λ and A [f ] = f . To thisend take a connection form ′ ′ ′ ′ ′ put f γ1 , ..., γj , γ1 , ..., γi = f ω0 (γ1 ) , ..., ω0 γj , γ1, ..., γi . Then f ′ fulfils the desired conditions. • AR Through isomorphism aj,i the differential dj,i 0 becomes a differentiation of values with respect to the differential d∇◦ι : C i (k, Γ (ξ)) → C i+1 (k, Γ (ξ))

(26)

( ι : k ֒→ g, is the inclusion), d˜∇◦ι : C j g/k, C i (k, Γ (ξ)) → C j g/k, C i+1 (k, Γ (ξ)) , d˜∇◦ι (f ) ([γ1 ] , ..., [γj ]) = d∇◦ι (f ([γ1 ] , ..., [γj ])) . j,i In conclusion, the differentials dj,i A,∇,0 and dAc ,∇,0 becomes (through the isomorphisms j,i aj,i A and aAc ) differentials of values with respect to

d∇+ : Λig ∗ ⊗ ξ → Λi+1g ∗ ⊗ ξ, namely d˜∇+ : Ωj M, Λig ∗ ⊗ ξ → Ωj M, Λi+1g ∗ ⊗ ξ , d˜∇+ (f ) (X1 , ..., Xj ) = d∇+ (f (X1 , ..., Xj )) . Analogously we obtain a differential d˜c,∇+ for compact supports. Remark 5.3. According to K.Mackenzie [17, Th.2.5, p.201] the homomorphisms di∇+ : Λig ∗ ⊗ ξ → Λi+1g ∗ ⊗ ξ are locally of constant rank, and consequently, there are welli g i i defined vector bundles Z i = ker di∇+ , B i = Im di−1 ∇+ and H (g , ξ) = Z /B such that Γ (H i (gg , ξ)) = H i (Γ (Λgg ∗ ⊗ ξ) , d∇+ ) . Clearly, H i (gg , ξ)x = H i Λgg ∗x ⊗ ξx , d∇+x . Therefore i g , ξ) , H Ωj M, Λig ∗ ⊗ ξ , d˜∇+ ∼ = Ωj M, H∇ + (g i g , ξ) . H Ωjc M, Λig ∗ ⊗ ξ , d˜c,∇+ ∼ = Ωjc M, H∇ + (g


685

From the above we obtain isomorphisms of C ∞ (M)-modules ∼ = j,∗ j,∗ i j i ∗ aj,i A # : H EA,0 , dA,∇,0 → Ω M, Λ g ⊗ ξ , ∼ = j,∗ j,∗ i j i ∗ aj,i : H E , d M, Λ g ⊗ ξ . → Ω c Ac # Ac ,0 Ac ,∇,0 Now we pass to consideration of the modules Zs , Ds , Es and ZA,s , DA,s , EA,s and ZAc ,s , DAc ,s , EAc ,s for three spectral sequences for graded, filtered, differential spaces (19), (21), (23), respectively. Immediately by definitions we get j j j Lemma 5.4. (1) ZA,s = Zsj ∩ Ω (A, ξ) , (2) DA,s = Dsj ∩ Ω (A, ξ) , (3) ZAj c ,s = ZA,s ∩ Ωc (A, ξ) . j,i Fix an auxiliary connection λ : T M → A and for f ∈ ZA,1 ⊂ Ωj+i (A, ξ) we define f¯j ∈ Ωj M, Λig ∗ ⊗ ξ

by the formula f¯j (X1 , ..., Xj ) (σ1 , ..., σi ) = f (λX1 , ..., λXj , σ1 , ..., σi ) = (−1)ji f (σ1 , ..., σi , λX1 , ..., λXj ) . j,i Lemma 5.5. If f ∈ ZA,1 then f¯j (X1 , ..., Xj ) ∈ Γ (Λig ∗ ⊗ ξ) is a d∇+ -cocycle independent of the choice of λ. j,i Proof. For f ∈ ZA,1 ⊂ Ωjj+i (A, ξ) ⊂ Asj+i we take a cochain fj ∈ C j (g, C i (g, Γ (ξ))) defined by fj (γ1 , , , , γj ) (γ1′ , ..., γi′ ) = f (γ1 , , , , γj , γ1′ , ..., γi′ ) , see [9]. From the equalities j,i j+i+1 ZA,1 = f ∈ Ωjj+i ; d∇ f ∈ Ωj+1 = Z1j,i ∩ Ωj+i (A, ξ)

we get (see [H-S]) that ι∗j (fj (γ1 , , , , γj )) ∈ C j (k, Γ (ξ)), where ι∗j : C j (g, Γ (ξ)) → C j (k, Γ (ξ)) , ι∗j (g) = g|k × ... × k, is a ∇ ◦ ι : k → LΓ(ξ) -cocycle and that this cocycle depends only on the equivalence class [γj ′ ] ∈ g/k ∼ = X (M) , i.e. on the anchors of the elements γj ′ , i.e. on #A (γj ′ ) . But ∗ ιj (fj (γ1 , , , , γj )) is C ∞ (M)-linear ι∗j (fj (γ1 , , , , γj )) ∈ Γ (Λig ∗ ⊗ ξ) therefore the condition d∇◦ι ι∗j (fj (γ1 , , , , γj )) = 0 is equivalent to d∇+ ι∗j (fj (γ1 , , , , γj )) = 0. The equality ι∗j (fj (γ1 , , , , γj )) = f¯j (#A (γ1 ) , ..., #A (γj )) proves the lemma. We recall that j,i j,i j+1,i−1 j,i EA,1 = ZA,1 / ZA,0 + DA,0 j+i+1 j+i = f ∈ Ωjj+i (A, ξ) ; d∇ f ∈ Ωj+1 / Ωj+1 + d∇ Ωj+i−1 j

686


and analogously for EAj,ic ,1 . Lemma 5.6. The homomorphisms j,i i g , ξ) , [f ] 7→ (−1)ji f¯j , ΨA,1 : EA,1 → Ωj M, H∇ + (g i g , ξ) , [f ] 7→ (−1)ji f¯j , ΨAc ,1 : EAj,ic ,1 → Ωjc M, H∇ + (g are isomorphisms of C ∞ (M)-modules. Proof. Clearly, we need to notice only that ΨA,1 is a composition of isomorphisms ΨA,1 :

j,i j,i σA,0 EA,1 −→ ∼ =

H

i

j,∗ EA,0 , dj,∗ A,0

(aj,i A )# j i g −→ Ω M, H (g , ξ) + ∇ ∼ =

and analogously for compact supports. j,i From the above lemmas we see that the canonical homomorphism EAj,ic ,1 → EA,1 is a monomorphism. • AR There exists a representation [precisely, a Lie derivation]

Li : g → LC i (k,Γ(ξ)) defined by the formula X Liγ f (σ1 , ..., σi ) = ∇γ (f (σ1 , ..., σi )) − f (σ1 , ..., [[γ, σt ]], ..., σi ) . t

Liγ commutes with R-differential operator d∇◦ι , see (26), induces a representation in cohomology i L#,i : g → LH∇◦ι (k,Γ(ξ)) and k ⊂ ker L#,i (because Liσ f = d∇◦ι (ισ f ) if f is a d∇◦ι -cocycle.). It produces a representation #,i i L : g/k → LH∇◦ι (k,Γ(ξ)) . Noticing that a Lie derivation of a C ∞ (M)-linear cochain is C ∞ (M)-linear too, we can pass to Γ (Λig ∗ ⊗ ξ) . Additionally we observe that Liγ : Γ (Λig ∗ ⊗ ξ) → Γ (Λig ∗ ⊗ ξ) is a covariant derivative operator with the anchor #A (γ) and Liγ is C ∞ (M)-linear with respect to γ. In conclusion we obtain a representation of the Lie algebroid A in the vector bundle Λig ∗ ⊗ ξ LiA : A → A Λig ∗ ⊗ ξ . Lemma 5.7. The representation LiA coincides with the adjoint representaion of A in Λig ∗ cross ∇, LiA = adA ⊗ ∇. Proof. The adjoint representation adA : A → A (gg ) , adA (γ) (σ) = [[γ, σ]], induces the one in the associated bundle Λig ∗ (denoted also adA ) and its tensor product with ∇ is just equal to LiA .


687

The representation LiA induces the one in cohomology i g , ξ) L#,i A : A → A H∇+ (g

i such that g ⊂ ker L#,i A (indeed, (LA )σ (f ) = d∇+ (ισ f ) for a d∇+ -cocycle f ). Therefore, we obtain a flat covariant derivative i g , ξ) ∇i : T M → A H ∇ (27) + (g

by the formula ∇iX ([f ]) = L#,i A

([f ]) = λX

LiA

(f ) λX

where for a d∇+ -cocycle f ∈ Λig ∗ ⊗ ξ X LiA λX (f ) (σ1 , ..., σi ) = ∇λX (f (σ1 , ..., σi )) − f (σ1 , ..., [[λX, σt ]], ..., σi ) . t

Remark 5.8. For the trivial representation ∇ = ∂A we get a flat structure in the cohomology bundle H i (gg ). If the structure Lie algebras g x are unimodular then H n (gg ) = Λng ∗ and the induced flat covariant derivative ∂An : T M → A (Λng ∗ ) is defined by X ((∂An )X f ) (σ1 , ..., σn ) = X (f (σ1 , ..., σn )) − f (σ1 , ...[[λX, σi ]], . . . , σn ) . i

This flat structure coincides with the flat structure in Λng ∗ defined in the paper [16] via some system A = {ϕñU } of local trivializations with locally-constant transitive functions. We recall that ϕñU : U × Λn g∗ → Λng ∗ (g is the typical fiber of g ) is determined by a local trivialization ϕU : AU → T U × g of the Lie algebroid A in the following way: ϕU induces a local trivialization ϕ+ U : g U → U × g of the adjoint Lie Algebra Bundle g and we put + ∗ n n (ϕ˜U )x = Λ ϕU x . j,i j+1,i j,i j,i j+1,i Now, we carry over the differentials dj,i A,1 : EA,1 → EA,1 , dAc ,1 : EAc ,1 → EAc ,1 , to i i g , ξ) and Ωjc M, H∇ g , ξ) , respectively, via the isomorphisms the spaces Ωj M, H∇ + (g + (g ΨA,1 and ΨAc ,1 . Since the canonical homomorphism i i g , ξ) → H∇◦ι Γ H∇ (k, Γ (ξ)) + (g

is not a monomorphism unless the Lie algebra bundle g is trivial, we can not infer the form of this differentials immediately from the level of R-cochains and its spectral sequence i (20). In comparising of the cohomology classes from H∇◦ι (k, Γ (ξ)) having representative ∞ of C (M)-linear cochains we must see whether these representatives differ by a C ∞ (M)linear cochain. Proposition 5.9. The following diagrams are commutative j,i EA,1

 Ψj,i y A,1

dj,i A,1

−−−→

j+1,i EA,1  Ψj+1,i y A,1

(−1)i d∇i i i g g , ξ) (g , ξ) Ωj M, H∇ −−−−−→ Ωj+1 M, H∇ + + (g

688


EAj,ic ,1  Ψj,i y Ac ,1

dj,i A ,1

c −−− →

EAj+1,i c ,1 Ψj+1,i y Ac ,1

(−1)i d i i i g , ξ) −−−−−∇→ Ωj+1 g Ωjc M, H∇ M, H (g , ξ) + (g + c ∇ j,i Proof. The calculations identical as in the R-linear cochains [9] yield for f ∈ ZA,1 ⊂ Z1j,i the following formulae (j+1)i (−1)i d∇i ◦ Ψj,i [ρf (X1 , ..., Xj+1)] , A,1 [f ] (X1 , ..., Xj+1 ) = (−1)

j+1,i ΨA,1 ◦ dj,i A,1 [f ] (X1 , ..., Xj+1 ) = (−1)(j+1)i ρf (X1 , ..., Xj+1) − d∇+ (−1)j f¯j+1 (X1 , ..., Xj+1 ) where ρf ∈ Ωj+1 (M, Λig ∗ ⊗ ξ) and ρf (X1 , ..., Xj+1) is a d∇+ -cocycle defined by ρf (X1 , ..., Xj+1) (σ1 , ..., σi ) =

j+1 X

+

j+1 X

t=1

+

t=1 X

(−1)t+1 ∇λXt f¯j X1 , ...tˆ..., Xj+1 (σ1 , ..., σi ) + (−1)

t

i X

f¯j X1 , ...tˆ..., Xj+1 (σ1 , ..., [[λXt , σs ]], . . . , σi ) +

s=1 r+s ¯ (−1) fj

([Xr , Xs ] , X1 , ...ˆ r ...ˆ s...Xj+1 ) (σ1 , ..., σi ) .

r<s

The cochain f¯j+1 (X1 , ..., Xj+1 ) is C ∞ (M)-linear, i.e. belongs to the module Ωj+1 (M, Λi−1g ∗ ⊗ ξ) . This gives j+1,i j,i (−1)i d∇i ◦ Ψj,i A,1 − ΨA,1 ◦ dA,1 (f ) (X1 , ..., Xj+1 ) = (−1)(j+1)i d∇+ (−1)j f¯j+1 (X1 , ..., Xj+1) = 0.

If f has a compact support, the same hold for ρf and f¯j+1 and we get the commutativity of the second diagram. The next theorem is the main goal of this section. It describes the second terms of the sepectral sequences (22) and (24) (see also [17]). Theorem 5.10. The homomorphisms j,i j i g , ξ) , [f ] 7→ (−1)ji f¯j , ΨA,2 : EA,2 → H∇ i M, H∇+ (g j i g , ξ) , [f ] 7→ (−1)ji f¯j , ΨAc ,2 : EAj,ic ,2 → H∇ i ,c M, H∇+ (g are isomorphisms of C ∞ (M)-modules.


689

Proof. Clearly, we need to notice only that ΨA,2 is a composition of isomorphisms ΨA,2 :

j,i j,i σA,1 EA,2 −→ ∼ =

H

j

∗,i EA,1 , d∗,i A,1

(Ψj,i A,1 )# j i g −→ H M, H , ξ) + (g i ∇ ∇ ∼ =

and analogously for compact supports.

6

Algebroids and pairings

Assume that A is a transitive Lie algebroid with three representations ∇r : A → A (ξr ) , r = 1, 2, 3, and a pairing F : ξ1 × ξ2 → ξ3 compatible with the representations (∇1 , ∇2 , ∇3 ) , i.e. fulfilling the property analogous to (5) in which we must replace X by γ ∈ Γ (A) . Then the multiplication of cochains ∧ : Λj g ∗ ⊗ ξ1 × Λig ∗ ⊗ ξ2 → Λj+ig ∗ ⊗ ξ3 is compatible with j+i (a) suitable representations LjA , LiA , LA j+i j+i j+i LA,γ (f ∧ g) = LA,γ (f ) ∧ g + f ∧ LA,γ (g) ,

f ∈ Γ (Λj g ∗ ⊗ ξ1 ) , g ∈ Γ (Λig ∗ ⊗ ξ2 ) , (b) differentials d∇+1 , d∇+2 , d∇+3 d∇+3 (f ∧ g) = d∇+1 (f ) ∧ g + (−1)j f ∧ d∇+2 (g) , f, g as above. The latter equality gives the pairing of cohomology vector bundles j j+i i g , ξ 1 ) × H∇ g , ξ 2 ) → H∇ g , ξ3 ) ∧ : H∇ + (g + (g + (g 2

1

(28)

3

#,i #,j+i which is compatible with the suitable representations L#,j and finally with A , LA , LA j i j+i the flat covariant derivatives ∇ , ∇ , ∇ j+i ∇X ([f ] ∧ [g]) = ∇jX ([f ]) ∧ [g] + [f ] ∧ ∇iX [g] .

We assume in the sequel that n = rank g (and we recall that m = dim M). Together with three representations ∇r one consider three graded filtered differential j,i 1 spaces Ω (A, ξ1 ) , Ωc (A, ξ2 ) , Ωc (A, ξ3 ) (21), (23), and theirs spectral sequences 1EA,s , dA,∇,s , 2 j,i 2 EAc ,s , dAc ,∇,s , 3EAj,ic ,s ,3 dAc ,∇,s . Using monomorphy of ρ0 and of ρc,0 , Lemma 5.1, we see immediately from the case of R-linear cochains [9] that the following diagram commutes. 1 j,i EA,0

′ ′

× 2EAj c,i,0   j,i j′ ,i′ yaA ×aAc

∧

−−−→

′ ′ 3 j+j ,i+i EAc ,0

  j,i j′ ,i′ yaA ×aAc

(−1)i′ j ·∧ ′ ′ ′ ′ Λi+i g ∗ ⊗ ξ3 . Ωj M, Λig ∗ ⊗ ξ1 × Ωjc Λi g ∗ ⊗ ξ2 −−−−−→ Ωj+j c

690


Passing twice to cohomology and using definitions of suitable homomorphisms we get the commutativity of the diagram 1 j,i EA,2

′ ′

′ ′ 3 j+j ,i+i EAc ,2

∧

× 2EAj c,i,2  1 j,i 2 j′ ,i′ y ΨA,2 × ΨAc ,2

−−−→

 3 j+j′ ,i+i′ y ΨAc ,2

(29)

(−1) ·∧ j j′ j+j ′ i+i′ i i′ g , ξ 1 ) × H∇ g g H∇ −−−−−→ H∇ i M, H + (g i′ ,c M, H∇+ (g , ξ2 ) i+i′ ,c M, H∇+ (g , ξ3 ) . ∇ 1

i′ j

2

3

The main theorem of Chapter 1 one gets the very important n g , ξ3 ) , ∇n ∼ (or (M) , ∂ or ) [in partiConclusion 6.1. If ξ3 is a line bundle and H∇ + (g 3 cular, ∇+ = ∇ ⊗ id according to Lemma 3.5] and the pairing of cohomology vector trad 3 x bundles i n−i n g , ξ 1 ) × H∇ g , ξ 2 ) → H∇ g , ξ3 ) ∧ : H∇ + (g + (g + (g 1

3

2

is nondegenerate, then the same holds for the pairing j H∇ i

i.e.

i M, H∇ + 1

(gg , ξ1 ) ×

m−j H∇ n−i ,c

n−i M, H∇ + 2

(gg , ξ2) →

m H∇ n ,c

n M, H∇ + 3

RM# (gg , ξ3 ) → R. (30)

∗ j m−j i n−i g , ξ2 ) . g , ξ1 ) ∼ H∇ = H∇n−i ,c M, H∇ + (g i M, H∇+ (g 1

2

Diagram (29) assert that the nondegenerate pairing (30) is ±equal to the multiplication of the second term of the spectral sequences 1 j,i EA,2

∧

∼ =

× 2EAm−j,n−i −→ 3EAm,n →R c ,2 c ,2

so the last is nondegenerate as well, 1 j,i EA,2

m−j,n−i ∗ ∼ , = 2EAc ,2

and the main theorem of Chapter 4 gives that the multiplication of cohomology classes ∼ =

j m+n−j m+n (A, ξ3 ) → R h·, ·iH : H∇ (A, ξ1 ) × H∇ (A, ξ2 ) → H∇ 3 ,c 1 2 ,c

is nondegenerate too, i.e. j m+n−j H∇ (A, ξ1 ) ∼ = H∇2 ,c (A, ξ2 ) 1

7

∗

.

Evens-Lu-Weinstein pairing for transitive Lie algebroids

7.1 Nondegeneracy of Evens-Lu-Weinstein pairing for transitive Lie algebroids We prove that for transitive Lie algebroid A the duality of Evens-Lu-Weinstein [4] m+n−j m+n or H j (A) × HD (A, Qor or ,c A ) → HD or ,c (A, QA ) → R


691

∼ =

m+n or is nondegenerate, i.e. HD or ,c (A, QA ) → R and m+n−j H j (A) ∼ = HDor ,c (A, Qor A)

∗

.

For arbitrary (nonregular in general) Lie algebroid A on a manifold M the authors [4] introduced a vector bundle QA = Λtop A ⊗ Λtop T ∗ M (the notation Λtop refers to the highest exterior power). Geometrically, sections of QA can be thought of as transverse measures to characteristic foliation Im #A to any Lie algebroid A [4]. For Poisson manifolds, the Evens-Lu-Weinstein pairings takes the form of the pairing on the Poisson homology; for more applications see [4]. Ibidem, there is an example of nonregular Lie algebroid A over a compact oriented manifold for which the m+n−j pairing H j (A) × HD,c (A, QA ) → R is not necessarily nondegenerate. J.Huebschmann in [10] has generalized the construction of the bundle QA and the modular class θA to Lie-Rinehart algebras, an algebraic generalization of Lie algebroids. We slightly modify the Weinstein construction to consider nonoriented manifolds: Qor A = QA ⊗ or (M) . For an oriented manifold M we can identify Qor A = QA . In [4] a representation D : A → A (QA ) was introduced by Dγ (Y ⊗ ϕ) = Lγ (Y ) ⊗ ϕ + Y ⊗ L#A (γ) (ϕ) , Y ∈ Γ (Λtop A) , ϕ ∈ Γ (Λtop T ∗ M) = Ωm (M) , where Lγ (Y ) = [γ, Y ] ([γ, Y ] denotes the Schouten bracket) and L#A (γ) (ϕ) is the usual Lie derivative of a differential form ϕ. We recall that for Y = γ1 ∧ .... ∧ γt X Lγ (Y ) = γ1 ∧ ... ∧ [[γ, γi ]] ∧ ... ∧ γt . i

There is some interest to consider the representation D in the context of intrinsic characteristic classes of Lie algebroids [3], [5]. We modify the representation D to D or = D ⊗ ∂Aor : A → A (Qor A). In the sequel we will be interested only in the transitive case. A choice of a connection λ : T M → A enables us to identify Λm+n A = Λng ⊗ Λm T M, ε ∧ (Λm λ)(X) = ε ⊗ X,

692


and n m m ∗ n Qor A = Λ g ⊗ Λ T M ⊗ Λ T M ⊗ or (M) = Λ g ⊗ or (M) .

(31)

Lemma 7.1. (a) D + : g → End (QA ) is defined by Dσ+ = (∇trad )σ = tr (adσ ) · id,

σ ∈ Γ (gg ) .

n n n ∗ or g , QA ) = Λng ∗ ⊗ QA , HD g , Qor (b) HD + (g or+ (g A ) = Λ g ⊗ QA .

Proof. (a) Consider locally defined nonsingular section of QA of the form εU ⊗ XU ⊗ ϕU , εU ∈ Γ (Λng U ) , XU ∈ Γ (Λm T MU ) , ϕU ∈ Γ (Λm T ∗ MU ) , and assume that hXU , ϕU i = 1. For σ ∈ Γ (gg ) we have #A (σ) = 0 and [[σ, λWi ]] ∈ Γ (gg ) . Therefore if εU = σ1 ∧ ... ∧ σn , XU = W1 ∧ ... ∧ Wm , σi ∈ Γ (gg ) , Wi ∈ Γ (T MU ) , Dσ (εU ⊗ XU ⊗ ϕU ) = Lσ (εU ⊗ XU ) ⊗ ϕU = Lσ (σ1 ∧ ... ∧ σn ∧ λW1 ∧ ... ∧ λWm ) ⊗ ϕU X = σ1 ∧ ... ∧ [[σ, σi ]] ∧ ... ∧ σn ∧ λW1 ∧ ... ∧ λWm ⊗ ϕU i

= tr (adσ ) · εU ⊗ XU ⊗ ϕU . (b) Follows immediately from Proposition 3.2. The vector bundle Λng ∗ ⊗ QA is trivial. Indeed, the classical homomorphism c : Λng ∗ ⊗ QA = Λng ∗ ⊗ Λng ⊗ Λm T M ⊗ Λm T ∗ M → M × R

(32)

c (ε∗ ⊗ ε ⊗ X ⊗ ϕ) = hε∗ , εi · hX, ϕi is an isomorphism. Therefore ∼ =

c ⊗ id : Λng ∗ ⊗ Qor A −→ or (M) .

(33)

Let A (c) : A (Λng ∗ ⊗ QA ) → A (M × R) be the induced isomorphism of Lie algebroids [12], A (c) (u) (f ) = c u c−1 ◦ f , u ∈ Γ (A (Λng ∗ ⊗ QA )) , f ∈ C ∞ (M) . Let

n g , QA )) = A (Λng ∗ ⊗ QA ) ∇D : T M → A (HD + (g

be the induced flat adjoint covariant derivative (27) for D. Analogously we have ∇D T M → A (Λng ∗ ⊗ Qor A). Lemma 7.2. The compositions A(c)

∇D : T M → A (Λng ∗ ⊗ QA ) −→ A (T M × R) , ∼ ∇

D or

= A(c⊗id)

: T M → A (Λng ∗ ⊗ Qor A (or (M)) , A ) −→ ∼ =

or

:


693

are trivial representations ∂ and ∂ or , respectively, so n g , QA ) , ∇D ∼ (T M × R, ∂) , HD + (g n D or g , Qor HD ) , ∇ ∼ (or (M) , ∂ or ) . or+ (g A Proof. It is necessary to check it locally. Take locally defined nonsingular sections εU ∈ Γ (Λng U ) , XU ∈ Γ (Λm T MU ) and theirs duals ε∗U ∈ Γ (Λng ∗U ) , ϕU ∈ Γ (Λm T ∗ MU ) , hε∗U , εU i = 1, hXU , ϕU i = 1. On the set U arbitrary section of the bundle Λng ∗ ⊗ QA = Λng ∗ ⊗ Λng ⊗ Λm T M ⊗ Λm T ∗ M is of the form f · ε∗U ⊗ εU ⊗ XU ⊗ ϕU , f ∈ C ∞ (U) . For X ∈ X (MU ), XU = W1 ∧ ... ∧ Wm (W1 , ..., Wm is a base of vector fields on U) and ϕU = W1∗ ∧ ... ∧ Wm∗ , Wi∗ is the dual basis, and εU = σ1 ∧ ... ∧ σn (σi is a base of the P P vector bundle g on U), we write [[λX, σi ]] = j gij · σj , [[λX, λWi ]] = k hki · σk + aki · λk , so [X, Wi ] = aki · Wk . Then DλX (εU ⊗ XU ⊗ ϕU ) = LλX (εU ⊗ XU ) ⊗ ϕU + εU ⊗ XU ⊗ LX ϕU X X X = gii · σ ∧ λX ⊗ ϕU + σ ∧ aii · λX ⊗ ϕU + σ ∧ λX ⊗ −aii ϕU =

Xi

i

gii

· ε U ⊗ XU ⊗ ϕ U .

i

Therefore ∗ ∇D X (f · εU ⊗ εU ⊗ XU ⊗ ϕU ) (εU )

= DλX (f · εU ⊗ XU ⊗ ϕU ) − X − f · ε∗U (σ1 ∧ ... ∧ [[λX, σi ]] ∧ ... ∧ σn ) · εU ⊗ XU ⊗ ϕU i

= ∂X f · εU ⊗ XU ⊗ ϕU + f · DλX (εU ⊗ XU ⊗ ϕU ) − f ·

X

gii · εU ⊗ XU ⊗ ϕU

i

= ∂X f · εU ⊗ XU ⊗ ϕU = ∂X f · (ε∗U ⊗ εU ⊗ XU ⊗ ϕU ) (εU ) . Finally D ∗ A (c) ◦ ∇D X (f ) = c ∇X (f · εU ⊗ εU ⊗ XU ⊗ ϕU ) = c (∂X f · (ε∗U ⊗ εU ⊗ XU ⊗ ϕU )) = ∂X f. For the proof of the second part we notice that for local ∂ or -constant section σ0 of or (MU ) one has or D A (c ⊗ id) ◦ ∇D (f ⊗ σ ) = A (c) ◦ ∇ 0 X X (f ) ⊗ σ0 = ∂X f ⊗ σ0

or = ∂X (f ⊗ σ0 ) .

694


Theorem 7.3. For an arbitrary transitive Lie algebroid A m+n or ∼ HD or ,c (A, QA ) = R,

and the Evens-Lu-Weinstein cohomology pairing m+n−j m+n or ∼ H j (A) × HD (A, Qor or ,c A ) → HD or ,c (A, QA ) = R

is nondegenerate, i.e. m+n−j H j (A) ∼ = HDor ,c (A, Qor A)

∗

.

Proof. Theorem 3.4 and Lemma 7.1 show that the pairing n−i n g , Qor g , Qor H i (gg ) × HD or+ (g A ) → HD or+ (g A)

is nondegenerate. On account of Theorem 2.3 and Conclusion 6.1 we assert that the pairing j m−j i n−i m n g ) × H∇ g , Qor g , Qor H∇ R, or,c M, H or+ (g D M, H (g A ) → H∇D or,c (M, HD or+ (g A )) → D ∼ =

is nondegenerate. Equivalently, this is a multiplication of the second terms of the HochshildSerre spectral sequences of graded filtered differential spaces Ω (A) with the trivial diffeor rential and Ωc (A, Qor A ) with the differential D . The fundamental Theorem 4.4, see also mentioned above Conclusion 6.1, completes the proof.

7.2 Remarks on the top group of cohomology Analyzing the proof of Theorem 4.4 and composing isomorphism (33) with isomorphism ∼ = m+n or (4) we can define the isomorphism I : HD or ,c (A, QA ) −→ R as a composition m,n σA ,∞

c m,n m,n m+n or or I : HD (HDor ,c (A, Qor EAm,n = or ,c (A, QA ) = HD or ,c (A, QA ) = E0 A )) ←− c ,∞ ∼

=

=

Ψm,n Ac ,2 EAm,n −→ c ,2 ∼ =

=

n ∗ m H∇ D or ,c (M, Λ g

m n g , Qor H∇ D or ,c (M, HD or+ (g A )) =

⊗

Qor A)

(c⊗id)#

−→ ∼ =

H∂mor ,c

R or,#

M R. (M, or (M)) −→ ∼

=

We compare this isomorphism with the one defined by direct formula in [E-L-W] resctricting our interest to transitive Lie algebroids. Immediately from the definition of mn ¯ m,n Ψm,n (see Theorem 5.10), Ψ [f ] = (−1) fm , and definition of σAm,n we observe Ac ,2 Ac ,2 c ,∞ that I1 :

m+n HD or ,c

(A, Qor A)

=

m,n m,n σAc ,∞ E0 ←− ∼ =

EAm,n c ,∞

=

m,n m,n ΨAc ,2 EAc ,2 −→ ∼ =

m n ∗ or H∇ D or ,c (M, Λ g ⊗ QA )

mn ¯ is given by the formula looking analogously to Ψm,n fm , or equivalently Ac ,2 , I1 [f ] = (−1) m+n ∗ m ∗ n ∗ (under the identification Λ A = Λ T M ⊗ Λ g given by the help of a connection λ : T M → A) by I1 ((ϕ ⊗ ε∗ ) ⊗ q) = (−1)mn [ϕ ⊗ (ε∗ ⊗ q)]


695

∗ n ∗ or where ϕ ∈ Ωm c (M) , ε ∈ Γ (Λ g ) , q ∈ Γ (QA ) . Therefore if q = ε ⊗ X ⊗ µ ⊗ e, ε ∈ Γ (Λng ) , X ∈ Γ (Λm T M) , µ ∈ Γ (Λm T ∗ M) = Ωm (M) , e ∈ Γ (or (M)) then

(c ⊗ id)# ◦ I1 ((ϕ ⊗ ε∗ ) ⊗ ε ⊗ X ⊗ µ ⊗ e) = (−1)mn [ϕ · hε∗ , εi · hX, µi ⊗ e] . So, for f = (ϕ ⊗ ε∗ ) ⊗ ε ⊗ X ⊗ µ ⊗ e we get Z or Z or mn mn ∗ I [f ] = (−1) hε , εi · hX, µi · ϕ ⊗ e = (−1) hε∗, εi · hX, ϕi · µ ⊗ e M M Z or mn = (−1) hϕ ⊗ ε∗ , X ⊗ εi · µ ⊗ e M

which is concordant up to the sign with the definition of Evens-Lu-Weinstein [4] given by them only for oriented compact manifold (but for any Lie algebroid, not necessary transitive). ∼ = m+n or The fact HD or ,c (A, QA ) −→ R for transitive Lie algebroids is not proved in [4]. Below we prove this immediately without use of the spectral sequences. (a) on oriented manifolds. The authors of [4] introduced an isomorphism of vector bundles ρ˜ : Λtop A∗ ⊗ Λtop A ⊗ Λtop T ∗ M → Λtop T ∗ M, ρ˜ (Ψ ⊗ Y ⊗ µ) = hΨ, Y i · µ and proved a version of Stokes Theorem (to be sure for compact manifold but without troubles we can extend it to differential forms with compact support on arbitrary oriented manifold). Theorem 7.4 (Stokes Theorem [4]). Let rank A = r. For r − 1-form Ψ′ ∈ Γ (Λr−1A∗ ) we have ρ˜ (dD (Ψ′ ⊗ Y ⊗ µ)) = (−1)r−1 ddR ι#A (Ψ′ yY ) µ . Consequently, if the form Ψ′ ⊗ Y ⊗ µ has compact support then Z ρ˜ (dD (Ψ′ ⊗ Y ⊗ µ)) = 0. M

Put ρ˜r−1 : Λr−1 A∗ ⊗ QA → Λm−1 T ∗ M, (Ψ′ ⊗ Y ⊗ µ) 7→ (−1)r−1 ι#A (Ψ′ yY ) µ, and notice the commutativity of the diagram ρ˜r−1 c

Ωcr−1 (A, QA ) −−−→ Ωcm−1 (M)   d d yD y dR ρ˜c

Ωrc (A, QA ) −−−→ Ωm c (M) .

From this we deduce that ρ˜c induces an R-linear homomorphism in cohomology r ρ˜c,# : HD,c (A, QA ) → Hcm (M) .

696


Since ρ˜ is an isomorphism ρ˜c,# is an epimorphism. Lemma 7.5. If A is transitive Lie algebroid, then ρ˜c,# is an isomorphism. Proof. One can easily see the lemma provided that ρ˜r−1 is an epimorphism. It is a simple matter to show that ρ˜xr−1 is an epimorphism at every point x ∈ M using transitivity of the Lie algebroid A i.e. using the fact that the anchor (#A )x : Ax → Tx M is an epimorphism. This finishes the proof that ρ˜c,# :

r HD,c

(A, QA ) →

Hcm

R#

M (M) −→ R ∼

=

is an isomorphism. (b) on nonoriented manifolds. We prove this analogously multiplying the vector bundles by or (M) and use the Stokes theorem for densities [1].

7.3 Exceptional property of the Evens-Lu-Weinstein representation Assume A is a transitive Lie algebroid. Before the next theorems we must give algebroid’s equivalent of some lemmas from Chapter 1. For any A-connection ∇ : A → A (ξ) and a 1-form ω ∈ Ω1 (A) we define a new A-connection ∇ω : A → A (ξ) , ∇ωγ ν = ∇γ ν + ω (γ) · ν. ω

The curvature tensors R∇ , R∇ ∈ Ω2 (A, ξ) of the connections ∇ω and ∇ are related via the formula ω R∇ = R∇ + dA ω ⊗ id. Therefore, if ∇ is flat (it means, ∇ is a representation) then ∇ω is flat if and only if ω is closed. Each A-connection ∇ : A → A (M × R) in the trivial vector bundle M × R is of the form ∂Aω , indeed, we need to put ω (γ) = ∇γ (1) . For a line bundle ξ and a representation ∇ : A → A (ξ) the differential equation ∇ν = 0 is locally uniquelly integrable provided that it is locally integrable. Lemma 7.6. For a line bundle ξ and a representation ∇ : A → A (ξ) the differential equation ∇ν = 0 is locally integrable if and only if ∇+ = 0. This last condition is ˜ ◦ #A for some usual flat covariant equivalent to the projectability of ∇, i.e. that ∇ = ∇ ˜ on M in the vector bundle ξ. derivative ∇ Proof. ”=⇒” Assume that ∇ν = 0 is locally integrable. If ν is locally defined nonsingular ∇-constant section of ξ then arbitrary section is equal to ν1 = f · ν for a smooth function f and for σ ∈ Γ (gg ) ∇+ σ (f · ν) = ∂#A (σ) f · ν + f · ∇σ ν = 0. ”⇐=” Assume that ∇+ = 0. Take x0 ∈ M and u ∈ ξx . Locally (x0 ∈ U ∼ = Rm ) 0

∇U = ∂Aω U : AU → A (ξU ) = A (U × R) for a closed 1-form ω ∈ Ω1 (AU ) . By assumption


697

∇+ = 0, ω (σ) · ν = 0 for all σ ∈ Γ (gg ) and ν ∈ Γ (ξ) , so ω (σ) = 0 and ω is projectable on U, ω = #∗A (¯ ω ) for some ω ¯ ∈ Ω1 (U) . Since the anchor #A is an epimorphism, the pullback of the differential forms #∗A is a monomorphism. Therefore, since 0 = dA ω = dA (#∗A (¯ ω )) = #∗A (ddR ω ¯ ) we get ddR ω ¯ = 0. Clearly, then ω ¯ = df for some function ∞ −f f ∈ C (U) . It is easy to see that the section σ = e of the bundle U × R ∼ = ξ U is ∇U -constant. Similar considerations show that for trivial vector bundle ξ = M × R and a representation ∂Aω the following conditions are equivalent (1) ∇+ = 0, (2) ω is projectable (i.e. ω = #∗A (¯ ω ) for some ω ¯ ∈ Ω1 (M) . On the other hand, if ω is exact, i.e. 0 = [ω]A ∈ H 1 (A) , then ∇+ = 0, which impies that the differential equation ∇ν = 0 is locally uniquelly integrable. By the definition the 0-group of cohomology can be written similarly to (1). 0 H∇ (A, ξ) = {ν ∈ Γ (ξ) ; ∇ν = 0} . 0 Proposition 7.7. (1) H∇ (A, ξ) = 0 if ξ is nontrivial. (2) For the trivial vector bundle ξ = M × R and ∇ = ∂Aω for closed 1-form ω ∈ Ω1 (A) we have 0 H∇ (A, ξ) 6= 0 ⇐⇒ [ω]A = 0. 0 In particular, if H∇ (A, ξ) 6= 0 then ∇+ = 0.

Proof. (1) Evidently, since each section of nontrivial line bundle ξ is singular and by Lemma 7.6 the set {x; ∇ν = 0} is open-closed. (2) This result may be proved in the same way as in the case of A = T M, i.e. as the formula (3), see also Example 2.6 from Chapter 1. Proposition 7.7(1) generalizes observation (••) from section 2.1. Proposition 7.8. Let ξ be a line bundle and fix an isomorphism ϕ : ξ ⊗ ξ → M × R. Let us assume that ϕ transforms a given A-representation ∇ : A → A (ξ) to A-representation ∂Aω for a closed 1-form ω ∈ Ω1 (A) . Then there exists a linear isomorphism j or ∼ m+n−j (A, ξ)∗ . HD or ⊗∇−ω (A, QA ⊗ ξ) = H∇,c

In particular 0 or ∼ m+n (A, ξ)∗ . HD or ⊗∇−ω (A, QA ⊗ ξ) = H∇,c

Proof. Consider the multiplication by reals or ρ : Qor A × (M × R) → QA .

ρ is compatible with (D or , ∂Aω , (D or )ω ) . The canonical nondegenerate pairing or F : Qor A ⊗ ξ × ξ → QA ⊗ ξ ⊗ ξ

698


is compatible with (D or ⊗ ∇, ∇, D or ⊗ ∇ ⊗ ∇) , so the composition id⊗ϕ ρ F or or or F˜ : Qor A ⊗ ξ × ξ → QA ⊗ ξ ⊗ ξ −→ QA ⊗ (M × R) → QA

is compatible with (D or ⊗ ∇, ∇, (D or )ω ) which implies that it is also compatible with (D or ⊗ ∇−ω , ∇, D or ) . Therefore, for each point x ∈ M, the pairing F˜x : Qor ⊗ ξx × ξx → A,x or or −ω + + or+ QA,x is compatible with the representations (D ⊗ ∇ )x , ∇x , Dx of the isotropy or or Lie algebra g x in the vector spaces QA,x ⊗ ξx , ξx , QA,x , respectively. From this it follows that the differentials d(Dor ⊗∇−ω )+x , d∇+x , d(Dor )+x fulfil condition (6) from Lemma 3.1. Of course, d(Dor )+x = dtrad ⊗ id satisfies condition (3) from the mentioned lemma. Since n−i ∗ g x ⊗ ξx → Λng ∗x ⊗ Qor ∧ : Λig ∗x ⊗ Qor A,x ⊗ ξx × Λ A,x is nondegenerate, the generalized Chern-Hirzebruch-Serre Lemma 3.1 asserts that induced pairing in cohomology l.(7.2)

i n−i n g , Qor g , ξ) → HD g , Qor H(D or+ (g or ⊗∇−ω )+ (g A ⊗ ξ) × H∇+ (g A ) → or (M)

is nondegenerate at every point x ∈ M. The fundamental Theorem 4.4, see also Conclusion 6.1, shows that the pairing j m+n−j or m+n or HD (A, ξ) → HD or ,c (A, QA ) or ⊗∇−ω (A, QA ⊗ ξ) × H∇,c

is nondegenerate. This ends the proof. n Conclusion 7.9. If ξ is not isomorphic to Qor A (i.e. ξ is not isomorphic to Λ g ⊗or (M) , m+n (A, ξ) = 0. see (31), then for an arbitrary connection ∇ : A → A (ξ) we have H∇,c or Proof. If ξ is not isomorphic to Qor A the vector bundle QA ⊗ξ is not trivial so Proposition 0 or 7.7 gives HD or ⊗∇−ω (A, QA ⊗ ξ) = 0. Proposition 7.8 proves our theorem.

The next theorem is one of the importest theorems of the paper. Compare this theorem and Theorem 7.3 with Theorem 5.4 form [10]. Theorem 7.10. For a line bundle ξ and a representation ∇ : A → A (ξ) the following conditions are equivalent: m+n (a) H∇,c (A, ξ) 6= 0, m+n−j m+n m+n (b) H∇,c (A, ξ) = R and the pairing H j (A) × H∇,c (A, ξ) → H∇,c (A, ξ) ∼ = R is ∗ m+n−j j ∼ nondegenerate, i.e. H (A) = H∇,c (A, ξ) , or or (c) (ξ, ∇) ∼ (QA , D ) . Proof. (c) =⇒ (b) by Lemma 7.2, (b) =⇒ (a) is evident. m+n (a) =⇒ (c). Let H∇,c (A, ξ) 6= 0. By Conclusion 7.9 ξ ∼ = Qor A . It remains to compare or the representations. Consider then a flat bundle (ξ = QA , ∇) and any linear isomorphism or Qor M × R, ∇ ⊗ ∇ ∼ ∂Aω . A ⊗ QA → ∼ =


699

By Proposition 7.8 0 or or ∼ m+n HD (A, Qor or ⊗∇−ω (A, QA ⊗ QA ) = H∇,c A ) 6= 0, or or −ω therefore there exists a nonsingular global section ν ∈ Γ (Qor A ⊗ QA ) which is D ⊗ ∇ ω −ω constant. Additionally, ∇ ⊗ ∇ ∼ ∂A implies ∇ ⊗ ∇ ∼ ∂A which means that there exists or −ω a second nonsingular section ν ′ ∈ Γ (Qor -constant. The bundle A ⊗ QA ) which is ∇ ⊗ ∇ or or ′ QA ⊗ QA is 1-dimensional, so ν = f · ν for a nonsingular function f ∈ C ∞ (M) . Write locally ν = να′ ⊗ να for nonsingular sections να′ , να of Qor A . Then 0 = D or ⊗ ∇−ω γ (ν) = Dγor (να′ ) ⊗ να + να′ ⊗ ∇−ω γ (να ) , 0 = ∇ ⊗ ∇−ω γ (f · ν) = ∇γ (f · να′ ) ⊗ να + f · να′ ⊗ ∇−ω γ (να ) .

Multiplying first equation by f and then substracting the second we get f · Dγor (να′ ) − ∇γ (f · να′ ) ⊗ να = 0. The nonsingularity of να yields the equation f · Dγor (να′ ) = ∇γ (f · να′ ) . The bundle Qor A is 1-dimensional, so f · Dγor (¯ ν ) = ∇γ (f · ν¯) (34) for all ν¯ ∈ Γ (Qor A ) . Define a linear isomorphism or ϕ : Qor ¯ 7→ f · ν¯. A → QA , ν or or By (34) one has that (Qor A , ∇) ∼ (QA , D ) .

7.4 Characterization of transitive Lie algebroids with Poincaré duality The last aim is to characterize two classes of transitive Lie algebroids. (A) 6= 0 - the top group of real compact cohomology is not trivial. (1) H∂m+n A ,c or This condition is equivalent to (Qor A , D ) ∼ (M × R, ∂A ) . These classes fulfil the Poincaré duality: the pairing

H j (A) × Hcm+n−j (A) → Hcm+n (A) ∼ =R ∗ is not degenerate, see Theorem 7.3, i.e. H j (A) ∼ = (Hcm+n−j (A)) .

(2) H∂m+n (A, or (M)) 6= 0 - the top group of or (M)-valued compact cohomology is or A ,c not trivial. or or This condition is equivalent to (Qor A , D ) ∼ (or (M) , ∂A ) . In this class the multiplication of cohomology classes

∼ H j (A) × H∂m+n−j (A, or (M)) → H∂m+n or ,c or ,c (A, or (M)) = R A

A

m+n−j is not degenerate, see Theorem 7.3, i.e. H j (A) ∼ = H∂ or ,c (A) A

∗

.

700


Before the characterization of these classes we reduce the Evens-Lu-Weinstein reor presentation (Qor A , D ) to equivalent simple form (only for transitive Lie algebroids of course). We recall that the adjoint representation adA : A → A (gg ) induces a top-power top g ) by representaion adtop A : A → A (Λ X X adtop σ1 ∧ ... ∧ [[γ, σi ]] ∧ ... ∧ σn = aii · σ1 ∧ ... ∧ σn A γ (σ1 ∧ ... ∧ σn ) = i

where [[γ, σi ]] =

P

j

i

aji · σj .

Lemma 7.11. There exist isomorphisms of flat vector bundles top (QA , D) ∼ = Λtopg , adA , or ∼ top or (Qor g ⊗ or (M) , adtop ⊗ ∂ . A,D ) = Λ A A

Proof. It is necessary to show the first assertion, because the second follows from first by tensor product with or (M). Fix arbitrary a connection λ : T M → A and a linear isomorphism ϕλ K ¯ : Λm+n A ⊗ Λm T ∗ M ←− K Λng ⊗ Λm T M ⊗ Λm T ∗ M −→ Λ ng ∼ ∼ =

=

ε ∧ λX ⊗ ϕ ←−p ε ⊗ X ⊗ ϕ 7−→ ε · hX, ϕi. Taking a local basis σ1 , ..., σn of g , W1 , ..., Wm of T M and the duals W1∗ , ..., Wm∗ we ¯ (σ1 ∧ ... ∧ σn ∧ λW1 ∧ ... ∧ λWm ⊗ W1∗ ∧ ... ∧ Wm∗ ) = σ1 ∧ ... ∧ σn . To prove our see that K ¯ K

lemma it is necessary to show the compatibility D ∼ adnA on these nonsingular sections only, i.e. ¯ (Dγ (σ1 ∧ ... ∧ σn ∧ λW1 ∧ ... ∧ λWm ⊗ W1∗ ∧ ... ∧ Wm∗ )) (adnA )γ (σ1 ∧ ... ∧ σn ) = K P P k P Let [[γ, σi ]] = j aji · σj , [[γ, λWj ]] = k a ˜j · σk + r brj · λWr , then [#A (γ) , Wj ] = P r r bj · λWr . The right side of the above equation is equal to ¯ K

X

aii · σ1 ∧ ... ∧ σn ∧ λW1 ∧ ... ∧ λWm ⊗ W1∗ ∧ ... ∧ Wm∗ + X j + σ1 ∧ ... ∧ σn ∧ bj ∧ λW1 ∧ ... ∧ λWm ⊗ W1∗ ∧ ... ∧ Wm∗ + X j +σ1 ∧ ... ∧ σn ∧ λW1 ∧ .... ∧ λWm ⊗ − bj W1∗ ∧ ... ∧ Wm∗ X ¯ aii · σ1 ∧ ... ∧ σn ∧ λW1 ∧ ... ∧ λWm ⊗ W1∗ ∧ ... ∧ Wm∗ =K X ¯ (σ1 ∧ .... ∧ σn ∧ λW1 ∧ ... ∧ λWm ⊗ W ∗ ∧ ... ∧ W ∗ ) = aii · K 1 m = (adnA )γ · σ1 ∧ ... ∧ σn .

Conclusion 7.12. (1) H∂m+n (A) 6= 0 A ,c

⇐⇒

(Λng ⊗ or (M) , adnA ⊗ ∂Aor ) ∼ (M × R, ∂A ) ,


701

(2) H∂m+n or ,c (A, or (M)) 6= 0 A

⇐⇒

(Λng ⊗ or (M) , adnA ⊗ ∂Aor ) ∼ (or (M) , ∂Aor ) .

The following proposition generalizes Proposition 2.2. The proof is analogous. Proposition 7.13. For a representation ∇ : A → A (ξ) in a line vector bundle ξ the following conditions are equivalent: (a) (ξ, ∇) ∼ (or (M) , ∂Aor ) , (b) (ξ ⊗ or (M) , ∇ ⊗ ∂Aor ) ∼ (M × R, ∂A ) . In the sequel we need the notion of a modular class of a Lie algebroid [19], [4]. Firstly, we recall the characteristic classes of a representation ∇ : A → A (ξ) in a line vector bundle ξ. If ξ is trivial as a line bundle and s ∈ Γ (ξ) is a nonsingular section of ξ we define a 1-cocycle θ ∈ Ω1 (A) with respect to dA defined by ∇γ ν = θs (γ) · s. The class θ∇ = [θ] ∈ H 1 (A) is independent on the choice of s and is called characteristic class of A associated to the representation ∇. For a general ξ, we define θ∇ = 21 θ∇⊗∇ (∇ ⊗ ∇ is a flat representation in trivial line bundle ξ ⊗ ξ). We add that if ξ is trivial, the last equation holds. For next propositions and theorems we need the following lemma. Lemma 7.14. If ξ is a line bundle and {ϕα } is a collection of local trivialiations with the transition functions λαβ : Uα × Uβ → R, ϕβ = ϕα · λαβ , then there exist functions fα > 0 such that the local trivializations ϕ¯α = ϕα · fα ¯ αβ = sgn λαβ . (ϕ¯α,x = ϕα,x · fα (x)) have transition functions λ In conclusion, each line bundle ξ possesses a system of local trivializations with transition functions equaling to ±1 and then a family {sα } of nonsingular ±sections i.e. with transition functions equaling just to ±1. Proof. Consider a line bundle ξ with a collection of local trivialiations {ϕα } and transition functions λαβ . The tensor product ξ ⊗ ξ is a trivializable vector bundle with local system of trivializations {ϕα ⊗ ϕα } . Choice a global trivialization ρ : ξ ⊗ ξ → M × R such that ρα := ρ (ϕα ⊗ ϕα (1 ⊗ 1)) > 0. We put 1 fα = √ > 0. ρα ¯ αβ for the collection We show that {fα } is a required family. The transition functions λ ¯ αβ = λαβ · fβ so that sgn λ ¯ αβ = of new local trivializations {ϕ¯α := ϕα · fα } are equl to λ fα 2 fβ 2 2 ¯ sgn λαβ . On the other hand, ρβ = ρ (ϕβ ⊗ ϕβ (1 ⊗ 1)) = λαβ · ρα so λαβ = λαβ · fα = ¯ αβ = sgn λ ¯ αβ = sgn λαβ . λ2 · ρa = 1 and next ¯ λαβ = 1. Finally λ αβ

ρβ

702


Lemma 7.15. For an arbitrary line bundle ξ the characteristic class θ∇ of a representation ∇ : A → A (ξ) can be computed via any family of local nonsingular ±sections {sα } (see Lemma 7.14) of ξ in the following way: the 1-differential A-form θ ∈ Ω1 (A) defined by θ (γ)|Uα sα = ∇γ (sα ) is a correctly defined dA -cocycle and its cohomology class is equal to θ∇ . θ∇ = 0 if and only if there exists a family of local nonsingular ∇-constant ±sections {sα } , ∇sα = 0. For transitive Lie algebroid A if θ∇ = 0 then ∇+ = 0, so the isotropy Lie algebras g x are unimodular. The simple proof will be omitted. We have θ∂Aor = 0. The modular class of a Lie algebroid A is by definition the characteristic class θA of the associated representation D : A → A (QA ) . According to Lemma 7.11 for a transitive Lia algebroid A we have θA = θadnA where adnA : A → A (Λng ) is the representation induced by the adjoint one adA : A → A (gg ) . For real coefficients we have the following characterization of Lie algebroids in the case of the nontriviality of the top group of cohomology. Let {(Uα , xa )} be a coordinate open cover for the manifold M, with transition functions gαβ = xα ◦ x−1 β . Each map xa m determines canonically a local trivialization x¯α of the line bundle Λ T M and the family {¯ xα } has the transition functions J (gαβ ) . Theorem 7.16. The following conditions are equivalent (a) H∂m+n (A) 6= 0, A ,c m+n (b) H∂A ,c (A) ∼ = R and H (A) is a Poincaré algebra, i.e. the pairing H j (A)×Hcm+n−j (A) ∗ → Hcm+n (A) ∼ = (Hcm+n−j (A)) , = R is nondegenerate, H j (A) ∼ or (c) (Qor A , D ) ∼ (M × R, ∂A ) , (d) (Λng ⊗ or (M) , adnA ⊗ ∂Aor ) ∼ (M × R, ∂A ) , (e) (Λng , adnA ) ∼ (or (M) , ∂Aor ) , that is the holonomy homomorphism of (Λng , adnA ) is the same as for the orientation bundle (or (M) , ∂Aor ) . (f ) A is orientable vector bundle and θA = 0 (in particular, g x are unimodular). Proof. (a) ⇐⇒ (b) ⇐⇒ (c) follows immediately from Theorem 7.10 for (ξ, ∇) = (M × R, ∂A ) , (c) ⇐⇒ (d) by Lemma 7.11, (d) ⇐⇒ (e) see Proposition 7.13, (e) =⇒ (f) indeed, θA = θadnA = θ∂Aor = 0. The bundle Λm+n A ∼ = Λng ⊗ Λm T M ∼ = m or (M) ⊗ Λ T M is trivial line bundle because it possesses a local system of trivializations with positive transition functions |J (gαβ )| . (f) =⇒ (e) It is necessary to find a local system of nonsingular adnA -constant sections {σα } of Λng with transition functions sgn J (gαβ ) . Fix a system of local trivializations {ψα } of Λm+n A with positive transition functions γαβ > 0. We can choose a system of local trivializations {ϕα } of the line bundle Λng


703

in such a way that ϕα ⊗ x¯α form a system of local trivializations of the line bundle Λm+n A ∼ = Λng ⊗ Λm T M compatible with {ψα } , i.e. such that ϕα ⊗ x¯α = gα · ψα , gα > 0. This implies that the transition functions λαβ of the system {ϕα } have the sign of J (gαβ ) . Indeed,    ϕα ⊗ x¯α · λαβ · J (gαβ ) = ψα · gα · λαβ · J (gαβ ) , ϕβ ⊗ x¯β =   ψβ · gβ = ψα · γαβ · gβ . Therefore, since gα , gβ and γαβ are positive we have λαβ · J (gαβ ) > 0, i.e. sgn λαβ = sgn J (gαβ ) . By Lemma 7.14 there exists functions fα > 0 such that the local trivializations ϕ¯α = ϕα · fα ¯ αβ = sgn λαβ = sgn J (gαβ ) . The family (ϕ¯α,x = ϕα,x · fα (x)) have transition functions λ of ±sections σ ¯α = ϕ¯α (1) determine a 1-cocycle θ ∈ Ω1 (A) with respect to dA defined by θ (γ)|Uα · σ ¯α = (adnA )γ (¯ σα ) whose cohomology class is the characteristic class of the n adjoint representation adA , [θ] = θadnA . Since θadnA = θA = 0 one has θ = dA f for some function f ∈ C ∞ (M) , i.e. θ (γ) = (dA f ) (γ) = ∂#A (γ) f. Put σα = e−f · σ ¯α . Then the transition functions of {σα } are equal to sgn J (gαβ ) and the sections σα are adnA -constant. In case of oriented manifold the above theorem yields: Theorem 7.17. If M is a oriented manifold then the following conditions are equivalent (a) H∂m+n (A) 6= 0, A ,c m+n (b) H∂A ,c (A) ∼ = R and H (A) is a Poincaré algebra, i.e. the pairing H j (A)×Hcm+n−j (A) ∗ → Hcm+n (A) ∼ = R is nondegenerate, H j (A) ∼ = (Hcm+n−j (A)) , (c) (Λng , adnA ) ∼ (M × R, ∂A ) , i.e. there exists a global nonsingular section ε ∈ Γ (Λng ) which is adnA -constant, that is, A is a TUIO-Lie algebroid, see [13], (d) g is orientable and θA = 0. The independent proof of the implication (c) =⇒ (b) one can be found in [13]. Finally we give a characterization of Lie algebroids whose the top group of cohomology with coefficients in the orientation bundle or (M) is not trivial. Theorem 7.18. The following conditions are equivalent: (a) H∂m+n or ,c (A, or (M)) 6= 0, A

704


∼ (b) H∂m+n or ,c (A, or (M)) = R and the pairing A

∼ H j (A) × H∂m+n−j (A, or (M)) → H∂m+n or ,c or ,c (A, or (M)) = R A

(c) (d) (e) (f ) (g)

A

∗ m+n−j is not degenerate, i.e. H j (A) ∼ = H∂Aor ,c (A, or (M)) , or or (Qor A , D ) ∼ (or (M) , ∂A ) , (Λng ⊗ or (M) , adnA ⊗ ∂Aor ) ∼ (or (M) , ∂Aor ) , (Λng , adnA ) ∼ (M × R, ∂A ) , g is orientable and there exists a global nonsingular section ε ∈ Γ (ξ) which is adnA constant (i.e. A is a TUIO-Lie algebroid, see [13]), g is orientable and θA = 0.

Proof. Only the implication (d) =⇒ (e) needs a proof. Since (or (M) ⊗ or (M) , ∂Aor ⊗ ∂Aor ) ∼ (M × R, ∂A ) one has (Λng , adnA ) ∼ (Λng ⊗ or (M) ⊗ or (M) , adnA ⊗ ∂Aor ⊗ ∂Aor ) (d)

∼ (or (M) ⊗ or (M) , ∂Aor ⊗ ∂Aor )

∼ (M × R, ∂A ) . For an orientable manifold we get Theorem 7.17.

7.5 Remarks on Example 5.3 from [4]. In the cited paper there is an example of nonregular Lie algebroid for which the E-L-W cohomological pairing is not necessary nondegenerate. In the text of Example 5.3 from [4] there are some inaccuracies (concerning dimensional of the group of cohomology) which we remove here. We prove additionally that there is no line representation for which the cohomological pairing is nondegenerate and we prove that the E-L-W representation is not exceptional. The example is the Lie transformation algebroid A = g × M → M associated with the infinitesimal action γ : g → X (M) of a finitely dimensional Lie algebra g on a manifold M. The anchor is given by ρ (v, x) = γ (v)x , and Lie bracket by [[a, b]] (x) = [a (x) , b (x)] + γ (a (x))x (b) − γ (b (x))x (η) , d a, b ∈ C ∞ (M, g) ∼ on R (N ∈ N) = Γ (g × M) and x ∈ M. The vector field X = xN dx defines an action of the 1-dimensional Lie algebra g = R on M = R by γ : R → X (R) , γ (t) = t · X. Let A be the transformation Lie algebroid associated with γ. Then d Γ (A) = C ∞ (R) , #x : Ax = R → Tx M, t 7→ t · xN · dx , [[a, b]] = xN · (a · b′ − b · a′ ) , Ω0 (A) = C ∞ (R) ,

Ω1 (A) = Γ (A∗ ) = C ∞ (R, R∗ ) = Ω1 (R) ∼ = C ∞ (R) , f dx 7→ f,

(35)


705

and, clearly, Ω2 (A) = 0. Lemma 7.19. H 1 (A) ∼ = RN . Proof. By definition dA : C ∞ (R) → Ω1 (A) ∼ = C ∞ (R) , dA (f ) (a) = # (a) (f ) = a·xN ·f ′ , and therefore dA (f ) = xN · f ′ and H 1 (A) = C ∞ (R)/{xN ·f ′ , f ∈C ∞ (R)} = C ∞ (R)/xN ·C ∞ (R) ∼ = RN . Indeed, the classes of functions x0 , x1 , ..., xN −1 form a basis of C ∞ (R)/xN ·C ∞ (R) because the PN −1 k classes are linearly independent and for any f ∈ C ∞ (R) the equality [f ] = i=0 ak x

holds where ak =

f (k) (0) . k!

0 Proposition 7.20. For each linear representation ∇ : A → A (ξ) we have (1) H∇,c (A, ξ) = 1 0, (2) H∇,c (A, ξ) 6= 0. Therefore, for each representation ∇ of A in a line bundle ξ the cohomological pairing 1 0 1 H (A) × H∇,c (A, ξ) → H∇,c (A, ξ) is not nondegenerate even in a weak manner‡.

Proof. (1): The line bundle ξ over R is trivial ξ = M ×R (M = R) so each representation ∇ : A → A (ξ) is of the form ∇ = ∂Aω for some 1-form ω ∈ Ω1 (A). Let ω (a) = g · a for g ∈ C ∞ (R) . Then (∂Aω )a (f ) = (∂A )a (f ) + ω (a) · f = a · xN · f ′ + a · g · f and 0 H∇,c (A, ξ) = f ∈ Cc∞ (R) ; xN · f ′ + g · f = 0 = 0. · y. by the uniqueness of the Cauchy problem for the differential equation y ′ = − g(x) xN 1 ∞ (2) H∇,c (A, ξ) = Cc (R)/{xN ·f ′ +g·f ; f ∈Cc∞ (R)} 6= 0. To prove this we find a compactly supported function h ∈ Cc∞ (R) . such that the differential equation xN · y ′ + g · y = h

(36)

has no global solution y ∈ Cc∞ (R) . Case g (0) = 0. For any h such that h (0) 6= 0 there is no solution of (36). Case g (0) 6= 0. Let |g (x)| ≥ δ > 0 for |x| ≤ ε, ε > 0. Take any function h ∈ Cc∞ (R) such that h ≥ 0, h 6= 0 and supp h ⊂ [α, β] ⊂ (ε, ∞) . The elementary theory of linear differential equations [the formula solving the Cauchy problem in the form of denoted integrals] yields easily that no global compactly supported solution of (36) exists. Consider the E-L-W representation D : A → A (QA ) .. We see that QA = A ⊗ T ∗ R ∼ = 1 ∞ ∼ M × R [M = R] so Γ (QA ) = Γ (A) ⊗ Ω (R) = C (M) by 1 ⊗ f dx 7→ f and that ′ D is equivalent to ∂Aω for ω ∼ = xN (with respect to isomorphism (35)). According to Proposition (7.20) the top group of cohomology of A for trivial and for E-L-W representations are nontrivial. We prove that this representations are not isomorphic so the E-L-W representation is not exceptional. ‡

A pairing F : V × W → U is called weakly non-degenerated if both null spaces N1 = {v ∈ V ; F (v, ·)} and N2 = {w ∈ W ; F (·, w)} are zero

706


Proposition 7.21. The A-flat line bundles (M × R, ∂A ) and (QA , D) [M = R] are not isomorphic. Proof. Let ϕ : M ×R →QA be a linear homomorphism compatible with ∂A and D. ϕ is of the form ϕ (f ) = 1⊗g ·f ·dx for some g ∈ C ∞ (M) . The equality Da (ϕ (f )) = ϕ ((∂A )a f ) ′ ′ yields a · xN · f g = a · xN · f ′ · g, therefore xN · g = 0 which produces g ≡ 0 and that ϕ is not an isomorphism.

References [1] R. Bott and L. Tu: Differential forms in algebraic topology, GTM 82, Springer-Verlag, 1982. [2] S.S. Chern, F. Hirzebruch and J-P. Serre: “On the index of a fibered manifold”, Proc. AMS, Vol. 8, (1957), pp. 587–596. [3] M. Crainic: “Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes”, preprint, arXiv:math.DG/0008064, Commentarii Mathematici Helvetici, to appear. [4] S. Evens, J-H. Lu and A. Weinstein: “Transverse measures, the modular class and a cohomology pairing for Lie algebroids”, Quart. J. Math. Oxford, Vol. 50, (1999), pp. 417–436. [5] R.L. Fernandes: “Lie algebroids, holonomy and characteristic classes”, preprint DG/007132, Advances in Mathematics, Vol. 170, (2002), pp. 119–179. [6] F. Guedira and A. Lichnerowicz: “Géometrie des algébres de Lie locales de Kirillov”, J. Math. Pures Appl., Vol. 63, (1984), pp. 407–484. [7] W. Greub, S. Halperin and R. Vanstone: Connections, Curvature and Cohomology, New York and London, Vol. I, 1971; Vol. II, 1973. [8] S. Haller and T. Rybicki: “Reduction for locally conformal symplectic manifolds”, J. Geom. Phys., Vol. 37, (2001), pp. 262–271. [9] G. Hochschild and J.-P. Serre: “Cohomology of Lie algebras”, Ann. Math., Vol. 57, (1953), pp. 591–603. [10] J. Huebschmann: “Duality for Lie-Rinehart algebras and the modular class”, J.Reine Angew: Math, Vol. 510, (1999), pp. 103–159. [11] V. Itskov, M. Karashev and Y. Vorobjev: “Infinitesimal Poisson Cohomology”, Amer. Math. Soc. Transl. (2), Vol. 187, (1998). [12] J. Kubarski: “Invariant cohomology of regular Lie algebroids”, In: X. Masa, E. Macias-Virgos, J. Alvarez Lopez (Eds.): Proceedings of the VII International Colloquium on Differential Geometry ANALYSIS AND GEOMETRY IN FOLIATED MANIFOLDS, Santiago de Compostela, Spain, July 1994, World Scientific, Singapure, 1995, pp. 137–151. [13] J. Kubarski: “Fibre integral in regular Lie algebroids”, In: Proceedings of the Conference: New Developments in Differential Geometry, Budapest 1996, Budapest, Hungary, 27-30 July 1996, Kluwer Academic Publishers, 1999. [14] J. Kubarski: “Poincaré duality for transitive unimodular invariantly oriented Lie algebroids”, Topology and Its Applications, Vol. 121(3), (2002), pp. 333–355.


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[15] R. Kadobianski, J. Kubarski, V. Kushnirevitch and R. Wolak: “Transitive Lie algebroids of rank 1 and locally conformal symplectic structures”, Journal of Geometry and Physics, Vol. 46, (2003), pp. 151–158. [16] J. Kubarski and A.S. Mishchenko: “Lie algebroids: spectral sequences and signature”, Sbornik: Mathematics, Vol. 194(7), (2003), pp. 1079–1103. [17] K. Mackenzie: Lie Groupoids and Lie Algebroids in Differential Geometry, London Math. Soc., Lecture Notes, Series 124, Cambridge University Press, 1987. [18] M. Spivak: A Comprehensive Introduction to Differential Geometry, Vol. I, 2nd Ed., Publish or Perish Inc., Berkeley, 1979. [19] A. Weinstein: “The modular automorphism group of a Poisson manifold”, J. Geom. Phys., Vol. 23, (1997), pp. 379–394.

CEJM 2(5) 2004 708–724

A geometric theory of harmonic and semi-conformal maps Anders Kock∗ Department of Mathematics, University of Aarhus, Nordre Ringgade 1, DK-8000 Aarhus C, Denmark

Received 15 December 2003; accepted 14 April 2004 Abstract: We describe for any Riemannian manifold M a certain scheme ML , lying in between the first and second neighbourhood of the diagonal of M . Semi-conformal maps between Riemannian manifolds are then analyzed as those maps that preserve ML ; harmonic maps are analyzed as those that preserve the (Levi-Civita-) mirror image formation inside ML . c Central European Science Journals. All rights reserved. ° Keywords: Harmonic, conformal, synthetic differential geometry MSC (2000): 51K10, 53A30, 53C43

Introduction For any Riemannian manifold M , we describe a subscheme ML ⊆ M × M , which encodes information about conformal as well as harmonic maps out of M in a succinct geometric way. Thus, a submersion φ : M → N between Riemannian manifolds is semi-conformal (=horizontally conformal) iff φ × φ maps ML into NL (Theorem 7.1); and a map φ : M → N is a harmonic map if it “commutes with mirror image formation for ML ”, where mirror image formation is one of the manifestations of the Levi-Civita parallelism (derived from the Riemannian metric). The mirror image preservation property is best expressed in the set theoretic language for schemes, which we elaborate on in Section 1. Then it just becomes the statement: for (x, z) ∈ ML ⊆ M × M , φ(z 0 ) = (φ(z))0 , where the primes denote mirror image formation in x (respectively in φ(x)). In particular, when the codomain is R (the real line with standard metric), this characterization of harmonicity ∗


A. Kock / Central European Journal of Mathematics 2(5) 2004 708–724

709

reads φ(z 0 ) = 2φ(x) − φ(z), that is, φ(x) equals the average value of φ(z) and φ(z 0 ), for any z with (x, z) ∈ ML . The last section deals with harmonic morphisms between Riemannian manifolds, meaning harmonic maps which are at the same time semi-conformal. This paper has some overlap with [7], but provides a simplification of the construction of ML , and hence also of the proofs. Theorems 7.1 and 9.1 below are new. A novelty in the presentation is a systematic use of the log-exp bijections that relate the infinitesimal neighbourhoods like ML with their linearized version in the tangent bundle. The first section is partly expository; it tries to present a (rather primitive) version of the category of (affine) schemes, and the “synthetic” language in which we talk about them. The paper grew out of a talk presented at the 5th conference “Geometry and Topology of Manifolds”, Krynica 2003; I want to thank the organizers for the invitation.

1

The language of schemes

Let M be a smooth manifold. In the ring C ∞ (M × M ), we have the ideal I of functions vanishing on the diagonal M ⊆ M × M . Kähler observed that differential 1-forms on M may be encoded as elements in I/I 2 (the module of Kähler differentials) (here, I 2 is the ideal of functions vanishing to the second order on the diagonal). Similarly, elements of I 2 /I 3 encode quadratic differential forms on M . Using the language of schemes will allow us to discuss elements of I/I 2 or of I 2 /I 3 in a more geometric way. We summarize here what we need about schemes. First, note that every smooth manifold M gives rise (in a contravariant way) to a commutative R-algebra, the ring C ∞ (M ) of (smooth R-valued) functions on it. Grothendieck’s bold step was to think of any commutative R-algebra as the ring of smooth functions on some “virtual” geometric object A, the affine scheme defined by A. So A = C ∞ (A), by definition, and the category of affine schemes Sch is just the opposite (dual) of the category Alg of (commutative R-)algebras, Sch = (Alg)op . The category of affine schemes contains the category of smooth manifolds as a full subcategory: to the manifold M , associate the scheme C ∞ (M ) (which we shall not notationally distinguish from M , except for the manifold R, where we write R for C ∞ (R)). Some important schemes associated to a manifold M are its infinitesimal “neighbourhoods of the diagonal” M(k) , considered classically by Grothendieck [3], Malgrange [12], Kumpera and Spencer [10] and others. For each natural number k, M(k) ⊆ M × M is the subscheme of M × M given by the algebra C ∞ (M × M )/I k+1 , where I is the ideal of functions vanishing on the “diagonal” M ⊆ M × M ; thus I k+1 is the ideal of functions vanishing to the k + 1’st order on the diagonal. We have M ⊆ M(1) ⊆ M(2) ⊆ . . . ⊆ M × M , with M ⊆ M × M identified with the submanifold consisting of “diagonal” points (x, x).

710


Now, by definition, C ∞ (M × M )/I 3 = C ∞ (M(2) ), so in the language of schemes, we arrive at the following way of speaking: elements of C ∞ (M × M )/I 3 are functions on M(2) ; and elements in I 2 /I 3 ⊆ C ∞ (M × M )/I 3 are functions on M(2) which vanish on M(1) . (A similar geometric language was presented in [8] for the elements of I/I 2 (=the Kähler differentials): they are functions on M(1) vanishing om M(0) = M , i.e. they are combinatorial differential 1-forms in the sense of [4].) Synthetic differential geometry adds one feature to this aspect of scheme theory, namely extended use of set theoretic language for speaking about objects in (sufficiently nice) categories, like Sch. Thus, since M(k) is a subobject of M × M , the synthetic language talks about M(k) as if it consisted of pairs of points of M ; we shall for instance call such pair “a pair of k’th order neighbours” and write x ∼k y for (x, y) ∈ M(2) . For instance, the fact that M(k) is stable under the obvious twist map M × M → M × M , we express by saying “x ∼k y implies y ∼k x”. The “set” (scheme) of points y ∈ M with x ∼k y, we also denote Mk (x), the k’th order neighbourhood, or k’th monad, around x. The relation ∼k is reflexive and symmetric, but not transitive; rather x ∼k y and y ∼l z implies x ∼k+l z. - Any map f preserves these relations: x ∼k y implies f (x) ∼k f (y). A quadratic differential form on M , i.e. an element of I 2 /I 3 , can now be expressed: it is a function g(x, y), defined whenever x ∼2 y, and so that g(x, y) = 0 if x ∼1 y. If further g is positive definite, then we may directly think of g(x, y) ∈ R as the square distance between x and y. n For M = Rn , M(k) is canonically isomorphic to M × Dk (n): (x, y) ∈ R(k) iff y − x ∈ Dk (n); here, Dk (n) is the “infinitesimal” scheme corresponding to a certain well known Weil-algebra: Recall that a Weil algebra is a finite dimensional R-algebra, where the nilpotent elements form a (maximal) ideal of codimension one. The most basic Weil algebra is the ring of dual numbers R[²] = R[X]/(X 2 ) = C ∞ (R)/(x2 ); the corresponding affine scheme is often denoted D, and is to be thought of as a “disembodied tangent vector” (cf. Mumford [13], III.4, or Lawvere, [11]). The reason is that maps of schemes D → M (M a manifold, say) by definition correspond to R-algebra maps C ∞ (M ) → R[²], and such in turn correspond, as is known, to tangent vectors of M. Note that since R[²] is a quotient algebra of C ∞ (R), D is, by the duality, a subscheme of R; this subscheme may be described synthetically as {d ∈ R | d2 = 0}, reflecting the fact that R[²] comes about from C ∞ (R) by dividing out x2 . More generally, for k and n positive integers, Dk (n) is the scheme corresponding to the Weil algebra which one gets from R[X1 , . . . , Xn ] by dividing out by the ideal generated by monomials of degree k + 1; or, equivalently, from C ∞ (Rn ) by the ideal of functions that vanish to order k + 1 at 0 = (0, . . . , 0) (it is also known as the “algebra of k-jets at


711

0 in Rn ”). – In particular, D1 (1) is the ring of dual numbers described above. Just as D is the subscheme of R described by D = {x ∈ R | x2 = 0}, Dk (n) may be described in synthetic language as {(x1 , . . . , xn ) ∈ Rn | xi1 · . . . · xik+1 = 0 for all i1 , . . . , ik+1 }. The specific Weil algebras which form the algebraic backbone of the present paper are the following (first studied for this purpose in [7]). For each natural number n ≥ 2, we consider the algebra C ∞ (Rn )/IL , where IL is the ideal generated by all x2i − x2j and all xi xj where i 6= j. The linear dimension of this algebra is n + 2; a basis may be taken to be (the classes mod IL of) the functions 1, x1 , . . . , xn , x21 + . . . + x2n . The corresponding affine scheme we denote DL (n) or DL (Rn ); the letter “L” stands for “Laplace”, for reasons that will hopefully become clear. Using synthetic language, DL (n) may be described DL (n) = {(x1 , . . . , xn ) ∈ Rn | x2i = x2j ; xi xj = 0 for i 6= j}. Note that D1 (n) ⊆ DL (n) ⊆ D2 (n). The inclusion D1 (n) ⊆ DL (n) corresponds to the quotient map C ∞ (Rn )/IL → C ∞ (Rn )/I1 which in turn comes about because IL ⊆ I1 . The kernel of this quotient map has linear dimension 1; a generator for it is the (class mod IL of) x21 + . . . + x2n . The following is a tautological translation of this fact: Proposition 1.1. Any function f : DL (n) → R which vanishes on D1 (n) is of the form c · (x21 + . . . + x2n ) for a unique c ∈ R. The subscheme Dk (n) ⊆ Rn can be described in coordinate free terms; in fact, it is just the k-monad Mk (0) around 0. More generally, for any finite dimensional vector space V , we can give an alternative description of Mk (0), which we also denote Dk (V ). We only give this description for the case k = 1 and k = 2, which is all we need: We have that u ∈ D1 (V ) iff for any bilinear B : V × V → R, B(u, u) = 0; this then also holds for any bilinear V × V → W , with W a finite dimensional vector space. Similarly u ∈ D2 (V ) iff for any trilinear C : V × V × V → R , C(u, u, u) = 0; this then also holds for any trilinear V × V × V → W , with W a finite dimensional vector space. Any function f : D2 (V ) → W (with W a finite dimensional vector space) can uniquely be written in the form u 7→ f (0)+L(u)+B(u, u) with L : V → W linear and B : V ×V → W bilinear symmetric. If V is equipped with a positive definite inner product, we shall in the following Section also describe a subscheme DL (V ) with D1 (V ) ⊆ DL (V ) ⊆ D2 (V ); for V = Rn with standard inner product, it will be the DL (n) already described.

2

L-neigbours in inner-product spaces

For a 1-dimensional vector space V , we say that a ∈ V is L-small if it is 2-small, i.e. if a ∼2 0.

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Given an n-dimensional vector space V (n ≥ 2) with a positive definite inner product < −, − >. We call a vector a ∈ V L-small if for all u, v ∈ V < a, u >< a, v >=

1 < a, a >< u, v > . n

(1)

The “set” (scheme) of L-small vectors is denoted DL (V ). It is clear that if a ∈ DL (V ), then λa ∈ DL (V ) for any scalar λ. But DL (V ) will not be stable under addition; it is not hard to prove that if a and b are L-small vectors, then a + b is L-small precisely if for all u, v ∈ V < a, u >< b, v > + < a, v >< b, u >=

2 < a, b >< u, v > . n

(2)

Let us analyze these notions for the case of Rn , with its standard inner product. We claim Proposition 2.1. The vector t = (t1 , . . . , tn ) belongs to DL (Rn ) iff t21 = . . . = t2n ; and ti tj = 0 for i 6= j.

(3)

(So DL (Rn ) equals the DL (n) described above, or in [7] equation (8).) Proof. If t ∈ DL (Rn ), we have in particular for each i = 1, . . . , n, t2i =< t, ei >< t, ei >=

1 < t, t >, n

where e1 , . . . , en is the standard (orthonormal) basis for Rn . The right hand side here is independent of i. – Also, if i 6= j, ti tj =< t, ei >< t, ej >=

1 < t, t >< ei , ej >= 0, n

since < ei , ej >= 0. Conversely, assume that (3) holds. Let u and v be arbitrary vectors, u = (u1 , . . . , un ), and similarly for v. Then X X < t, u >< t, v >= ( ti ui )( tj vj ) i

=

X

ti tj ui vj = t21

j

X

i,j

ui v i ,

i

using (3) for the last equality sign. But this is t21 < u, v >, and since, again by (3) t21 = we conclude < t, u >< t, v >=

1 2 1 (t1 + . . . + t2n ) = < t, t >, n n 1 n

< t, t >< u, v >.


713

As a Corollary, we get that for v ∈ V (an n-dimensional inner-product space), v ∈ DL (V ) iff for some, or for any, orthonormal coordinate system for V , the coordinates of v satisfy the equations (3). From the coordinate characterization of DL (V ) also immediately follows that DL (V ) ⊆ D2 (V ). Here is an alternative characterization of L-small vectors, for inner product spaces V of dimension ≥ 2 (the word “self-adjoint” may be omitted, but we shall need the Proposition in the form stated). Proposition 2.2. The vector a belongs to DL (V ) if and only if for every self adjoint linear map L : V → V of trace zero, < L(a), a >= 0 Proof. We pick orthonormal coordinates, and utilize the “coordinate” description of DL (Rn ). Assume a ∈ DL (Rn ), and assume L is given by the symmetric matrix [cij ] with P cii = 0. Then X < L(a), a >= cij aj ai ; ij

since ai aj = 0 if i 6= j, only the diagonal terms survive, and we get < L(a), a >= P P P 2 2 cii , since all the a2i are equal to a21 . Since cii = 0, we get 0, as claimed. i cii ai = a1 Conversely, let us pick the L given by the symmetric matrix with cij = cji = 1(i 6= j), and all other entries 0. Then 0 =< L(a), a >= ai aj + aj ai , whence ai aj = 0. Next let us pick the L given by the matrix cii = 1, cjj = −1 (i 6= j) and all other entries 0. Then 0 =< L(a), a >= ai ai − aj aj , whence a2i = a2j . So a ∈ DL (Rn ). We now consider the question of when a linear map f : V → W between inner product spaces preserves L-smallness, i.e. when f (DL (V )) ⊆ DL (W ). Let us call an m × n matrix semi-conformal if the rows are mutually orthogonal, and have same (strictly positive) square norm. (This square norm is then called the square dilation of the matrix, and is typically denoted Λ.) The rank of a semi-conformal matrix is m, since its rows, being orthogonal, are linearly independent. It thus represents a surjective linear map Rn → Rm . We have Proposition 2.3. Let f : V → W be a surjective linear map between inner product spaces. Then t.f.a.e. 1) f (DL (V )) ⊆ DL (W ) 2) In some, or any, pair of orthonormal bases for V , W , the matrix expression for f is a semi-conformal matrix

714


In case these conditions hold, the common square norm Λ of the rows of the matrix is characterized by: for all z ∈ DL (V ) 1 1 < f (z), f (z) >= Λ < z, z >, m n (where n = dim(V ), m = dim(W )). Proof. Assume 1). Pick orthonormal bases for V and W , thereby identifying V and W with Rn and Rm , with standard inner product. Let the matrix for f be A = [aij ]. For all z ∈ DL (V ), we have by assumption that X ( aij zj )2 is independent of i. j

We calculate this expression: X X X X ( aij zj )2 = ( aij zj )( aij 0 zj 0 ) = a2ij zj2 j

j

j0

(4)

j

since the condition z ∈ DL (V ) implies that zj zj 0 = 0 for j 6= j 0 , so all terms where j 6= j 0 are killed. Also zj2 = z12 , so bringing this factor outside the sum, we get X 1 X 2 X 2 z )( a ). = z12 ( a2ij ) = ( n k k j ij j

(5)

P Since this is independent of i, then so is j a2ij , by the uniqueness assertion in Proposition 1.1. - The proof that the rows of A are mutually orthogonal is similar (or see the proof for Theorem 3.2 in [7]). – Conversely assume 2), and assume that z ∈ DL (Rn ). We prove that A · z ∈ DL (Rm ). The square of the i’th coordinate here is X X ( aij zj )2 = z12 a2ij (6) j

j

by the same calculation as before. But now the sum is independent of i, by assumption on the matrix A. – Similarly, if i 6= i0 , the inner product of the i’th and i0 ’th row of A · z is X X X ( aij zj )( ai0 j 0 zj 0 ) = z12 ( aij ai0 j ), j

j0

j

using again the special equations that hold for the zj ’s; but now the sum in the parenthesis is 0 by the assumed orthogonality of the rows of A. Let Λ be the common square norm of the rows of the matrix for f . Then for z ∈ DL (V ), X 1 1 XX < f (z), f (z) >= ( aij zj )( aij 0 zj 0 ), m m i j j0 and multiplying out, only the terms where j = j 0 survive, since z ∈ DL (V ). Thus we get XX X 1 XX 2 2 1 1 a2ij ) = z12 ( Λ) ( aij zj ) = z12 ( m i m m i j i j


715

P but this is z12 Λ, since there are m indices i. On the other hand, z12 = 1/n( j zj2 ). We have the following “coordinate free” version of Proposition 1.1 (derived from it by picking orthonormal coordinates): Proposition 2.4. Let f1 , f2 : DL (V ) → R be functions which agree on D1 (V ). Then there exists a unique number c ∈ R so that for all z ∈ DL (M ) we have f1 (z) − f2 (z) = c· < z, z > .

Consider a map f : D2 (V ) → W with f (0) = 0 and a symmetric bilinear B : V ×V → W . Let b : V → W denote the “quadratic” map u 7→ B(u, u). Lemma 2.5. The map f takes DL (V ) into DL (W ) if and only if f + b does. Proof. This is a simple exercise in degree calculus. Assume f has the property. To prove that f + b does, let a ∈ DL (V ), and let u, v be arbitrary “test” vectors in W . We consider < f (a) + b(a), u >< f (a) + b(a), v >. Using bilinearity of inner product, this comes out as four terms, one of which is < f (a), u >< f (a), v >, and three of which vanish for degree reasons, thus for instance < b(a), u >< f (a), v >=< B(a, a), u >< f (a), v > which contains a in a trilinear way, so vanishes since a ∈ DL (V ) ⊆ D2 (V ). So the left hand side in the test equation for L-smallness of f (a) + b(a) equals the left hand side in the test equation for L-smallness of f (a). The right hand sides of the test equation is dealt with in a similar way.

3

Riemannian metrics

Recall from [6], [7] that a Riemannian metric g on a manifold M may be construed as an R-valued function defined on the second neighbourhood M(2) of the diagonal, and vanishing on M(1) ⊆ M(2) ; we think of g(x, y) as the square distance between x and y. Also g should be positive definite, in a sense which is most easily expressed when passing to a coordinatized situation. Since our arguments are all of completely local (in fact infinitesimal) nature, there is no harm in assuming that one chart covers all of M , meaning that we have an embedding of M as an open subset of Rn , or of an abstract n-dimensional vector space V . In this case, each Tx M gets canonically identified with V : to u ∈ V , associate the tangent vector t at x given by d 7→ x + d · u for d ∈ D. The vector u is called the principal part of t. In this case g is of the form g(x, z) = G(x; z − x, z − x), where G : M × V × V → R is bilinear symmetric in the two last arguments. We require each G(x; −, −) to be positive definite, i.e. G(x; −, −) provides V with an inner product (depending on x). Since Tx M is canonically identified with V , each Tx M also acquires

716


an inner product; this inner product can in fact be described in a coordinate free way, in terms of g alone, cf. [7] formula (4).

4

Symmetric affine connections, and the log-exp-bijection

According to [5], an affine connection ∇ on a manifold M is a law ∇ which allows one to complete any configuration (with x ∼1 y, x ∼1 z) z. ¥ 1¥

.y ¥ »»» » » x .¥» 1 into a configuration .∇(x, y, z)

1 »»¥ »» ¥1 z .»» ¥¥ ¥ 1 ».¥ y ¥ »»» » ¥ » x. 1

(with z ∼1 ∇(x, y, z) ∼1 y), to be thought of as an “infinitesimal parallelogram according to ∇”. There is only one axiom assumed: ∇(x, x, z) = z; ∇(x, y, x) = y. If ∇(x, y, z) = ∇(x, z, y) for all x ∼1 y, x ∼1 z, we call the connection symmetric. In a coordinatized situation, i.e. with M identified with an open subset of a finite dimensional vector space V , the data of an affine connection ∇ may be encoded by a map Γ : M × V × V → V , bilinear in the two last arguments, namely ∇(x, y, z) = y − x + z + Γ(x; y − x, z − x), so that Γ measures the discrepancy between “infinitesimal parallelogram formation according to ∇” and the corresponding parallelograms according to the affine structure of the vector space V . This Γ is the “union of” the Christoffel symbols; and ∇ is symmetric iff Γ(x; −, −) is. A fundamental result in differential geometry is the existence of the Levi-Civita connection associated to a Riemann metric g. This result can be formulated synthetically, without reference to tangent bundles or coordinates, namely: given a Riemann metric g on a manifold, then there exists a unique symmetric connection ∇ on M with the property that for any x ∼1 y, the map ∇(x, y, −) : M1 (x) → M1 (y) preserves g, i.e. for z ∼1 x, u ∼1 x, g(∇(x, y, z), ∇(x, y, u)) = g(z, u).


717

(This latter condition is equivalent to: the differential of ∇(x, y, −) at x is an innerproduct preserving linear map Tx M → Ty M .) There is, according to [9] Theorem 4.2, an alternative way of encoding the data of a symmetric affine connection on M , namely as a “partial exponential map”, meaning a bijection (for each x ∈ M ) M2 (x) ∼ = D2 (Tx M ) ⊆ Tx (M ), with certain properties. We describe how such bijection expx : D2 (Tx M ) → M2 (x) is related to the connection ∇ (and this equation characterizes expx completely): expx ((d1 + d2 )t) = ∇(x, t(d1 ), t(d2 )), where t ∈ Tx M and d1 , d2 ∈ D (this implies (d1 + d2 )t ∈ D2 (Tx M )). Since ∇(x, y, x) = y, it follows by taking d2 = 0 that exp(d1 t) = t(d1 ), so that the partial exponential map M2 (0) → M2 (x) is an extension of the “first order” partial exponential map M1 (0) → M1 (x), as considered in [8]; the first order exponential map is “absolute” in the sense that its construction does not depend on a metric g on M . In the coordinatized situation with M ⊆ V an open subset of a vector space V , the second order exponential map corresponding to ∇ is given as follows. Note first that since now M is an open subset of V , Tx (M ) may be identified with V canonically, via the usual notion of “principal part” of a tangent vector to V . Let u ∈ D2 (V ). Then 1 expx (u) = x + u + Γ(x; u, u). 2 This is an element in M ⊆ V , since M is open, in fact, it is an element of M2 (x). The inverse of expx we of course have to call logx ; in the coordinatized situation M ⊆ V , it is given as follows: let y ∼2 x; then y = x + u with u ∈ D2 (V ), and 1 logx (x + u) = u − Γ(x; u, u). 2 The fact that the map logx thus described is inverse for expx is a simple calculation using bilinearity of Γ(x; −, −), together with Γ(x; u, Γ(x; u, u)) = 0, and Γ(x; Γ(x; u, u), Γ(x; u, u)) = 0, and these follow because they are trilinear (respectively quatrolinear) in the arguments where u is substituted. –The following gives an “isometry” property of the log-exp-bijection. (It does not depend on the relationship between the metric g and the affine connection/partial exponential.) Proposition 4.1. For z ∼2 x, g(x, z) =< logx z, logx z >. Proof. We work in a coordinatized situation M ⊆ V , so that g is encoded by G : M × V × V → R, and the connection is encoded by Γ : M × V × V → V , with both G and Γ bilinear in the two last arguments. Let z ∼2 x, so z is of the form x + u with u ∈ D2 (V ). Then on the one hand g(x, z) = G(x; u, u),

718


and on the other hand, logx (z) = u − 1/2Γ(x; u, u) so that 1 1 < logx z, logx z >= G(x; u − Γ(x; u, u), u − Γ(x; u, u)), 2 2 and expanding this by bilinearity, we get G(x; u, u) plus some terms which vanish because they are tri- or quatro-linear in u.

5

Mirror image

Using the (second order) partial exponential map, we can give a simple description of the infinitesimal symmetry ([7]) which any Riemannian manifold has. Let z ∼2 x in M . Its mirror image z 0 in x is defined by z 0 := expx (− logx (z)). In the coordinatized situation M ⊆ V , we can utilize the formulae for log and exp given in terms of Γ to get the following formula for mirror image formation. If z = x + u with u ∈ D2 (V ), we get z 0 = x − u + Γ(x; u, u). This is a calculation much similar to the one above, namely, cancelling terms of the form Γ(x; Γ(x; u, u), u) or Γ(x; Γ(x; u, u), Γ(x; u, u)), these being tri- or quatro-linear in u. A similar calculation will establish that z 00 = z. Note also that if u ∈ D1 (V ), and z = x + u, then z 0 = x − u. From this follows Lemma 5.1. Given x ∈ M . Let f : M → R. The function f˜ : M2 (x) → R defined by f˜(z) = f (z 0 ) + f (z) − 2f (x) vanishes on M1 (x). For, if df denotes the differential of f at x, and z = x + u with u ∈ D1 (V ), the right hand side here is (f (x) + df (−u)) + (f (x) + df (u)) − 2f (x), and this is 0 since df is linear.

6

L-neighbours in a Riemannian manifold

We consider a Riemannian manifold (M, g), and the various structures on M derived from it, as in the previous sections. In particular, we have the partial exponential map exp, and its inverse log. Using these maps, we shall transport the L-neighbour relation from the inner-product spaces Tx M back to a relation in M . Explicitly,


719

Definition 6.1. Let x ∼2 z in M . We say that x ∼L z if logx (z) is L-small in the inner product space Tx M (with inner product derived from g). Note that this is not apriori a symmetric relation, since log(x, z) and log(z, x) are not immediately related – they belong to two different vector spaces Tx M and Tz M ; in a coordinatized situation M ⊆ V , both these vector spaces may be canonically identified with V , but the notion of exp and log depend on inner products, and V in general gets different inner products from Tx M and Tz M . In [7], the question of symmetry of the relation ∼L was left open (and the relation ∼L was defined in a different, more complicated way). We state without proof: Proposition 6.2. The L-neighbour relation is symmetric. This fact will not be used in the present paper. It depends on the fact that parallel transport according to ∇ preserves L-smallness, being an isometry. The following is the fundamental property of L-neighbours, and provides the link to the Laplace operator and harmonic functions, and more generally to harmonic morphisms. It is identical to Theorem 2.4 in [7], but the argument we give presently is more canonical (does not depend on chosing a geodesic coordinate system): Theorem 6.3. For any f : ML (x) → R, there exists a unique number c so that for all z ∈ ML (x), f (z) + f (z 0 ) − 2f (x) = c · g(x, z). (7)

Proof. Consider the composite of expx with the function f˜ of z described by the left hand side of (7), f˜ expx R DL (Tx M ) ML (x) It is a function defined on DL (Tx M ) ⊆ Tx M . It follows from Lemma 5.1 that this function vansihes on D1 (Tx M ), and thus is constant multiple of the square-norm function Tx M → R, by Proposition 2.4, f˜(expx (u)) = c· < u, u > . Apply this to u = logx z for z ∼L x; we get f˜(z) = f˜(expx (logx (z))) = c· < logx z, logx z >, which is c · g(x, z) by Proposition 4.1. For any function f : M → R, we can for each x ∈ M consider the corresponding c, characterized by (7); this gives a function c : M → R, and we define ∆(f ) to be n times this function, in other words, the function ∆(f ) is characterized by: for each pair x ∼L z f (z) + f (z 0 ) − 2f (x) =

∆(f )(x) g(x, z), n

(8)

720


where z 0 denotes the mirror image of z in x. (This ∆ operator can be proved to be the standard Laplace operator, cf. [7].) We call f a harmonic function if ∆(f ) = 0. Thus harmonic functions are characterized by the average value property: for any x ∼L z, f (x) is the average of f (z) and f (z 0 ). This property can also be expressed: for any z ∼L x, f (z 0 ) is the mirror image of f (z) in f (x), where mirror image of b in a for a, b ∈ R means 2a − b. This is also the mirror-image formation in R w.r.to the standard Riemannian metric given by g(a, b) = (b − a)2 . This observation prompts the following definition: Definition 6.4. Let (M, g) and (N, h) be Riemannian manifolds, and let φ : M → N a map. We say that φ is a harmonic map if it preserves mirror image formation of L-neighbours x, z, φ(z 0 ) = φ(z)0 , where the prime denotes mirror image formation in x w.r.to g and in φ(x) w.r.to h, respectively. Note that even if z is an L-neighbour of x, φ(z) may not be an L-neighbour of φ(x), but it will be a 2-neighbour of φ(x), so that the notion of mirror image of it makes sense. – The notion may be localized at x: φ is a harmonic map at x if for all z ∼L x, φ(z 0 ) = φ(z)0 .) A stronger notion than harmonic map is that of harmonic morphism; this is a map which is as well a harmonic map, and is also semi- (or horizontally) conformal in the sense of the next section. (The terminology is not very fortunate, but classical, cf. [1].)

7

Semi-conformal maps

We consider again two Riemannian manifolds (M, g) and (N, h), and a submersion φ : M → N . It defines a ”vertical” foliation, whose leaves are the (components of) the fibres of φ, and hence the transversal distribution consisting of Ker(dfx )⊥ ⊆ Tx M . (This “horizontal” distribution can also be described in purely combinatorial terms without reference to the tangent bundle.) Recall (from [1], say) that φ is called semi-conformal (or horizontally conformal) at x ∈ M , with square-dilation Λ > 0, if the linear map dfx : Tx M → Tφ(x) N is semiconformal with square-dilation Λ > 0, in the sense of Section 2. (This property can also be expressed combinatorially.) The following is a generalization of Theorem 3.2 in [7] (which dealt with the case of a diffeomorphism φ). Theorem 7.1. Let φ : M → N be a submersion, and let x ∈ M . Then t.f.a.e.: 1) φ is semi-conformal at x (for some Λ > 0) 2) φ maps ML (x) into ML (φ(x)).


721

Proof. Consider the diagram M2 (x)

φ-

M2 (φ(x)) logφ(x)

logx ?

D2 (Tx M )

? - D2 (Tφ(x) N )

fdφx

where f is the unique map making the diagram commutative, and where dφx is (the restriction of) the differential of φ. It does not make the diagram commutative, but when restricted to M1 (x), it does, by the very definition of differentials. So f and dφx agree on M1 (x), and hence differ by a quadratic map b. It then follows from Lemma 2.5 that f maps DL (Tx M ) into DL (Tφ(x) )if and only if dφx does. By definition, ML (x) comes about from DL (Tx M ) by transport along the log-exp-bijection, so φ preserves ML iff f preserves DL . On the other hand, by the Proposition 2.3, semi-conformality of dφx is equivalent to dφx preserving DL . We may summarize the results of the last two sections by stating the following (which may be taken as definitions of these notions, but couched in purely geometric/combinatorial language): let φ : M → N be a submersion between Riemannian manifolds. Then • φ is a harmonic map if it preserves mirror image formation of L-neighbours • φ is a semi-conformal map if it preserves the notion of L-neighbour • φ is a harmonic morphism if it has both these properties. If the codomain is R, any 2-neigbour is an L-neighbour, so any map to R is automatically semi-conformal, so for codomain R, harmonic map and harmonic morphism means the same thing. Such a map/morphism is in fact exactly a harmonic function M → R. All three notions make sense “pointwise”: φ is a harmonic at x ∈ M if it preserves mirror image formation of L-neighbours of x. For this to make sense, we don’t need φ to be defined on all of M , because the property only depends on the 2-jet of φ at x, meaning the restriction of φ to M2 (x).

8

Sufficiency of harmonic 2-jets

By 2-jets, we understand in this Section 2-jets of R-valued functions; so a 2-jet at x ∈ M is a map M2 (x) → R. If M is a Riemannian manifold, we say that such a 2-jet f is harmonic if it preserves mirror image formation of L-neigbours of x, f (z 0 ) = 2f (x)−f (z), for all z ∼L x. Among such harmonic 2-jets, we have in particular those of the form M2 (x)

logx-

Tx M

p

- R,

(9)

where the last map p is linear. For, by construction of mirror image in terms of logx ,

722


logx (z 0 ) = − logx (z), and this mirror image formation is preserved by p (here, we don’t even need z ∼L x, just z ∼2 x). Another type of harmonic 2-jet are those of the form M2 (x)

logx-

Tx M

q

-R

(10)

where q is a “quadratic map of trace 0”, meaning q(u) =< L(u), u > for some selfadjoint L : Tx M → Tx M of trace zero. For, z ∼L x means by definition that logx (z) ∈ DL (Tx M ), and quadratic trace zero maps kill DL , by Proposition 2.2. These two special kinds of harmonic jets are the only ones that we shall use in the proof of the following “recognition Lemma”: Lemma 8.1. There are sufficiently many harmonic 2-jets to recognize mirror image formation in x, and to recognize L-neighbours of x. Precisely, if z and z˜ are 2-neighbours of x, and f (˜ z ) = 2f (x) − f (z) for all harmonic 0 2-jets f , then z˜ = z ; and if z is a 2-neighbour of x such that f (z) = 0 for all harmonic 2-jets f which vanish on M1 (x), then z ∼L x. Proof. The first assertion follows because logx (z 0 ) = − logx (z), and because there are sufficiently many linear p : Tx M → R to distinguish any pair of vectors (Tx M being finite-dimensional). The second assertion follows because logx maps ML (x) bijectively onto DL (Tx M ), and the latter is recognized by quadratic trace zero maps, by Proposition 2.2. There is a partial converse: Proposition 8.2. Let f : M2 (x) → R be a harmonic 2-jet which vansihes at M1 (x). Then it vanishes at ML (x). Proof. Let b denote the composite f ◦ expx : D2 (Tx M ) → R. The vanishing assumption on f implies that there is a unique quadratic map Tx M → R extending b. It suffices to prove that b(u) = 0 for any u ∈ DL (Tx M ). Let z denote expx (u); then z ∈ ML (x). Harmonicity of f at x implies f (z)+f (z 0 ) = 0 by Theorem 6.3, and hence b(u)+b(−u) = 0. But b is a even function, being quadratic, hence b(u) = 0.

9

Characterization Theorem

The following Theorem is now almost immediate in view of the combinatorial/geometric description of harmonic maps and semi-conformal maps. It is a version of the Characterization Theorem of Fuglede and Ishihara, cf. [1] Theorem 4.2.2. Theorem 9.1. Given a submersion φ : M → N between Riemannian manifolds, and let x ∈ M . Then t.f.a.e. 1) φ is a harmonic morphism at x


723

2) for any harmonic 2-jet f at φ(x), f ◦ φ : M → R is a harmonic 2-jet. (The Theorem in the classical form talks about harmonic germs at φ(x), rather than harmonic 2-jets. The “upgrading” of our version to the classical one thus depends on a rather deep existence theorem: any harmonic 2-jet comes about by restriction from a harmonic germ, see Appendix of [1]. Such existence results are beyond the scope of our methods.) Proof. Assume that φ is a harmonic morphism at x, and let f be a harmonic 2-jet. Let z ∼L x. Then φ(z 0 ) = (φ(z))0 , since φ is a harmonic map; also φ(z) ∼L φ(x) since φ is semi-conformal. So f preserves the mirror image of φ(z). So both φ and f preserve the relevant mirror images, hence so does the composite f ◦ φ : M2 (x) → R; hence it is a harmonic 2-jet. Conversely, suppose φ has f ◦ φ harmonic for all harmonic 2-jets f at φ(x). Let z ∼L x. To prove φ(z 0 ) = (φ(z))0 , it suffices, by the Recognition Lemma (applied to N ) to prove that all harmonic 2-jets f at φ(x) take φ(z 0 ) to the mirror image of φ(z). But by assumption f ◦ φ is harmonic at x, so preserves mirror image. – Also, to prove φ(z) ∼L φ(x), it suffices by the Recognition Lemma to prove that any harmonic 2-jet at φ(x), vanishing on M1 (φ(x)), kills φ(z). But by assumption, f ◦φ is a harmonic 2-jet, and it vanishes at M1 (x), so by Proposition 8.2, it kills z. So f (φ(z)) = 0, so φ(z) ∼L φ(x). This proves the Theorem.

References [1] P. Baird and J.C. Wood: Harmonic Morphisms Between Riemannian Manifolds, Oxford University Press, 2003. [2] E. Dubuc: “C ∞ schemes“, Am.J.Math, Vol. 103, (1981), pp. 683–690. [3] A. Grothendieck: Techniques de construction en géometrie algébrique, Sem. H.Cartan, Paris, 1960-61, pp. 7–17. [4] A. Kock: Synthetic Differential Geometry, Cambridge University Press, 1981. [5] A. Kock: “A combinatorial theory of connections“, Contemporary Mathematics, Vol. 30, (1984), pp. 132–144. [6] A. Kock: “Geometric construction of the Levi-Civita parallelism“, Theory and Applications of Categories, Vol. 4(9), (1998). [7] A. Kock: “Infinitesimal aspects of the Laplace operator“, Theory and Applications of Categories, Vol. 9(1), (2001). [8] A. Kock:“First neighbourhood of the diagonal, and geometric distributions”, Universitatis Iagellonicae Acta Math., Vol. 41. (2003), pp. 307–318. [9] A. Kock and R. Lavendhomme: “Strong infinitesimal linearity, with applications to strong difference and affine connections“, Cahiers de Top. et Géom. Diff., Vol. 25, (1984), pp. 311–324. [10] A.Kumpera and D. Spencer: “Lie Equations“, Annals of Math. Studies, Vol. 73, Princeton, 1972.

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[11] F.W. Lawvere: “Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body“, Cahiers de Top. et Géom. Diff., Vol. 21, (1980), pp. 377–392. [12] B. Malgrange: “Equations de Lie“, I, J. Diff. Geom., Vol. 6, (1972), pp. 503–522. [13] D. Mumford: The Red Book of varieties and schemes, Springer L.N.M. 1358, 1988. [14] A. Weil: “Théorie des points proches sur les varietés différentiables“, Colloque Top. et Géom. Diff., Stasbourg 1953.

CEJM 2(5) 2004 725–731

On the topology of spherically symmetric space-times J. Szenthe∗ Department of Geometry, E¨ otv¨ os University Pázmány Péter stny. 1/C, H-1117 Budapest, Hungary

Received 15 December 2003; accepted 21 March 2004 Abstract: Spherically symmetric space-times have attained considerable attention ever since the early beginnings of the theory of general relativity. In fact, they have appeared already in the papers of K. Schwarzschild [12] and W. De Sitter [5] which were published in 1916 and 1917 respectively soon after Einstein’s epoch-making work [7] in 1915. The present survey is concerned mainly with recent results pertainig to the toplogy of spherically symmetric spacetimes. Definition. By space-time a connected time-oriented 4-dimensional Lorentz manifold is meant. If (M, ) is a space-time, and Φ : SO(3) × M → M an isometric action such that the maximal dimension of its orbits is equal to 2, then the action Φ is said to be spherical and the space-time is said to be spherically symmetric [8]; [11]. Likewise, isometric actions Ψ : O(3) × M → M are also considered ([10], p. 365; [4]) which will be called quasi-spherical if the maximal dimension of its orbits is 2 and then the space-time is said to be quasi-spherically symmetric here. Each quasi-spherical action yields a spherical one by restricting it to the action of SO(3); the converse of this statement will be considered elsewhere. The main results concerning spherically symmetric space-times are generally either of local character or pertaining to topologically restricted simple situations [14], and earlier results of global character are scarce [1], [4], [6], [13]. A report on recent results concerning the global geometry of spherically symmetric space-times [16] is presented below. c Central European Science Journals. All rights reserved.

Keywords: Lorentz mainfolds, general relativity, isometric actions MSC (2000): 53C50, 57S25, 83C40

∗


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J. Szenthe / Central European Journal of Mathematics 2(5) 2004 725–731

Some basic concepts and facts

First some well-known fundamental facts are given. A spherically symmetric action cannot have 1-dimensional orbits by a simple observation concerning the dimension of the possible isotropy subgroups (see e. g. [11], p. 261). The 2-dimesional orbits are imbedded submanifolds, which are diffeomorphic either to the 2-sphere S2 or to the real projective plane P 2 (R) and carry a Riemannian metric of constant positive curvature induced by the Lorentz metric of the manifold M as a simple argument shows [16]. As SO(3) is a compact connected Lie group, the general theory of such group actions applies to spherical actions and accordingly the orbits of spherical actions can be classified as principal, exceptional and singular ones (see e. g. [3], pp. 179-182]. The principal and exceptional orbits of a smooth action of a compact connected Lie group on a connected smooth manifold are those orbits which have maximal dimension. Accordingly, the principal and exceptional orbits of a spherical action are its 2-dimensional orbits which are compact imbedded submanifolds of constant positive curvature, and consequently they are diffeomorphic either to S2 or to P 2 (R). Moreover, the principal orbits of a smooth action of a connected compact Lie group on an orientable manifold M are orientable by a fundamental result (see e. g. [3], p. 185). Therefore the principal orbits of a spherical action on an orientable space-time are orientable, and therefore they are diffeomorphic to S2 . Furthermore, if a compact connected Lie group acts smoothly on a connected smooth manifold M such that H1 (M; Z2 ) = 0, then the maximal dimensional orbits of the spherical action are orientable by a well-known fundamental result (see e. g. [3], pp 188-189). Therefore, if a spherically symmetric space-time M is such that H1 (M; Z2 ) = 0, then the principal orbits are diffeomorphic to S2 and there are no exceptional orbits. By the principal orbit type theorem ([3], pp. 179-180) the union of the principal orbits of a smooth action of a connected compact group G on a connected smooth manifold M is a connected open dense set in M. Accordingly, the union M ∗ of the principal orbits of a spherical action Φ : SO(3) × M → M will be called the principal part of the spherically symmetric space-time. The 0-dimensional orbits of a spherical action are obviously its singular orbits and also its fixed points. The connected components of the set of 0-dimensional orbits of a spherical action were shown to be timelike geodesics [16], which are called the axes of the action. Example 1.1. The Einstein-De Sitter space-time. Put M = R+ × R3 and consider the 2 smooth function φ : R+ ∋ τ 7→ τ 3 ∈ R+ . Then a 4-dimesional Lorentz manifold is defined on M by the warped product R+ ×φ R3 where a time orientation is given by R+ , thus the Einstein-De Sitter space-time is obtained (see e. g. [10], pp. 352-357). If Ξ : SO(3) × R3 → R3 is the canonical action, then a


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spherical action Φ : SO(3) × M ∋ (g, (τ, z)) 7→ (τ, Ξ(g, z)) ∈ M is obtained. Now R+ × {0} ⊂ M is an axis of the action ( see also [11], p. 261).

2

A construction of transverse submanifolds

Definition 2.1. Let z ∈ M ∗ be a point of the principal part of a spherically symmetric space-time M. Then the orbit G(z), being a submanifold on which the Lorentz metric induces a Riemannian metric, has a non-degenerate tangent space Tz G(z). Therefore, the subspace Tz⊥ G(z) ⊂ Tz M, the orthogonalizer of Tz G(z) is also a complement to Tz G(z). Thus a smooth 2-dimensional distribution N is defined on M ∗ by N : M ∗ ∋ z 7→ Tz⊥ G(z)ßTz M which is called the normal distribution induced by the spherical action Φ. The following result has been obtained in [16] concerning the normal distribution. Theorem 2.2. If SO(3) × M → M is a spherical action, then normal distribution N induced by Φ is involutive. Definition 2.3. Let Φ : G × M → M be a spherical action, then the maximal integral manifolds of the normal distribution N induced by Φ are called the leaves of the spherically symmetric space-time. Fundamental properties of leaves are given in the following theorem which was proved in [16]. Theorem 2.4. The leaves of a spherically symmetric space-time M are totally geodesic submanifolds of M. If L is a leaf then the isotropy subgroups of the spherical action in all the points of the leaf L are equal. Definition 2.5. Let z ∈ M ∗ be a point of the principal part of a spherically symmetric space-time and L ⊂ M ∗ the leaf of the spherically symmetric space-time passing through z, and H < SO(3) the subgroup which is the isotropy subgroup of the spherical action Φ in points of L. Then the connected components of the fixed point set F (H) of the subgroup H are closed totally geodesic submanifolds in M (see e. g. [9], II. , p. 61). The connected component P of F (H) containing the point z is called a tranverse submanifold of the spherically symmetric space-time. The above concept a of transverse submanifold is a special case of a more general one introduced in the affine case earlier [15] as follows: Let ∇ be a covariant derivation on a smooth manifold M and Φ : G × M → M a smooth action leaving ∇ invariant. If

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h < g is an isotropy subalgebra of Φ which has minimal dimension and Z(h) a connected component of the zero set of h such that Tx Z(h) ∩ Tx G(x) = {0x } holds for some x ∈ Z(h), then Z(h) is called a transverse submanifold of to the action Φ. In fact, if (M, ) is a spherically symmetric space-time and ∇ the Levi-Cività covariant derivation, then the spherical action leaves ∇ invariant. Moreover, if x ∈ M ∗ then the corresponding isotropy subalgebra gx = h has minimal dimension and the connected component of Z(h) containing x is the transverse submanifold P for which Tx Z(h ∩ Tx G(x) = Tx P ∩ Tx G(x) = {0x } holds considering the fact that the action Φ is spherical and the linear isotropy representation does not leave non-zero tangent vectors of principal orbits invariant. Proposition 2.6. Let z ∈ M ∗ be a point in the principal part of the spherically symmetric space-time, L ⊂ M ∗ the leaf of M passing through z, and P ⊂ M the transverse submanifold defined by L. Then L ⊂ P is an open and dense set in P . A proof of the above proposition based on the principal orbit type theorem has been given in [16]. Theorem 2.7. Let P ⊂ M be a transverse submanifold of a spherically symmetric spacetime, then M = Φ(SO(3), P ) holds. If P ′ ⊂ M is also a transverse submanifold, then there is a g ∈ SO(3) such that P ′ = Φ(g, P ) holds. A proof of the above theorem was given in [16]. Theorem 2.8. If A ⊂ M is an axis of a spherically symmetric space-time then it is included in all the transverse submanifolds of M. The above theorem which has a fundamental role in the classification of spherically symmetric space-times was also established in [16].

3

Normal spherically symmetric space-times and their classifications

Definition 3.1. A spherically symmetric space-time (M, ) is said to be normal if M is oriented, the spherical action Φ has no exceptional orbits, and the leaves are simply connected and non-compact submanifolds. The principal orbits of a normal spherically symmetric space-time are orientable, since M is oriented ([3], p. 185); therefore they are diffeomorphic to the 2-sphere. Consequently, a principal orbit G(z), z ∈ M ∗ with its Riemannian metric induced by the Lorentz metric of M is isometric to a standard


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2-sphere of radius ρ(z). Thus a function ρ : M ∗ → R+ is obtained which is called the radial function of the normal spherically symmetric spacetime. As to the fundamental properties of radial functions, the following proposition has been proved in [16]. Proposition 3.2. The radial function of a normal spherically symmetric space-time is smooth and invariant under the spherical action. Concerning the axes in normal spherically symmetric space-times, the following proposition has been established in [16]. Proposition 3.3. A normal spherically symmetric space-time can have at most 1 axis. Definition 3.4. A normal spherically symetric space-time is said to be of the EinsteinDe Sitter type if it has one axis and of the Schwarzschild type if it has no axis. Thus, if M is of the Einstein-De Sitter type and A ⊂ M is its axis, then M = A ∪ M∗ holds; and if M is of the Schwarzschild type then M = M ∗ is valid. Theorem 3.5. If M is a normal spherically symmetric space-time then its principal part is canonically isometric to the warped product L ×ρ S2 where L ⊂ M ∗ is a leaf of the space-time. The above theorem which presents a solution to the fundamental question raised by C.J.S. Clarke [4], was also given in [16].

4

The existence of not normal spherically symmetric spacetimes

In what follows a not normal spherically symmetric space-time is presented which was originally given by C.J.S. Clarke [4]. Let HS2 ßS2 be a closed hemisphere with pole o ∈ HS2 and (θ, φ), 0 ≤ θ ≤ π2 , 0 ≤ φ < 2π its polar coordinate system where o is given by ( π2 , φ), 0 ≤ φ < 2π. Then a c = R2 × HS2 . Let (τ, r) be 4-dimensional smooth manifold with boundary is given by M

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c→M c the canonical coordinate system of R2 . A properly discontinuous action Γ : Z × M is generated by the element 1 ∈ Z as follows: c ∋ (1, (τ, r, θ, φ)) 7→ (τ + π, −r, θ, φ)ßM. c Γ:Z×M

c/Z of the action Γ is canonically diffeomorphic to B × HS2 where B The orbit space M is the Moebius band and it is again a 4-dimensional smooth manifold with boundary. c let [(τ, r, θ, φ)] be the corresponding point of M c/Z. An equivalence For (τ, r, θ, φ) ∈ M c/Z as follows: 1. If 0 ≤ θ < π then [(τ, r, θ, φ)] ∼ [(τ ′ , r ′ , θ′ , φ′ )] relation ∼ is defined on M 2

if and only if

[(τ, r, θ, φ)] = [(τ ′ , r ′ , θ′ , φ′ )]. 2. If θ = π2 , then [(τ, r, π2 , φ)] ∼ [(τ ′ , r ′, π2 , φ′ )] if and only if π π π [(τ ′ , r ′ , , φ′)] = [(τ + , r, , φ)]. 2 2 2 c/Z)/ ∼ with the corresponding identification Consider now the quotient space M = (M

toplogy. Then M is a 4-dimensional manifold without boundary on which a smooth manifold structure is canonically defined. In order to obtain a spherical action on M, ¯ ∈ o(3) consider the corresponding consider the Lie algebra o(3) of SO(3). Then for each X infinitesimal generator X : S2 → T S2 of the canonical action of SO(3), and restrict it to the interior H0 S2 of the closed hemisphere, then extended to R2 × H0 S2 by the canonical imbeddings of H0 S2 in the product; thus a smooth vector field e : B × H0 S2 → T (B × H0 S2 ) X

b : M → T M. The is obtained, which in turn induces a a unique smooth vector field X b for X ¯ ∈ o(3) generate a spherical action vector fields X Φ : SO(3) × M → M

such that the tranverse submanifolds being diffeomorphic to the Moebius band B are not simply connected.

References [1] P.G. Bergmann, M. Cahen and A.B. Komar: “Spherically symmetric gravitational fields“, J. Math. Phys., (1965), pp 1–5. [2] G.D. Birkhoff: Relativity and Modern Physics, Cambridge, 1923. [3] G.E. Bredon: Introduction to Compact Transformation Groups, New York, 1972. [4] C.J.S. Clarke: “Spherical symmetry does not imply direct product“, Class. Quantum Grav., Vol. 4, (1987), pp. 37–40.


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[5] W. De Sitter: “On Einstein’s theory of gravitation and its astronomical consequences“, Mon. Not. Roy. Astr. Soc., Vol. 76, (1917), p. 699. [6] J. Ehlers: Relativity, Astrophysics and Cosmology, Dordrecht, 1973. [7] A. Einstein: “Zur allgemeinen Relativitätstheorie“, Sitzungsb. Preuss. Akad. Wiss. ; Phys.-Math. Kl., (1915), pp. 778–779. [8] S.W. Hawking and G.F.R. Ellis: The Large Scale Structure of Space-time, London, 1973. [9] S. Kobayashi and K. Nomizu: Foundations of Differential Geometry I, II, New York, 1963, 1969. [10] B. O’Neill: Semi-Riemannian Geometry with Applications to Relativity, New York, 1983. [11] R.K. Sachs and H. Wu: General Relativity for Mathematicians, New York, 1977. ¨ [12] K. Schwarzschild: “Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie“, Sitzungsb. Preuss. Akad. Wiss.; Phys.-Math. Kl., (1916), pp. 189–196. [13] R. Siegl: “Some underlying manifolds of the Schwarzschild solution“, Class. Quantum Grav., Vol. 9, (1992), pp. 239–240. [14] J.L. Synge: Relativity: The General Theory, Amsterdam, 1960. [15] J. Szenthe: “A construction of transverse submanifolds“, Univ. Iagell. Acta Math., Vol. 41, (2003), To appear. [16] J. Szenthe: “On the global geometry of spherically symmetric space-times“, Math. Proc. Camb. Phil. Soc., Vol. 137, (2004), pp. 297–306.

CEJM 2(5) 2004 732–753

Quaternionic and para-quaternionic CR structure on (4n+3)-dimensional manifolds Dmitri Alekseevsky1∗ , Yoshinobu Kamishima2† 1

Department of Mathematics, Hull University, Cottingham Road HU6 7RX, Hull, England 2 Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan

Received 15 December 2003; accepted 7 March 2004 Abstract: We define notion of a quaternionic and para-quaternionic CR structure on a (4n+3)dimensional manifold M as a triple (ω1 , ω2 , ω3 ) of 1-forms such that the corresponding 2-forms satisfy some algebraic relations. We associate with such a structure an Einstein metric on M and establish relations between quaternionic CR structures, contact pseudo-metric 3-structures and pseudo-Sasakian 3-structures. Homogeneous examples of (para)-quaternionic CR manifolds are given and a reduction construction of non homogeneous (para)-quaternionic CR manifolds is described. c Central European Science Journals. All rights reserved.

Keywords: Quaternionic K¨ ahler structure, contact structure, CR-structure, integrability, complex structure, Sasakian 3-structure, para-quaternions MSC (2000): 53C55, 57S25

1

Introduction

In this paper, we shall study the geometry of quaternionic CR-manifolds (respectively para-quaternionic CR -manifolds) i.e. manifolds M, which are equipped with the globally defined sp(1)-valued (or, respectively sp(1, R)-valued ) 1-form ω under some conditions. (Compare with [4] for the notion of quaternionic CR-structure and the related work.) First of all we simply extract the conditions of the real 1-form ωα (α = 1, 2, 3) repre∗ †


D. Alekseevsky, Y. Kamishima / Central European Journal of Mathematics 2(5) 2004 732–753

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senting a quaternionic CR-structure from [2] in which we have introduced an integrable, nondegenerate, quaternionic Carnot-Carathéodory (Q C-C) structure B on a (4n + 3)manifold M more generally rather than the global case. In §2, we define and study the basic properties of (para) quaternionic CR-manifolds. The real 1-forms {ωα }α=1,2,3 define a codimension 3-subbundle H of T M on which there exists a quaternionic or, respectively, para-quaternionic structure {Jγ }γ=1,2,3 . In §3 we describe relations between quaternionic CR structure, contact pseudo - (Riemannian) metric 3-structure, pseudo-Kcontact 3-structures and pseudo-Sasakian 3-structures. We associate with a quaternionic CR structure three (integrable ) CR structures J¯α , α = 1, 2, 3 and a pseudo-Riemannian metric g1 . We prove that these objects are consistent and define a pseudo-Sasakian 3structure. In particular, the metric g1 is an Einstein metric. More generally, we prove that quaternionic CR structures are equivalent to the pseudo-Sasakian 3-structure. As by-product, we get the following result. Theorem 3.3. A contact pseudo-metric 3-structure is automatically a pseudo-Sasakian 3-structure of type (3 + 4p, 4q). This is a generalization of the results, obtained by Tanno [17], Jelonek [11] and Kashiwada [13]. The key point is a generalization of the Hitchin Lemma (cf. Proposition 2.5) which implies the integrability of the almost complex structures J¯α (α = 1, 2, 3) on corresponding codimension 1-contact subbundle. In §4 we indicate another proof of the result, that the natural metric, associated with a (para)-quaternionic CR structure is Einstein (cf. Theorem 4.1). It is based on the O’Neill’s formulas for Riemannian submersion with totally geodesic fibers. In §5 we assume that the action of 3-dimensional Lie algebra of vector fields, associated with a (para)-quaternionic CR manifold can be integrated to an almost free action of the corresponding Lie group H 1 . We prove that in the case of proper free action the orbit space M/H 1 carries a (para)-quaternionic Kähler geometry. We gave some homogeneous examples of manifolds with (para)-quaternionic CR structure in §6. In §7 we show that the reduction method works smoothly for manifolds with a (para)-quaternionic CR structure and allows us to construct non homogeneous (para)-quaternionic CR manifolds starting from (para)-quaternionic CR manifolds with a proper group of symmetries. In §8 we consider ε-hyperKähler manifolds. We fix some notations and give a unified descriptions of two algebras of quaternions. We denote by Hε the associative 4-dimensional algebra over R with the standard basis 1, i1 = i, i2 = j, i3 = k and multiplication i2 = −1, j 2 = k 2 = ε, jk = −kj = −εi, where ε = ±1. Then i, j, k anti-commute and ij = k, ki = j. For ε = −1 we get the algebra H = H−1 of quaternions and for ε = +1 the algebra H′ = H+1 is the split-quaternions . We shall refer to this last algebra also p as algebra of paraquaternions. The algebra Hε admits a multiplicative “norm” |x| = (x¯ x) such that |xy| = |x||y| where x¯ = x0 − x1 i − x2 j − x3 k

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is the quaternionic conjugation of x = x0 + x1 i + x2 j + x3 k associated with the Hermitian scalar product < x, y >= x¯ y which is positively defined for ε = −1 and has split signature (2, 2) in the para case when ε = +1. The group H1ε of unit quaternions is isomorphic 0 to Sp1 for ε = −1 and SO2,1 = Sl2 (R) for ε = 1. (Here 0 indicates the connected component of the group). The left multiplication defines an irreducible representation of H1ε in Hε = R4 and the adjoint action Da : x 7→ axa−1 = ax¯ a gives the 3-dimensional 1 3 representation of Hε in the space ImHε = R of imaginary quaternions. Note that the linear group DH1ε is the connected group of automorphisms of Hε . The Lie algebra h1ε of H1ε is identified with the space ImHε of imaginary quaternions with respect to the Lie bracket [x, y] = xy − yx. The commutator relations with respect to the standard basis i, j, k are [i, j] = 2k, [j, k] = −2εi [k, i] = 2j. Let π : M → B is a principal bundle with the structure group H1ε = Sp1 or Sl2 (R). A connection in π is defined by a horizontal h1ε -valued 1-form ω on M which is an extension of the natural vertical parallelism, i.e. natural identification of vertical spaces Txv M with h1ε . With respect to the standard basis i, j, k of h1ε , the form ω = ω1 i + ω2 j + ω3 k splits into a triple (ω1 , ω2 , ω3 ) of scalar forms. Similarly, the curvature form ρ := dω + [ω ∧ ω] splits into a triple of scalar 2-forms (ρ1 , ρ2 , ρ3 ) where ρ1 = dω1 − 2εω2 ∧ ω3 , ρ2 = dω2 + 2ω3 ∧ ω1 , ρ3 = dω3 + 2ω1 ∧ ω2 . Assume that the curvature forms ρα , α = 1, 2, 3 are non-degenerate on the horizontal distribution H = Ker ω. Then we can construct three fields Jα of endomorphisms of H setting J1 = −ε(ρ3 |H )−1 ◦ ρ2 |H , J2 = (ρ1 |H )−1 ◦ ρ3 |H , J3 = (ρ2 |H )−1 ◦ ρ1 |H . We are interested in the case when these three endomorphisms Jα define a representation of the quaternion algebra Hε , that is they anti-commute and satisfy the quaternionic relations J12 = −εJ22 = −εJ32 = −1, J2 J3 = −εJ1 . We will show that a connection ω which satisfies these algebraic conditions for the curvature ρ is closely related with contact metric 3-structure (in particular, Sasakian 3 structures) and determines Einstein metrics. In the next section we give a more general definition of such structure in terms of three 1-forms ωα which satisfy the structure equations.

2

Quaternionic and para-quaternionic CR structure: definition and the main properties

Let ω = (ω1 , ω2 , ω3 ) be a triple of scalar 1-forms on a (4n + 3)-dimensional manifold M which are linearly independent, i.e. ω1 ∧ ω2 ∧ ω3 6= 0. We associate with ω a triple


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ρ = (ρ1 , ρ2 , ρ3 ) where ρ1 = dω1 − 2εω2 ∧ ω3 , ρ2 = dω2 + 2ω3 ∧ ω1 , ρ3 = dω3 + 2ω1 ∧ ω2 , and ε = +1 or −1. Definition 2.1. A triple of linearly independent 1-forms ω = (ω1 , ω2 , ω3 ) is called a εquaternionic CR structure (ε = +1 or −1) if the associated 2-forms ρα , α = 1, 2, 3 satisfy the following conditions : (1) They are non degenerate on the codimension three distribution H = Ker ω1 ∩Ker ω2 ∩ Ker ω3 and have the same 3-dimensional kernel V , (2) The three fields of endomorphisms Jα of the distribution H, defined by J1 = −ε(ρ3 |H )−1 ◦ ρ2 |H , J2 = (ρ1 |H )−1 ◦ ρ3 |H , J3 = (ρ2 |H )−1 ◦ ρ1 |H , anti-commute and satisfy the ε-quaternionic relations J12 = −εJ22 = −εJ32 = −1, J2 J3 = −εJ1 . For ε = −1, the ε-quaternionic CR structure is called also a quaternionic CR structure and ε = +1 quaternionic CR structure is called also para-quaternionic CR structure. The manifold M with an ε-quaternionic CR structure is called ε-quaternionic CR manifold. We will see that there exists a big similarity between quaternionic and para-quaternionic CR manifolds. The distribution H is called the horizontal distribution and V the vertical distribution of ε-quaternionic CR manifold. It follows from the definition that T M = V ⊕ H. Using this direct sum decomposition of the tangent bundle, we define an one-parameter family of pseudo-Riemannian metrics gt , t ∈ R+ on a ε-quaternionic CR manifold M by gt = gVt + gH

(1)

gVt = t(ω1 ⊗ ω1 − εω2 ⊗ ω2 − εω3 ⊗ ω3 ) X =t ε α ωα ⊗ ωα

(2)

where

gH = ρ1 ◦ J1 = ρ2 ◦ J2 = −ερ3 ◦ J3 , and ε1 = 1, ε2 = ε3 = −ε. Note that 1-forms ωα |x , α = 1, 2, 3 at a point x ∈ M form a basis of the dual space Vx∗ ⊂ Tx∗ M. We denote by ξα |x the dual basis of Vx . Then ξα , α = 1, 2, 3 are vertical vector fields such that ωβ (ξα ) = δαβ . We have gt ◦ ξ1 = tω1 ,

gt ◦ ξβ = −tεωβ

for β = 2, 3.

We will denote by LX the Lie derivative with respect to a vector field X.

(3)

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Lemma 2.2. (1) The vector fields ξα preserve the decomposition T M = V ⊕H and span a 3-dimensional Lie algebra hε of Killing fields of the metric gt for t > 0, which is isomorphic to sp(1, R) for ε = 1 and sp(1) for ε = −1. More precisely, the following commutator relations hold: [ξ1 , ξ2 ] = 2ξ3 , [ξ2 , ξ3 ] = −2εξ1 , [ξ3 , ξ1 ] = 2ξ2 . (2) The vector field ξα preserves the forms ωα and ρα for α = 1, 2, 3. Moreover, the following relations hold : Lξ2 ω3 = −Lξ3 ω2 = ω1 , Lξ3 ω1 = εLξ1 ω3 = −εω2 , Lξ1 ω2 = εLξ2 ω1 = ω3 , and similar relations for ρα . Proof. Using the formula LX = dιX + ιX d for the Lie derivative, we have for any horizontal vector field h ∈ Γ(H) and a vector field ξ ∈ {ξ1, ξ2 , ξ3 } : (Lξ ωα )(h) = ((dιξ + ιξ d)ωα )(h) = dωα (ξ, h) = (ρα − 2δα ωβ ∧ ωγ )(ξ, h) = 0. (δ1 = −ε, δ2 = δ3 = 1). Here (α, β, γ) is a cyclic permutation of (1, 2, 3). This shows that ξα preserves the horizontal distribution H. Now we prove that ξα preserves the vertical distribution V . Using the fact that ξα preserves the horizontal distribution we get for any h ∈ Γ(H): ρα ([ξβ , ξγ ], h) = dωα ([ξβ , ξγ ], h) + 2δα ωβ ∧ ωγ ([ξβ , ξγ ], h) 1 = − ωα ([[ξβ , ξγ ], h]) 2 1 = (ωα ([[ξγ , v], ξβ ]) + ωα ([[ξβ , h], ξγ ])) (by Jacobi identity) 2 = 0. This shows that [ξα , ξβ ] is a linear combination of ξ1 , ξ2 , ξ3 . To determine the coefficients, we calculate for ε = −1 0 = ρα (ξβ , ξγ ) = dωα (ξβ , ξγ ) + 2ωβ ∧ ωγ (ξβ , ξγ ) 1 = (−ωα ([ξβ , ξγ ]) + 2), 2 that is ωα ([ξβ , ξγ ]) = 2. Similarly, we can check that ωα ([ξα , ξβ ]) = 0. This proves the relations of [ξα , ξβ ] = 2δγ ξγ . The relations of (2) follow now immediately. Since ξα preserves V and ωα , it preserves also the vertical part gtV of the metric gt due to the equation (3). One can easily check that ξα preserves the field Jα considered as a field of endomorphisms of T M which is zero on V . This implies that ξα preserves also the horizontal part gH = ±ρα ◦ Jα of the metric gt . Using the Koszul formulas for the covariant derivative ∇t of the metric gt , we get


Corollary 2.3. (1) ∇tξα ξβ = −∇tξα ξβ = 21 [ξα , ξβ ] ∇tξα ξα = 0,

∇tξ1 ξ2 = ξ3 ,

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for α, β ∈ {1, 2, 3}, that is

∇tξ2 ξ3 = −εξ1 , ∇tξ3 ξ1 = ξ2 ;

(2) ∇tξ1 ω2 = ω3 , ∇tξ2 ω3 = −εω1 , ∇tξ3 ω1 = ω2 ; (3) ∇tξα ρα = 0, ∇tξ1 ρ2 = −∇tξ2 ρ1 = ρ3 , ∇tξ2 ρ3 = −∇tξ3 ρ2 = −ερ1 , ∇tξ3 ρ1 = −∇tξ1 ρ3 = ρ2 . Proof. (1) follows from the Koszul formulas, (2) follows from (1), since g t ◦ ξα = −εδα tωα and ∇t preserves gt (δ1 = −ε, δ2 = δ3 = 1). Now we calculate, for example, ∇tξ1 ρ2 as follows : ∇tξ1 ρ2 = ∇tξ1 (dω2 + 2ω3 ∧ ω1 ) = dω3 + 2∇ξ1 ω3 ∧ ω1 = dω3 − 2ω2 ∧ ω1 = ρ3 . The proof of other identities is similar. Now we extend endomorphisms Jα , α = 1, 2, 3 of the horizontal distribution H to endomorphisms J¯α of the all tangent bundle by the following conditions : J¯α ξα = 0, J¯α |H = Jα J¯1 ξ2 = −εξ3 , J¯1 ξ3 = εξ2 , J¯2 ξ3 = ξ1 , J¯2 ξ1 = εξ3 , J¯3 ξ1 = ξ2 , J¯3 ξ2 = εξ1 .

(4)

Note that the endomorphisms J¯α , α = 1, 2, 3 at a point x constitute the standard basis of the Lie algebra h1ε ⊂ End(Tx M). Using Lemma 2.2 and Corollary 2.3 we can prove Lemma 2.4. The vector filed ξα preserves the field of endomorphism J¯α for α = 1, 2, 3 and the Lie derivatives of J¯α with respect to ξβ are given by Lξ1 J¯2 = −Lξ2 J¯1 = −εJ¯3 , Lξ2 J¯3 = εLξ3 J¯2 = −J¯1 , Lξ3 J¯1 = εLξ1 J¯3 = J¯2 . The following proposition shows that the restriction of the field of endomorphisms ¯ Jα , α = 1, 2, 3 to the (non-holonomic) codimension one distribution Tα = Ker ωα are integrable. This means that the Nijenhuis tensor N(J¯α , J¯α )Tα = 0 or, equivalently, the eigendistributions Tα± of J¯α |Tα are involutive. We remark that J¯1 , J¯2 , J¯3 are gt -skew symmetric anticommuting endomorphisms with one-dimensional kernel Tα⊥ . Moreover, in the case ε = −1 , J¯α |Tα is a complex structure in Tα and in the case ε = 1 J¯1 |T1 is a complex structure and J¯2 |T2 and J¯3 |T3 are involutive endomorphisms (i.e. has square +1). Proposition 2.5. Let (M, ω1 , ω2 , ω3 ) be a ε-quaternionic CR manifold. Then the above defined field of endomorphisms J¯1 |T1 , J¯2 |T2 , J¯3 |T3 are integrable.

738


Proof. First of all, note the following formula (cf. [15]): LX (ιY dωα ) = ι(LX Y ) dωα + ιY LX dωα = ι[X,Y ] dωα + ιY LX dωα (∀X, Y ∈ T M).

(5)

Secondly, we remark that if X ∈ H, then ιX dω2 = −ειJ1 X dω3 , (6)

ιX dω3 = ιJ2 X dω1 , ιX dω1 = ιJ3 X dω2 .

Then the proof is based on the following Lemma which is a generalization of Lemma by Hitchin. Lemma 2.6. (i) Let T1 ⊗ C = T1+ + T1− be the eigenspace decomposition of the complexified distribution T1 ⊗ C = Kerω1 ⊗ C with respect to the endomorphism J¯1 with eigenvalues +i, −i. Then ι[X,Y ] dω2 = −εiι[X,Y ] dω3 ,

∀X, Y ∈ T1+ ;

(ii) For ε = 1, let Tα = Tα+ + Tα− be the eigenspace decomposition of Tα , α = 2, 3 with respect to the endomorphism J¯α with eigenvalues +1 and −1. Then ι[X,Y ] dω3 = ι[X,Y ] dω1 , ∀ X, Y ∈ T2+ ; ι[X,Y ] dω2 = ι[X,Y ] dω1 , ∀ X, Y ∈ T3+ . Proof. We prove (i). Let X ∈ T1+ such that J1 X = iX. Then LX dω2 = (dιX + ιX d)dω2 = d(ιX dω2 ) = −εd(ιJ1 X dω3 ) (by (6)) = −εi(dιX )dω3

(7)

= −εi(LX − ιX d)dω3 = −εiLX dω3 . Applying Y ∈ T1+ to the equation (5) we get LX (ιY dω2 ) = LX (−ειJ1 Y dω3 ) = −εiLX (ιY dω3 ) = −εi(ι[X,Y ] dω3 + ιY LX dω3 ) ((5))

(8)

= −εiι[X,Y ] dω3 + ιY LX dω2 ((7)) Since LX (ιY dω2 ) = ι[X,Y ] dω2 + ιY LX dω2 by (5), comparing this with (8), we obtain −εiι[X,Y ] dω3 = ι[X,Y ] dω2 .


3

739

Quaternionic CR structure and contact pseudo-metric 3-structure

We will show here that the quaternionic CR structure (that is (ε = −1)-quaternionic CR structure) is equivalent to the contact (pseudo-Riemannian) metric 3-structure and, moreover, any (pseudo-Riemannian) metric 3-structure is in fact Sasakian 3-structure. The last statement is a generalization of results, obtained by Tanno [17], [18], Jelonek [11] and Kashiwada [13]. We recall the classical definitions of contact metric 3-structure, normal contact metric structure and 3-Sasakian structure . Definition 3.1. (Tanno [17],[18], Blair [5])A contact pseudo-metric 3-structure {g, (ηα, ξα , φα ), α = 1, 2, 3} on a (4n + 3)-manifold M consists of a pseudo-Riemannian metric g of signature (3 + 4p, 4q); p + q = n, contact forms ηα , the dual vector fields ξα = g −1 (ηα ) and endomorphisms φα , which satisfy the following conditions: (1) g(ξα, ξβ ) = δαβ ; (2) φ2α (X) = −X + ηα (X)ξα , φα (ξα ) = 0, dηα (X, Y ) = g(X, φαY ); (3) g(φαX, φα Y ) = g(X, Y ) − ηα (X)ηα (Y ); (4) φα = φβ φγ − ξβ ⊗ ηγ = −φγ φβ + ξγ ⊗ ηβ , where (α, β, γ) is a cyclic permutation of (1, 2, 3). A contact pseudo-metric 3-structure {g, (ηα , ξα, φα )} is called a pseudo-K-contact 3-structure if (5) the vector fields ξα are Killing fields with respect to g. A pseudo-K-contact 3-structure is called pseudo-Sasakian 3-structure if (7) it is normal, i.e. if the following tensors Nηα (·, ·), (α = 1, 2, 3) vanish: N ηα (X, Y ) := Nφα (X, Y ) + (Xηα (Y ) − Y ηα (X))ξα

(9)

(∀ X, Y ∈ T M). Here Nφα (X, Y ) = [φα X, φα Y ] − [X, Y ] − φα [φα X, Y ] − φα [X, φα Y ] is the usual Nijenhuis tensor of a field of endomorphisms φα . Remark 3.2. (1) Some historical explanation may be needed. When g is a Riemannian metric (the case q = 0), the above set (g, ηα , ξα , φα ) is called a contact metric 3-structure. If, in addition, each ξα is Killing with respect to g, it is called a K-contact 3-structure. A K-contact 3-structure with normality condition is called a Sasakian 3-structure. Tanno had proved that a K-contact 3-structure on 7-dimensional manifold is always a Sasakian 3-structure. Later, this result was generalized by Jelonek, who proved that any pseudoRiemannian K-contact 3-structure in the case when it comes from a quaternionic Kähler metric of positive or negative scalar curvature [11] is a pseudo-Sasakian 3-structure. Recently, Kashiwada [13] has shown this result for contact (positive) metric 3-structures (not necessarily K-contact). It is natural to ask whether this will be true also for any contact pseudo-Riemannian metric 3-structures.

740


The following theorem gives an affirmative answer on this question. Theorem 3.3. The following three structures on a (4n + 3)-dimensional manifold M are equivalent: contact pseudo-metric 3-structures, quaternionic CR structures and pseudoSasakian 3-structures. Proof. Since any pseudo-Sasakian 3-structure is a contact metric 3-structure, we have to prove that i) if (g, ηα, ξα , φα ) is a contact pseudo-metric 3-structure, then ωα = ηα is a quaternionic CR structure, and ii) if (ωα ) is a quaternionic CR structure, then the structure (g = g1 , ηα = ωα , ξα , φα = J¯α ) defined by the equations (1), (3), (4) is a pseudo-Sasakian 3-structure. i) Let (g, ηα , ξα , φα ) be a contact pseudo-metric 3-structure. We have to prove that 1-forms ωα = ηα satisfy conditions (1), (2) of Definition 2.1. It follows from the definition that 2-forms dωα = dηα are non-degenerate on the codimension three distribution H = 3

∩ Ker ηα and T M = {ξ1 , ξ2 , ξ3} ⊕ H. The conditions (2),(3) of Definition 3.1 show that α=1 2-forms ρα = dηα + 2ηβ ∧ ηγ have the kernel V = span(ξ1 , ξ2, ξ3 ). This proves (1). To prove (2) it is sufficient to check that Jα = (ργ |H )−1 ◦ (ρβ |H ) = φα |H . From (2) of Definition 3.1, we have ρα (X, Y ) = dηα (X, Y ) = g(X, φa Y ) for X, Y ∈ H.

(10)

The left hand side is equal to ρβ (Jγ X, Y ). The right hand side can be rewritten as g(X, φαY ) = g(φγ X, φγ (φα Y )) (by (4) of Definition 3.1) = g(φγ X, φβ Y ) = ρβ (φγ X, Y ).

(11)

Since ρα |H is non-degenerate, we conclude that Jγ = φγ on H. Since φα |H satisfies the quaternionic relations, this proves (2). ii) Let now (ωα ) be a quaternionic CR structure. We have to check that the associated structure (g = g1 , ηα = ωα , ξα , φα = J¯α ) satisfy conditions (1)-(7) of Definition 3.1. The conditions (1)-(5) follow directly from the definition of a quaternionic CR structure. The condition (6) is proved in (2) of Lemma 2.2. Now we check the normality condition N ηα (X, Y ) = Nφα (X, Y ) + (Xηα (Y ) − Y ηα (X))ξα = 0 for φα := J¯α . By Proposition 2.5, Nφα (X, Y ) = 0, ∀X, Y ∈ Ker ηα = {ξβ , ξγ } ⊕ H. This shows that N ηα = 0 on Ker ηα . Since T M = {ξα} ⊕ Ker ηα , it remains to check that N ηα (ξα , X) = 0 for a local vector field X ∈ Ker ηα . We have N ηα (ξα , X) = −([ξα , X] + J¯α [ξα , J¯α X]).


741

Since Lξα J¯α = 0 by Lemma 2.4, we have N ηα (ξα , X) = −([ξα , X] + J¯α [ξα , J¯α X]) = −(Lξα X + J¯α Lξα J¯α X) = 0. Hence the normality condition N ηα = 0 holds.

4

Einstein metric associated with ε-quaternionic CR structure

We have proved that (ε = −1)-quaternionic CR structure ω = (ωα ) on a (4n + 3)dimensional manifold M defines a pseudo-Sasakian 3-structure (g, ηα , ξα, φα ) where g = P g1 = ωα ⊗ ωα + ρ1 ◦ J1 , ηα = ωα , φα = J¯α . It is known that the metric g of a Sasakian 3-structure is Einstein with the Einstein constant 2(2n + 1). Tanno [18] remarked that this result remains true also for pseudo-Sasakian structure. It is natural to expect that the result can be generalized also for the metric g1 associated with para-quaternionic CR structure. The following theorem shows that this is true. Theorem 4.1. Let (M, ω = (ωα )) be a ε-quaternionic CR manifold. Then the metric g = g1 is an Einstein metric. Proof. Lemma 2.2 implies that the orbits of the Lie algebra h1ε (that is maximal integrable submanifolds of the vertical distribution V ) are totally geodesic submanifolds of (M, gt ) for t > 0. To simplify the notations, we will assume that h1ε consists of complete vector fields. Then it defines an isometric action of the Lie group Hε1 with a discrete stabilizer. We will assume that the action of Hε1 is proper. Then the orbit space M/Hε1 is an orbifold . Deleting the singular points, we get a smooth fibration π : Mreg → B ⊂ M/Hε1 which is a Riemannian submersion with respect to the induced metric on B. For brevity, we will assume that π : M → B = M/Hε1 is a Riemannian submersion (with totally geodesic fibers). Then we can use O’Neill’s formulas which relate the Ricci curvature rict of (M, gt ) with the Ricci curvature rict V of the fiber and the Ricci curvature ricB of the base manifold B. Since O’Neill’s formulas are purely local, without loss of generality, it can be written as in [3]: rict (ξ, ξ ′) = rict V (ξ, ξ ′) + t2 g(Aξ, Aξ ′), rict (X, Y ) = ricB (X, Y ) − 2tg(AX , AY ), rict (X, ξ) = tg(δA)X, ξ), for any vectors ξ, ξ ′ ∈ Vx , X, Y ∈ Hx , where 1 AX Y = [X, Y ]v = (∇X Y )v , 2 g(Aξ, Aξ ′) =

X

g(AXi ξ, AXi ξ ′),

g(AX , AY ) =

X

(AX ξα , AY ξα ),

742


δA = −

X

(∇Xi A)Xi .

Here ∇ is the covariant derivative of the metric g1 , Xi (respectively, ξα ) is an orthonormal basis of Hx (respectively, Vx ) for x ∈ M and X v stands for the vertical part of a vector X. To apply these formulas, we calculate the Nomuzu operator Ltξα = −∇t ξα ∈ End(T M) associated with the Killing field ξα where ∇t is the covariant derivative of the metric gt . Lemma 4.2. Ltξ1 |H := −∇t ξ1 = tJ1 , Ltξ2 |H = tJ2 , Ltξ3 |H := −∇t ξ1 = −εJ3 , Ltξ1 |V = J¯1 |V , Ltξ2 |V = J¯2 |V , Ltξ3 |V = −εJ¯3 |V . In particular, for t = 1 Lξα := L1ξα = εα J¯α ,

(α = 1, 2, 3, ε1 = 1, ε2 = ε3 = −ε).

Proof. We recall the following Koszul formula for the covariant derivative : 2g(∇X Y, Z) = g([X, Y ], Z) − g([X, Z], Y ) − g(X, [Y, Z]) + X · g(Y, Z) + Y · g(X, Z) − Z · g(X, Y )

(12)

where X, Y, Z are vector fields on a Riemannian manifold (M, g). Applying this formula to the metric gt for Y = ξα and horizontal vector fields Y, Z, and using the formula gt ◦ ξα = tεα ωα , where ε1 = 1, ε2 = ε3 = −ε, we get 2gt (∇X ξα , Z) = −gt (ξα , [X, Z]) = −tεα ωα ([X, Z]) = 2εα dωα (X, Z) = 2εα ρα (X, Z). Now the result follows from the identities gH = ρ1 ◦ J1 = ρ2 ◦ J2 = −ερ3 ◦ J3 . Corollary 4.3. For X, Y ∈ H , the following formulas hold: (i) g1 (AX Y, ξα ) = g1 (Lξα X, Y ) = g1 (Jα X, Y ), (ii) g1 (AX , AX ) = 3g1 (X, X), g1 (Aξα , Aξβ ) = 4nεα δαβ , (iii) g1 ((δA)X, ξα ) = 0. Proof. (i)

(13)


743

g1 (AX Y, ξα ) = g1 (∇X Y, ξα ) = −g1 (Y, ∇X ξα ) = g1 (Jα X, Y ), (ii)

X εα (Ax ξα , AX ξα ) = εα g1 (Jα X, Jα X) = 3g1 (X, X), X X g1 (Aξα , Aξβ ) = g1 (AXi ξα , AXi ξβ ) = g1 (Jα Xi , Jβ Xi ) = 4nεα δαβ .

g1 (AX , AX ) =

X

The last equation can be checked, using the formulas for the covariant derivatives of ρα . Lemma 4.4. The Ricci curvature of the restriction gVt of the metric gt to the fiber F is 2 given by rict V = gVt . t Proof. We remark that the restriction gVt of the metric gt to the fiber F ≃ Hε1 is proportional to the standard metric B on Hε1 associated with Killing form, more precisely, 1 gVt = −t B. It is known that the Ricci curvature of a bi-invariant metric on a semisimple 8 Lie group is equal to − 14 B. This implies Lemma. To finish the proof of Theorem, we have to calculate the Ricci curvature ricB of the base manifold B = M/Hε1 . The answer is given in the following proposition, which is known in the case ε = −1 [18]. The proof in the case ε = 1 is similar. Proposition 4.5. The metric gB induces by the metric gt on the base manifold B = M/Hε1 is Einstein with Einstein constant λ = 4(n + 2), that is ricB = 4(n + 2)gB . Collecting all these results, we conclude that the for any t > 0 the Ricci operator of the metric gt act as a scalar on the vertical space V and horizontal space H. Moreover, for t = 1 it is a scalar operator and hence, g1 is an Einstein metric. This finishes the proof of Theorem.

5

ε-quaternionic CR manifolds and ε-quaternionic K¨ ahler manifolds

Recall that a (pseudo-Riemannian) quaternionic Kähler manifold (respectively, paraquaternionic Kähler manifold) is defined as a 4n-dimensional pseudo-Riemannian manifold (M, g) with the holonomy group H ⊂ Sp(1)Sp(p, q) (respectively, H ⊂ Sp(1, R) Sp(n, R)). This means that the manifold M admits a parallel 3-dimensional subbundle Q (quaternionic subbundle) of the bundle of endomorphisms which is locally generated by three skew-symmetric endomorphisms J1 , J2 , J3 which satisfy the quaternionic relations (respectively, para-quaternionic relations). To unify the notations, we shall call a quaternionic Kähler manifold also a (ε = −1)-quaternionic Kähler manifold and a paraquaternionic Kähler manifold a (ε = 1)-quaternionic Kähler manifold.

744


It is known that any ε-quaternionic Kähler manifold is Einstein and its curvature tensor has the form R = νR1 + W , where ν is a constant (proportional to the scalar curvature), R1 is the curvature tensor of the canonical metric on the (para) quaternionic projective space and W is a quaternionic Weyl tensor such that curvature operators W (X, Y ) commute with Q. Let (M, ω) be a ε-quaternionic CR manifold. We will assume that the Lie algebra 1 hε = span(ξα ) of vector fields is complete and generates a free action of the group Hε1 (or its quotient by a discrete central subgroup) on M. Then the orbit space B = M/Hε1 is a smooth manifold and π : M → B is a principal bundle. Moreover, the pseudoRiemannian metric g1 of (M, ω) induces a pseudo-Riemannian metric gB on B such that π : M → B is a Riemannian submersion with totally geodesic fibers. We will give a sketch of the proof of the following theorem. Theorem 5.1. Let (M, ω) be a ε-quaternionic CR manifold, ε = ±1. Assume that the Lie algebra h1ε generates a free action of the corresponding Lie group Hε1 on M. Then the orbit space B = M/Hε1 has the natural structure of ε-quaternionic K¨ ahler manifold. Conversely, let (B, gB , Q) be a ε-quaternionic K¨ ahler manifold and SB the Sasakian bundle of frames (J1 , J2 , J3 ) of the quaternionic bundle Q which satisfy ε-quaternionic relations. Then SB has the natural structure of a ε-quaternionic CR manifold. Proof. Let (M, ω) be a ε-quaternionic CR manifold and π : M → B the projection on the orbit space. The projection π induces an isometry π∗ of a horizontal space Hx at any point x ∈ M onto the tangent space Tπ(x) B. The images π∗ ◦ Jα of the endomorphisms Jα , α = 1, 2, 3 define a quaternionic subbundle Q of the bundle of endomorphisms of T B. It remains to check that this bundle is parallel with respect to the Levi-Civita connection of the induces metric gB . In the case when ε = −1 and the metric g1 on M is positively defined (hence, defines a Sasakian 3-structure ) this is a classical result by Ishihara [8]. His approach can be generalized to the case of pseudo-Sasakian 3-structure and also to the case ε = 1, see [2]. Here we indicate another approach, which is based on Swann characterization of a quaternionic Kähler manifolds in terms of fundamental 4-form. It works when n > 2. Swann [16] proved that a Riemannian manifold (M 4n , g), n > 2, with a quaternionic subbundle Q (i.e. a three-dimensional subbundle of End(T M) locally generated by three skew-symmetric almost complex structures (J1 , J2 , J3 ) which satisfy the quaternionic relations) is a quaternionic Kähler manifold if the globally defined fundamental 4-form P Ω= g ◦ Jα ∧ g ◦ Jα is closed. This result remains valid for pseudo-Riemannian case and can be generalized to the case of para-quaternionic Kähler manifolds. We can apply this result and finish the proof of the first part of the theorem as follows. P Consider 4-form Ω = ρα ∧ ρα on M. One can easily check that it is Hε1 -invariant and horizontal. Hence it is a pull-back of a 4-form ΩB on B. It is obvious that ΩB is the fundamental 4-form of the quaternionic subbundle Q. It remains to check that the form


745

Ω is closed. It can be done as follows. X dΩ = d (dωα + 2δα ωβ ∧ ωγ ) ∧ (dωα + 2δα ωβ ∧ ωγ ) = 2d

α X

dωα ∧ ωβ ∧ ωγ

α

=2

3 X

(14)

(dωα ∧ dωβ ∧ ωγ − dωα ∧ ωβ ∧ dωγ )

α=1

= 0, where (α, β, γ) is a cyclic permutations of (1, 2, 3) and δ1 = −ε, δ2 = δ3 = 1.

6

Homogeneous quaternionic and para-quaternionic CR manifolds

Here we describe some homogeneous examples of ε-quaternionic CR manifolds, which are the total spaces SB of the Sasakian bundle over symmetric ε-quaternionic Kähler manifolds B. The classification of ε-quaternionic Kähler symmetric spaces was given by D. Alekseevsky and V. Cortes [1]. Below we give a list of homogeneous ε-quaternionic CR manifolds associated with ε-quaternionic Kähler symmetric space of classical Lie groups. We have the following homogeneous ε-quaternionic CR manifolds of classical Lie groups: Cn : ε = +1, SH′ n,n = Spn+1 (R)/Spn (R); ε = −1, SHp,q = Spp+1,q /Spp,q An : ε = +1, SUp+1,q+1/Up,q ; ε = −1, SUp+2,q /Up,q ; BDn : ε = +1, SOp+2,q+2/SOp,q , ε = −1, SOp+4,q /SOp,q . We give more details concerning the geometry of the quaternionic CR manifold and para-quaternionic CR manifold.

6.1

Model space of quaternionic CR manifold

Put p + q = n. Let Hn+1 be the quaternionic arithmetic space of quaternionic dimension n + 1 with nondegenerate quaternionic Hermitian form < x, y >= x¯1 y1 + · · · + x¯p+1 yp+1 − x¯p+2 yp+2 − · · · − x¯n+1 yn+1 .

(15)

Denote by SHp,q the (4n + 3)-dimensional quadric: {(z1 , · · · , zp+1 , w1 , · · · , wq ) ∈ Hn+1 | ||(z, w)||2 = |z1 |2 + · · · + |zp+1 |2 − |w1|2 − · · · − |wq |2 = 1}.

(16)

746


Then the group Spp+1,q acts transitively on SHp,q with stabilizer isomorphic to Spp,q . According to the definition (cf. [2], [19],[12]), SHp,q = Spp+1,q /Spp,q is known as the quaternionic pseudo-Riemannian space of signature (3 + 4p, 4q) with constant positive curvature. On the other hand, if ω = −(¯ z1 dz1 + · · · + z¯p+1 dzp+1 − w ¯1 dw1 − · · · − w¯q dwq ).

(17)

is the sp(1)-valued 1-form on SHp,q , then we have shown in [2] that the set (ω, I, J, K) gives a quaternionic CR structure on SHp,q .See [2] for the proof. We shall construct the similar space form for para-quaternionic CR geometry using para-quaternionic arithmetic vector space H′ n .

6.2 Model space of para-quaternionic CR manifold We shall present an explicit model of the homogeneous space Spn+1 (R)/Spn (R) in terms of the algebra of para-quaternions H′ = C + Cj. Let M(n, H′ ) be the set of n × n-matrices with split-quaternion entries and I the identity matrix. First of all, we identify the group Spn (R) with the group Sp(n, H′ ) = {A ∈ M(n, H′ ) |∗ AA = I}.

(18)

The isomorphism of Sp(n, H′ ) onto Spn (R) is induced by the correspondence:    x+z y +w (x + yi) + (zj + wk) 7→  . −(y − w) x − z Let (H′ n+1 , < ·, · >) be the para-quaternionic arithmetic space of real dimension 4(n + 1) equipped with the inner product (i.e. a non-degenerate para-quaternionic Hermitian form) < z, w >= z¯1 w1 + · · · + z¯n+1 wn+1 .

(19)

The real part Re < z, w > of < z, w > is a non-degenerate symmetric bilinear form on H′ n+1 . The group of all invertible matrices GL(n+ 1, H′ ) is acting from the left and H′ ∗ = GL(1, H′ ) acting as the scalar multiplications from the right on H′ n+1 , which forms the group GL(n+1, H′ )·GL(1, H′ ) = GL(n+1, H′ ) × GL(1, H′ ). Let Sp(n+1, H′ )·Sp(1, H′ ) {±1}

be the subgroup of GL(n + 1, H′) · GL(1, H′ ) whose elements preserve the non-degenerate bilinear form Re < ·, · >. 2n+2,2n+1 Denote by ΣH the (4n + 3)-dimensional quadric: ′ {(z1 , · · · , zn+1 ) ∈ H′

n+1

| < z, z >= |z1 |2 + · · · + |zn+1 |2 = −1}

(20)


747

Note that if v = x + yi + zj + wk, then < v, v >= x2 + y 2 − z 2 − w 2 . In particular, 2n+2,2n+1 the group Sp(n + 1, H′ ) · Sp(1, H′ ) leaves ΣH invariant. Let < ·, · >x be the non′ degenerate para-quaternionic inner product on the tangent space Tx Hn+1 obtained by the parallel translation of < ·, · > to the point x ∈ H′ n+1 . Denote by {I, J, K} the standard para-quaternionic structure on H′ n+1 which operates as Iz = zi, Jz = zj, or Kz = zk. As usual, the set of endomorphisms {Ix , Jx , Kx } acts on Tx H′ n+1 at each point x. As < IX, IY >0 =< Xi, Y i >= −i < X, Y > i, < JX, JY >0 =< Xj, Y j >= −j < X, Y > j (similarly for K), it follows that Re < IX, IY >0 = Re < X, Y >0 , Re < JX, JY >0 = −Re < X, Y >0 , Re < KX, KY >0 = −Re < X, Y >0 , because ′ j 2 = k2 = 1. Then it is easy to see that gxH (X, Y ) = Re < X, Y >x (∀ X, Y ∈ Tx H′ n+1 ) is the standard pseudo-Euclidean metric of type (2n + 2, 2n + 2) on H′ n+1 such that ′

′

gxH (IX, IY ) = gxH (X, Y ), ′

′

gxH (JX, JY ) = −gxH (X, Y ), ′ gxH (KX, KY

)=

′ −gxH (X, Y

(21)

).

2n+2,2n+1 If Nx is the normal vector field at x ∈ ΣH in H′ n+1 , then gxH (IN, IN) = gxH (N, N) = ′ ′ ′ gxH (JN, JN) = gxH (KN, KN) = 1 by (21). There is the decomposition T H′ n+1 = ′ 2n+2,2n+1 2n+2,2n+1 {N} ⊕ T ΣH . Restricted g H to ΣH in H′ n+1 , we obtain a non-degenerate ′ ′ pseudo-Riemannian metric g of type (2n + 2, 2n + 1). ′

′

2n+2,2n+1 Definition 6.1. ΣH is referred to the para-quaternionic CR space form of type ′ (2n + 2, 2n + 1) endowed with the transitive group of isometries Sp(n + 1, H′ ) · Sp(1, H′ ): 2n+2,2n+1 ΣH = Sp(n + 1, H′ ) · Sp(1, H′ )/Sp(n, H′ ) · Sp(1, H′ ) ′

where Sp(n, H′ ) · Sp(1, H′ ) is the stabilizer of (1, 0, · · · , 0). 2n+2,2n+1 2n+2,2n+1 When Nx is the normal vector at x ∈ ΣH , note that Tx ΣH = Nx⊥ with ′ ′ ′ 2n+2,2n+1 respect to g H . In particular, IN, JN, KN ∈ T ΣH such that ′ 2n+2,2n+1 T ΣH = {IN, JN, KN} ⊕ {IN, JN, KN}⊥ . ′

By (21), 4n-dimensional subbundle {IN, JN, KN}⊥ is invariant under {I, J, K}. The 2n+2,2n+1 group Sp(1, H′ ) acts freely on ΣH : ′ ¯ · · · , zn+1 · λ). ¯ (λ, (z1 , · · · , zn+1 )) = (z1 · λ, There is the equivariant principal bundle: π

2n+2,2n+1 Sp(1, H′ )→(Sp(n + 1, H′ ) · Sp(1, H′ ), ΣH ) −→ ′ ′ ˆ 2n+2,2n+1 (PSp(n + 1, H ), Σ ′ ).

(22)

H

Let E(γ) = S

4n+1

×C

S1

2n+1

be the fiber bundle over the complex projective space CP2n

ˆ 2n+2,2n+1 with fiber C2n+1 . Then it is easy to see that E(γ) fibers over Σ with fiber H′ 2 diffeomorphic to the hyperbolic plane HR . On the other hand, let ω = −(¯ z1 dz1 + · · · + z¯n+1 dzn+1 )

(23)

748


2n+2,2n+1 be the ImH′ = sp(1, H′ ) = so(2, 1)-valued 1-form on ΣH . Let ξ1 , ξ2 , ξ3 be the ′ 2n+2,2n+1 vector fields on ΣH′ induced by the one-parameter subgroups {cos θ + i sin θ}θ∈R , {cosh θ − j sinh θ}θ∈R , {cosh θ − k sinh θ}θ∈R respectively. A calculation shows that

ω(ξ1) = −i, ω(ξ2 ) = j, ω(ξ3) = k.

(24)

2n+2,2n+1 2n+2,2n+1 By the formula of ω, if a ∈ Sp(1, H′ ), then the right translation Ra : ΣH →ΣH ′ ′ satisfies R∗a ω = a ¯ · ω · a. (25)

Therefore, ω is a connection form of the above bundle (22). Note that Sp(n+ 1, H′ ) leaves 2n+2,2n+1 ω invariant. Let Ω be the curvature form on ΣH : ′ dω + ω ∧ ω = Ω (dωα + 2εαωβ ∧ ωγ = Ωα , ε1 = −1, ε2 = 1, ε3 = 1).

(26)

2n+2,2n+1 To prove that (ΣH , ω) is a para-quaternionic CR-manifold, we check the conditions ′ (1), (2) of Definition 2.1 of §2.

ω = ω1 i + ω2 j + ω3 k.

(27)

It follows from (24) that ω1 (ξ1 ) = −1, ω2 (ξ2 ) = 1, ω3 (ξ3 ) = 1. A calculation shows that that ω 3 ∧ dω 2n = 6ω1 ∧ ω2 ∧ ω3 ∧ (−dω1 2 + dω2 2 + dω3 2 )n , 2n+2,2n+1 so each ωα is a non-degenerate contact form on ΣH . Using (25), It follows that ′ Lξ1 ω1 = 0, Lξ2 ω2 = Lξ3 ω3 = 0. As {ξ1 , ξ2 , ξ3 } generates the Lie algebra of SO(2, 1) = 3

2n+2,2n+1 Sp(1, H′ ), there is the decomposition T ΣH = {ξ1 , ξ2 , ξ3}⊕H where H = ∩ Ker ωα . ′ α=1

This proves Properties (1) of Definition 2.1. Let Ωα |H = σα (α = 1, 2, 3). Note that σα = dωα on H. Since Ker ω1 = {ξ2 , ξ3 } ⊕H and (X, Y ) 7→ dω1 (IX, Y ) is non-degenerate on H, σα is also non-degenerate on H. To prove the para-quaternionic relations (2) of Definition 2.1, we need the following lemma. Lemma 6.2. dω1 (X, Y ) = g(X, IY ), dω2 (X, Y ) = −g(X, JY ), dω3 (X, Y ) = g(X, KY ) where X, Y ∈ H. Proof. Given X, Y ∈ Hx , let u, v be the vectors at the origin obtained by the parallel translation of X, Y , respectively. Then by definition, g(X, Y ) = Re < u, v >. Furthermore, g(X, IY ) = Re(< u, v · i >) = Re(< u, v > ·i). (28) Similarly for J, K. Since dω = −(d¯ z1 ∧ dz1 + · · · + d¯ zn+1 ∧ dzn+1 ), for X, Y ∈ Hx , dω(X, Y ) = −(d¯ z1 ∧ dz1 + · · · + d¯ zn+1 ∧ dzn+1 )x (X, Y ) = −(d¯ z1 ∧ dz1 + · · · + d¯ zn+1 ∧ dzn+1 )(u, v) 1 1 = − (< u, v > − < v, u > = − (< u, v > −< u, v >). 2 2

(29)


749

Using (27), it is easy to check that g(X, IY ) = dω1 (X, Y ), g(X, JY ) = −dω2 (X, Y ), g(X, KY ) = −dω3 (X, Y ). As g|H is invariant under {I, J, K} by (21), the above equality implies that dω1 (IX, Y ) = dω2 (JX, Y ) = dω3(KX, Y ) = g(X, Y ).

(30)

By Definition 2.1, each Jγ is defined as ρ3 (J1 X, Y ) = −ρ2 (X, Y ), ρ1 (J2 X, Y ) = ρ3 (X, Y ), ρ2 (J3 X, Y ) = ρ1 (X, Y )(∀ X, Y ∈ H). Proposition 6.3. The set {Jα }α=1,2,3 forms a para-quaternionic structure on H. In fact, J1 = I, J2 = J, J3 = K. Proof. By definition, the endomorphism J3 is defined by ρ2 (J3 X, Y ) = ρ1 (X, Y ) (∀ X, Y ∈ H). Calculate that ρ1 (X, Y ) = dω1 − 2ω2 ∧ ω3 (X, Y ) = dω1 (X, Y ) = g(X, IY ) = −g(IX, Y ). Using the para-quaternionic relation of Definition 2.1, ρ2 (J3 X, Y ) = dω2(J3 X, Y ) = −g(J3 X, JY ) = g(JJ3 X, Y ) so that g(IX, Y ) = −g(JJ3 X, Y ). Hence, I = −JJ3 or JI = −J 2 J3 , i.e. K = J3 on H. The same argument yields that J = J2 , I = J1 . We have proved the following theorem. 2n+2,2n+1 Theorem 6.4. (ΣH , {ωα}α=1,2,3 , {I, J, K}) is a (4n + 3)-dimensional homogene′ ous para-quaternionic CR-manifold of type (2n + 2, 2n + 1). Moreover, there exists the equivariant principal bundle of the pseudo-Riemannian submersion over the homogeneous ˆ 2n+2,2n+1 para-quaternionic Kähler manifold Σ of signature (2n, 2n) (n ≥ 2): H′ π

2n+2,2n+1 Sp(1, R)→(Sp(n + 1, R) · Sp(1, R), ΣH , g) −→ ′ 2n+2,2n+1 ˆ ′ (PSp(n + 1, R), Σ , gˆ). H

Remark 6.5. If we note that (ξ1)x = x · (−i), then (ξ1 )x = −INx (similarly for ξ2 , ξ3 ). By definition, ω(Xx ) = − < x, u > where u ∈ Hn+1 is the vector at the origin by parallel translation of Xx . Then it follows that ω1 (Xx ) = Re(− < x, u > ·(−i)) = Re(i· < x, u >). Here we used Re(a · b) = Re(b · a). ¯ < x, y > by definition of < , >. We have On the other hand, < x · λ, y >= λ· < (ξ1 )x , Xx >= Re(< x · (−i), u >) = Re(i· < x, u >),

(31)

750


hence, ω1 (X) = g(ξ1 , X), Similarly we have that (ξ2)x = x · (j), (ξ3 )x = x · (k). It follows that < (ξ2 )x , Xx >= Re(< z(j), u >) = Re(−j < x, u >), < (ξ3 )x , Xx >= Re(−k < x, u >). Noting j 2 = k2 = 1, ω2 (Xx ) = Re(− < x, u > ·j), ω3 (Xx ) = Re(− < x, u > ·k). It follows that ω2 (Xx ) =< (ξ2 )x , Xx >= g((ξ2 )x , Xx ), ω3 (Xx ) = g((ξ3 )x , Xx ).

7

Reduction of ε-quaternionic CR manifold with a symmetry group

The reduction method is one of the most powerful methods of constructing of a new manifolds with some geometric structures , starting from a manifold M with such a structure which admits a non trivial Lie group G of symmetries. To apply this method, one has to define a G-equivariant “momentum map” µ from M into an appropriate G-module ˆ = µ−1 (0)/G of V . The new manifold with the structure S is defined as the quotient M zero-level set of µ by G. This method works if the structure S can be described in terms of differential form and structure equations can be written in terms of natural differential operators, for example, exterior differential. ε-quaternionic CR structure is one of such structures. Below we describe the reduction method for ε-quaternionic CR structures. Let (M, ωα , α = 1, 2, 3) be a ε-quaternionic CR manifold and G be a Lie group of its automorphisms, i.e. transformations which preserves 1-forms ωα . We denote by g∗ the dual space of the Lie algebra g of G and we will consider elements X ∈ g as vector fields on M. We define a momentum map as µ : M → R3 ⊗ g∗ , x 7→ µx , µx (X) = ω(Xx ) = (ω1 (Xx ), ω2 (Xx ), ω3 (Xx )) ∈ R3 . Lemma 7.1. The momentum map is a G-equivariant, where G acts on R3 ⊗ g∗ by the coadjoint representation on the second factor. Proof. For any φ ∈ G and X ∈ g we have µφx (Adφ Xx ) = ω(Adφ Xx ) = φ∗ ω(Xx ) = ω(Xx ) since ω = (ω1 , ω2 , ω3 ) is G-invariant form. Let µ−1 (0) be the preimage at the origin 0. It consists of all point x ∈ M such that the tangent space gx to the orbit Gx is horizontal: gx ⊂ Hx . In general, it is a stratified manifold. We calculate the tangent space Tx (µ−1 (0)) at a point x defined as the kernel of the differential dµx . We denote by gx the tangent space to the orbit Gx at x. Lemma 7.2. Tx (µ−1 (0)) = Vx + Hx′


751

where Hx = span{Jα gx, α = 1, 2, 3} ⊕ H ′ (Here ⊕ stands for gt -orthogonal direct sum). Proof. For Y ∈ Tx M and ξ ∈ g we have dµx (ξ)(Y ) = dιξ ω(Y ) = ((Lξ − ιξ d)ω)(Y ) = −ιξ dω(Y ) = (ρ − ω ∧ ω)(ξ, Y ) = ρ(ξ, Y ), since ω(ξ) = 0. If Y is vertical, then the last expression vanishes. If Y is horizontal, we can rewrite it as ρ(ξ, Y ) = (ρ1 (ξ, Y ), ρ2 (ξ, Y ), ρ3 (ξ, Y )) = (gt (J1 ξ, Y ), gt (J2 ξ, Y ), ±gt (J3 ξ, Y )). This proves Lemma. Corollary 7.3. (1) dim Gx ≤ dim Tx (µ−1 (0)) ≤ 3 dim Gx; (2) If the group G is one dimensional and it acts freely, then µ−1 (0) is a smooth regular (i.e. closed imbedded) submanifold of dimension 4n. We will assume that zero level set M ′ = µ−1 (0) of the momentum map is a smooth ′ submanifold of M and the group G acts on M ′ properly. We denote by Mreg the open G′ invariant submanifold of G-regular points of M (i.e. points which have a maximal possible ′ stabilizer K defined up to conjugation). It is well known that the orbit space Mreg /G is a smooth manifold. Theorem 7.4. Let (M, ωα ) be a ε-quaternionic CR manifold and G a connected Lie group ′ of its automorphisms. Assume that G acts properly on the manifold Mreg ⊂ µ−1 (0). Then the ε-quaternionic CR structure of M induces a ε-quaternionic CR structure ω ˆ α on the ′ ˆ orbit space M = Mreg /G. Proof. Denote by ω ′ and ρ′ the restriction of forms ω and ρ to the submanifold M ′ . Then dω ′ + ω ′ ∧ ω ′ = ρ′ . Since the forms ω ′ , ρ′ are G-invariant and horizontal, they can be ˆ such that π ∗ (ˆ ˆ projected down to forms ω ˆ , ρˆ on M ω ) = ω ′ , π ∗ (ˆ ρ) = ρ′ where π : M ′ → M is the natural projection. Then dˆ ω+ω ˆ ∧ω ˆ = ρˆ. ˆ = π∗ (Tx M ′ ). Now we describe the tangent space Tx M ′ and the tangent space Tπ(x) M We have Tx M = Vx + Hx , Hx = span{Jα gx} ⊕ Hx′ . ˆ Then Then Tx M ′ = V + Hx′ . We decompose Hx′ into an orthogonal sum Hx′ = gx ⊕ H. ˆ ≈ Tπ(x) M ˆ . This the projection π∗ has kernel gx and induces an isomorphism π∗ : V + H shows that 2-forms ρˆα have kernel π∗ (Vx ) and the associated endomorphisms Jˆα = ρˆ−1 ˆβ γ ◦ρ satisfy the quaternionic relations.

752

8


ε-hyperK¨ ahler structure on the cone over a ε-quaternionic CR manifold

A ε-hyperKähler manifold is defined as a 4n-dimensional ε-quaternionic Kähler manifold, whose holonomy group is a subgroup of Sp(p, q) for ε = −1 and Sp(n, R) for ε = 1. In this section we prove the following result. Theorem 8.1. Let (M, ωα ) be a ε-quaternionic CR manifold and gt the natural metric. Then the cone R+ × M with the cone metric g N = dt2 + t2 g1 is a ε-hyperK¨ ahler manifold. N + Conversely, if the cone metric g on the cone N = R × M over a manifold M is εhyperKähler with a parallel ε-hypercomplex structure Jα , then the manifold M has the canonical ε-quaternionic CR structure ωα = dt ◦ Jα such that g1 is the associated natural metric. Proof. We prove this theorem for ε = −1. The proof for ε = 1 is similar. Let (M, ωα ) be a ε-quaternionic CR manifold. We construct three exact 2-forms on the cone R+ × M by Ωα = d(t2 · ωα ) = 2tdt ∧ ωα + t2 dωα where t is the coordinate in R+ . Now we extend the almost complex structures J¯α (α = 1, 2, 3) to almost complex structures J¯α on R+ × M by d J¯α ξα = t · , dt d ξ α J¯α = − . dt t

(32)

Since J¯α satisfy the quaternionic relations and 2-forms Ωα are non degenerate and exact, hence, closed, it is sufficient to check that the metric g N can be written as g N = Ω1 (J¯1 ·, ·) = Ω2 (J¯2 ·, ·) = Ω3 (J¯3 ·, ·).

(33)

We check this as follows. Recall that g1 =

3 X

ωα ⊗ ωα + dωβ ◦ Jβ |H .

α=1

for β = 1, 2, 3. Now for any Y ∈ T (R+ × M) we calculate d 1 d Ωα (J¯α , Y ) = − (2tdt ∧ ωα + t2 dωα)(ξα , Y ) = dt(Y ) = g N ( , Y ). dt t dt Now we may assume that X, Y ∈ T M. For X = ξα we get d Ωα (J¯α ξα , Y ) = (2tdt ∧ ωα + t2 dωα )(t , Y ) = t2 ωα (Y ) = g N (ξα , Y ). dt

(34)

Similar, Ωα (J¯α ξβ , Y ) = Ωα (ξγ , Y ) = t2 dωα (ξγ , Y ) = −2t2 ωβ ∧ ωγ (ξα , Y.

(35)


753

It remains to consider the case when X, Y ∈ H are horizontal vectors. In this case Ωα (J¯α X, Y ) = t2 dωα (Jα X, Y ) = t2 g1 (X, Y ) = g N (X, Y ). This prove the first claim. The inverse statement can be checked similarly.

References [1] D.V. Alekseevsky, V. Cortés : “Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type“, Preprint Institut Eli Cartan, Vol. 11, (2004). [2] D.A. Alekseevsky and Y. Kamishima: “Locally pseudo-conformal quaternionic CR structure”, preprint. [3] A. Besse: Einstein manifolds, Springer Verlag, 1987. [4] O. Biquard: Quaternionic structures in mathematics and physics, World Sci. Publishing, Rome, 1999, River Edge, NJ, 2001, pp. 23–30. [5] D.E. Blair: “Riemannian geometry of contact and symplectic manifolds Contact manifolds”, Birkhäuser, Progress in Math., Vol. 203, (2002). [6] C.P. Boyer, K. Galicki and B.M. Mann: “The geometry and topology of 3-Sasakian manifolds”, Jour. reine ange. Math., Vol. 455, (1994), pp. 183–220. [7] N.J. Hitchin: “The self-duality equations on a Riemannian surface”, Proc. London Math. Soc., Vol. 55, (1987), pp. 59–126. [8] S. Ishihara: “Quaternion Kählerian manifolds”, J. Diff. Geom., Vol. 9, (1974), pp. 483–500. [9] S. Ishihara: “Quaternion Kählerian manifolds and fibred Riemannian spaces with Sasakian 3-structure”, K¯ odai Math. Sem. Rep., Vol. 25, (1973), pp. 321–329. [10] S. Ishihara and M. Konishi: “Real contact and complex contact structure”, Sea. Bull. Math., Vol. 3, (1979), pp. 151–161. [11] W. Jelonek: “Positive and negative 3-K-contact structures”, Proc. of A.M.S., Vol. 129, (2000), pp. 247–256. [12] R. Kulkarni: “Proper actions and pseudo-Riemannian space forms”, Advances in Math., Vol. 40, (1981), pp. 10–51. [13] T. Kashiwada: “On a contact metric structure”, Math. Z., Vol. 238, (2001), pp. 829– 832. [14] M. Konishi: “On manifolds with Sasakian 3-structure over quaternionic Kaehler manifolds”, |emphKodai Math. Jour., Vol. 29, (1975), pp. 194–200. [15] S. Kobayashi and K. Nomizu: Foundations of differential geometry I,II, Interscience John Wiley & Sons, New York, 1969. [16] A. Swann: “Aspects symplectiques de la géométrie quaternionique”, C. R. Acad. Sci. Paris, Seria I, Vol. 308, (1989), pp. 225–228. [17] S. Tanno: “Killing vector fields on contact Riemannian manifolds and fibering related to the Hopf fibrations”, |emphTôhoku Math. Jour., Vol. 23, (1971), pp. 313–333. [18] S. Tanno: “Remarks on a triple of K-contact structures”, Tˆ ohoku Math. Jour., Vol. 48, (1996), pp. 519–531. [19] J. Wolf: Spaces of constant curvature, McGraw-Hill, Inc., 1967.

CEJM 2(5) 2004 754–766

An introduction to finite fibonomial calculus Ewa Krot∗ Institute of Computer Science, Bialystok University, ul.Sosnowa 64, 15-887 Bialystok, Poland

Received 15 December 2003; accepted 1 July 2004 Abstract: This is an indicatory presentation of main definitions and theorems of Fibonomial Calculus which is a special case of ψ-extented Rota’s finite operator calculus [7]. c Central European Science Journals. All rights reserved.

Keywords: Fibonacci numbers, Fibonomial Calculus, Sheffer F- polynomials MSC (2000): 11C08, 11B37, 47B47

1

Fibonomial coefficients

The famous Fibonacci sequence {Fn }n≥0 (

Fn+2 = Fn+1 + Fn F0 = 0, F1 = 1

is attributed and referred to in the first edition (lost) of ”Liber Abaci” (1202) by Leonardo Fibonacci (Pisano)(see edition from 1228 reproduced as ”Il Liber Abaci di Leonardo Pisano publicato secondo la lezione Codice Maglibeciano by Baldassarre Boncompagni in Scritti di Leonardo Pisano” , vol. 1,(1857)Rome). In order to specify what a ”Fibonomial Calculus” is let us define for the sequence F = {Fn }n≥0 the following: (1) F -factorial: Fn ! = Fn Fn−1 ...F2 F1 , F0 ! = 1. ∗


E. Krot / Central European Journal of Mathematics 2(5) 2004 754–766

755

(2) F -binomial (Fibonomial) coefficients [5]: n nkF Fn Fn−1 . . . Fn−k+1 Fn ! = = = , k F kF ! Fk Fk−1 . . . F2 F1 Fk !Fn−k !

n = 1. 0 F

Some properties of nk F are: n , (symmetry); (a) nk F = n−k F n (b) Fn−k k F = Fn n−1 ; k F n (c) k F ∈ N for every n, k ∈ N ∪ 0.

2

Operators and polynomial sequences

Let P be the algebra of polynomials over the field K of characteristic zero. Definition 2.1. The linear operator ∂F : P → P such that ∂F xn = Fn xn−1 for n ≥ 0 is called the F -derivative. Definition 2.2. The F -translation operator is the linear operator E y (∂F ) : P → P of the form: X yk ∂ k F , y∈K E y (∂F ) = expF {y∂F } = F k! k≥0 Definition 2.3. ∀p∈P

p(x +F y) = E y (∂F )p(x)

x, y ∈ K

Definition 2.4. A linear operator T : P → P is said to be ∂F -shift invariant iff ∀y∈K

[T, E y (∂F )] = T E y (∂F ) − E y (∂F )T = 0

We shall denote by ΣF the algebra of F -linear ∂F -shift invariant operators. Definition 2.5. Let Q(∂F ) be a formal series in powers of ∂F and Q(∂F ) : P → P. Q(∂F ) is said to be ∂F -delta operator iff (a) Q(∂F ) ∈ ΣF (b) Q(∂F )(x) = const 6= 0 We refer to [7](see also references therein) for the proofs of most statements. The particularities of the case considered here are explained in the sequel, especially in Sections 4 and 5. There the scope of new possibilities is exhibited by means of new examples. Proposition 2.6. Let Q(∂F ) be the ∂F -delta operator. Then ∀c∈K

Q(∂F )c = 0.

756


Proposition 2.7. Every ∂F -delta operator reduces the degree of any polynomial by one. Definition 2.8. The polynomial sequence {qn (x)}n≥0 such that deg qn (x) = n and: (1) q0 (x) = 1; (2) qn (0) = 0, n ≥ 1; (3) Q(∂F )qn (x) = Fn qn−1 (x), n ≥ 0 is called ∂F -basic polynomial sequence of the ∂F -delta operator Q(∂F ). Proposition 2.9. For every ∂F -delta operator Q(∂F ) there exists the uniquely determined ∂F -basic polynomial sequence {qn (x)}n≥0 . Definition 2.10. A polynomial sequence {pn (x)}n≥0 (deg pn (x) = n) is of F -binomial (fibonomial) type if it satisfies the condition y

E (∂F )pn (x) = pn (x +F y) =

X n k≥0

k

pk (x)pn−k (y) ∀y∈K F

Theorem 2.11. The polynomial sequence {pn (x)}n≥0 is a ∂F -basic polynomial sequence of some ∂F -delta operator Q(∂F ) iff it is a sequence of F -binomial type. Theorem 2.12. (First Expansion Theorem) Let T ∈ ΣF and let Q(∂F ) be a ∂F -delta operator with ∂F -basic polynomial sequence {qn }n≥0 . Then X an T = Q(∂F )n ; an = [T qk (x)]x=0 . F ! n≥0 n Theorem 2.13. (Isomorphism Theorem) Let ΦF = KF [[t]] be the algebra of formal expF series in t ∈ K ,i.e.: fF (t) ∈ ΦF

if f

fF (t) =

X ak tk k≥0

Fk !

f or ak ∈ K,

and let the Q(∂F ) be a ∂F -delta operator. Then ΣF ≈ ΦF . The isomorphism φ : ΦF → ΣF is given by the natural correspondence: fF (t) =

X ak tk k≥0

Fk !

into

−→ T∂F =

X ak Q(∂F )k . F ! k k≥0

Remark 2.14. In the algebra ΦF the product is given by the fibonomial convolution, i.e.: ! ! ! X bk X ck X ak xk xk = xk F F F k! k! k! k≥0 k≥0 k≥0


where ck =

X k l≥0

l

757

al bk−l . F

Corollary 2.15. Operator T ∈ ΣF has its inverse T −1 ∈ Σψ iff T 1 6= 0. Remark 2.16. The F -translation operator E y (∂F ) = expF {y∂F } is invertible in ΣF but it is not a ∂F -delta operator. None of the ∂F -delta operators Q (∂F ) is invertible with respect to the formal series ”F-product”. Corollary 2.17. Operator R(∂F ) ∈ ΣF is a ∂F -delta operator iff a0 = 0 and a1 6= 0, where P ak k P x R(∂F ) = n≥0 Fann! Q (∂F )n or equivalently : r(0) = 0 & r′ (0) 6= 0 where r(x) = Fk ! k≥0

is the correspondent of R(∂F ) under the Isomorphism Theorem. Corollary 2.18. Every ∂F -delta operator Q (∂F ) is a function Q(∂F ) according to the expansion X qn Q (∂F ) = ∂Fn F ! n≥1 n This F -series will be called the F -indicator of the Q(∂F ). Remark 2.19. expF {zx} is the F -exponential generating function for ∂F -basic polynomial sequence {xn }∞ n=0 of the ∂F operator. Corollary 2.20. The F -exponential generating function for ∂F -basic polynomial sequence {pn (x)}∞ n=0 of the ∂F -delta operator Q (∂F ) is given by the following formula X pk (x) k≥0

Fk !

z k = expF {xQ−1 (z)}

where Q ◦ Q−1 = Q−1 ◦ Q = I = id. Example 2.21. The following operators are the examples of ∂F -delta operators: (1) ∂F ; (2) F -difference operator ∆F = E 1 (∂F ) − I such that (∆F p)(x) = p(x +F 1) − p(x) for every p ∈ P ; (3) The operator ∇F = I − E −1 (∂F ) defined as follows: (∇F p)(x) = p(x) − p(x −F 1) for every p ∈ P; P ak k+1 (4) F -Abel operator: A(∂F ) = ∂F E a (∂F ) = ∂ ; Fk ! F k≥0 P k+1 ∂F . (5) F -Laguerre operator of the form: L(∂F ) = ∂F∂F−I = k≥0

758

3


The Graves-Pincherle F -derivative

Definition 3.1. The xˆF -operator is the linear map xˆF : P → P such that xn+1 f or n ≥ 0. ( [∂F , xˆF ] = id.) xˆF xn = Fn+1 n+1 Definition 3.2. A linear map ’ : ΣF → ΣF such that T ’ = T xˆF − xˆF T = [T , xˆF ] is called the Graves-Pincherle F -derivative [3, 9]. Example 3.3. (1) ∂F ’=I = id; (2) (∂F )n ’=n∂Fn−1 According to the example above the Graves-Pincherle F -derivative is the formal derivative with respect to ∂F in ΣF i.e., T ’ (∂F ) ∈ ΣF for any T ∈ ΣF . Corollary 3.4. Let t (z) be the indicator of operator T ∈ ΣF . Then t′ (z) is the indicator of T ’∈ ΣF . Due to the isomorphism theorem and the Corollaries above, the Leibnitz rule holds . Proposition 3.5. (T S)’ = T ’ S + ST ’ ; T , S ∈ ΣF . As an immediate consequence of the Proposition 3.5 we get (S n )’= n S’S n−1 ∀S∈ΣF . >From the isomorphism theorem we assert that the following is true. Proposition 3.6. Q (∂F ) is the ∂F -delta operator iff there exists invertible S ∈ ΣF such that Q (∂F ) = ∂F S. The Graves-Pincherle F -derivative notion appears very effective while formulating expressions for ∂F -basic polynomial sequences of the given ∂F -delta operator Q (∂F ). Theorem 3.7. (F -Lagrange and F -Rodrigues formulas) [7, 10, 8] Let {qn }n≥0 be ∂F -basic sequence of the delta operator Q(∂F ), Q(∂F ) = ∂F P (P ∈ ΣF , invertible). Then for n ≥ 0: (1) qn (x) = Q (∂F )’ P −n−1 xn ; (2) qn (x) = P −n xn − Fnn (P −n ) ’xn−1 ; (3) qn (x) = Fnn xˆF P −n xn−1 ; (4) qn (x) = Fnn xˆF (Q (∂F )’ )−1 qn−1 (x) (← Rodrigues F -formula ). Corollary 3.8. Let Q(∂F ) = ∂F S and R(∂F ) = ∂F P be the ∂F -delta operators with the ∂F -basic sequences {qn (x)}n≥0 and {rn (x)}n≥0 respectively. Then: (1) qn (x) = R’(Q’)−1 S −n−1 P n+1 rn (x), n ≥ 0;


759

(2) qn (x) = xˆF (P S −1 )n xˆF−1 rn (x), n > 0. The formulas of the Theorem 3.7 can be used to find ∂F -basic sequences of the ∂F -delta operators from the Example 2.21. Example 3.9. (1) The polynomials xn , n ≥ 0 are ∂F -basic for F -derivative ∂F . (2) Using Rodrigues formula in a straightforward way one can find the following first ∂F -basic polynomials of the operator ∆F : q0 (x) = 1 q1 (x) = x q2 (x) = x2 − x q3 (x) = x3 − 4x2 + 3x q4 (x) = x4 − 9x3 + 24x2 − 16x q5 (x) = x5 − 20x4 + 112.5x3 − 250x2 + 156.5x q6 (x) = x6 − 40x5 + 480x4 − 2160x3 + 4324x2 − 2605x. (3) Analogously to the above example we find the following first ∂F -basic polynomials of the operator ∇F : q0 (x) = 1 q1 (x) = x q2 (x) = x2 + x q3 (x) = x3 + 4x2 + 3x q4 (x) = x4 + 9x3 + 24x2 + 16x q5 (x) = x5 + 20x4 + 112.5x3 + 250x2 + 156.5x q6 (x) = x6 + 40x5 + 480x4 + 2160x3 + 4324x2 + 2605x. (4) Using Rodrigues formula in a straightforward way one finds the following first ∂F basic polynomials of F -Abel operator: (a) A0,F (x) = 1 (a) A1,F (x) = x (a) A2,F (x) = x2 + ax (a) A3,F (x) = x3 − 4ax2 + 2a2 x (a) A4,F (x) = x4 − 9ax3 + 18a2 x2 − 3a3 x.

760


(5) In order to find ∂F -basic polynomials of F -Laguerre operator L(∂F ) we use formula (3) from Theorem 3.7: −n Fn 1 Fn xn−1 = xˆF xˆF (∂F − 1)n xn−1 = Ln,F (x) = n ∂F − 1 n n n Fn X Fn X n−k n−k n−1 k n k n = (−1) (−1) ∂F x = (n − 1)F xk−1 = xˆF xˆF k k n n k=0 k=0 n Fn X n−k k k k n (n − 1)F (−1) x . = k n k=1 Fk

4

Sheffer F -polynomials

Definition 4.1. A polynomial sequence {sn }n≥0 is called the sequence of Sheffer F polynomials of the ∂F -delta operator Q(∂F ) iff (1) s0 (x) = const 6= 0 (2) Q(∂F )sn (x) = Fn sn−1 (x); n ≥ 0. Proposition 4.2. Let Q(∂F ) be ∂F -delta operator with ∂F -basic polynomial sequence {qn }n≥0 . Then {sn }n≥0 is the sequence of Sheffer F -polynomials of Q(∂F ) iff there exists an invertible S ∈ ΣF such that sn (x) = S −1 qn (x) for n ≥ 0. We shall refer to a given labeled by ∂F -shift invariant invertible operator S Sheffer F -polynomial sequence {sn }n≥0 as the sequence of Sheffer F -polynomials of the ∂F -delta operator Q(∂F ) relative to S. Theorem 4.3. (Second F - Expansion Theorem) Let Q (∂F ) be the ∂F -delta operator Q (∂F ) with the ∂F -basic polynomial sequence {qn (x)}n≥0 . Let S be an invertible ∂F -shift invariant operator and let {sn (x)}n≥0 be its sequence of Sheffer F -polynomials. Let T be any ∂F -shift invariant operator and let p(x) be any polynomial. Then the following identity holds : P sk (y) ∀y∈K ∧ ∀p∈P (T p) (x +F y) = [E y (∂F )p] (x) = T Q (∂F )k S T p (x) . Fk ! k≥0

Corollary 4.4. Let sn (x)n≥0 be a sequence of Sheffer F -polynomials of a ∂F -delta operator Q(∂F ) relative to S.Then: S −1 =

X sk (0) k≥0

Fk !

Q(∂F )k .

Theorem 4.5. (The Sheffer F -Binomial Theorem) Let Q(∂F ), invertible S ∈ ΣF , qn (x)n≥0 , sn (x)n≥0 be as above. Then: y

E (∂F )sn (x) = sn (x +F y) =

X n k≥0

k

sk (x)qn−k (y). F


Corollary 4.6. sn (x) =

X n k≥0

k

761

sk (0)qn−k (x) F

Proposition 4.7. Let Q (∂F ) be a ∂F -delta operator. Let S be an invertible ∂F -shift invariant operator. Let {sn (x)}n≥0 be a polynomial sequence. Let P sk (a) ∀a∈K ∧ ∀p∈P E a (∂F ) p (x) = Q (∂F )k S∂F p (x) . Fk ! k≥0

Then the polynomial sequence {sn (x)}n≥0 is the sequence of Sheffer F -polynomials of the ∂F -delta operator Q (∂F ) relative to S. Proposition 4.8. Let Q (∂F )and S be as above. Let q(t) and s(t) be the indicators of Q (∂F ) and S operators. Let q−1 (t ) be the inverse F -exponential formal power series inverse to q(t). Then the F -exponential generating function of the Sheffer F -polynomials sequence {sn (x)}n≥0 of Q (∂F ) relative to S is given by X sk (x) k≥0

Fk !

zk =

−1 s q −1 (z) expF {xq −1 (z)}.

Proposition 4.9. A sequence {sn (x)}n≥0 is the sequence of Sheffer F -polynomials of the ∂F -delta operator Q (∂F ) with the ∂F -basic polynomial sequence {qn (x)}n≥0 iff sn (x +F y) =

X n k≥0

k

sk (x) qn−k (y) . F

for all y ∈ K Example 4.10. Hermite F -polynomials are Sheffer F -polynomials of the ∂F -delta opa∂ 2 erator ∂F relative to invertible S ∈ ΣF of the form S = expF { 2F }. One can get them by formula (see Proposition 4.2 ): Hn,F (x) = S −1 xn =

X (−a)k k≥0

2k Fk !

n−2k . n2k F x

Example 4.11. Let S = (1 − ∂F )−α−1 . The Sheffer F -polynomials of ∂F -delta operator L(∂F ) = ∂F∂F−1 relative to S are Laguerre F -polynomials of order α . By Proposition 4.2 we have (α) Ln,F (x) = (1 − ∂F )α+1 Ln,F (x), From the above formula and using Graves-Pincherle F -derivative we get X Fn ! α + n (α) (−x)k Ln,F (x) = Fk ! n − k k≥0 for α 6= −1.

762


Example 4.12. Bernoullie’s F -polynomials of order 1 are −1 Sheffer F -polynomials of ∂F F }−I . Using Proposition 4.2 one -delta operator ∂F related to invertible S = expF {∂ ∂F arrives at X 1 X 1 n k−1 n −1 n Bn,F (x) = S x = xn−k+1 = ∂F x = k − 1 F ! F k k F k≥1 k≥1 X 1 n xn−k = Fk+1 k F k≥0 Theorem 4.13. (Reccurence relation for Sheffer F -polynomials) Let Q, S, {sn }n≥0 be as above. Then the following reccurence formula holds: Fn+1 S′ −1 sn+1 (x) = xˆF − [Q(∂F )′ ] sn (x); n ≥ 0. n+1 S Example 4.14. The reccurence formula for the Hermite F -polynomials is: Hn+1,F (x) = xˆF Hn,F (x) − a ˆF Fn Hn−1,F (x) Example 4.15. The reccurence relation for the Laguerre F -polynomials is: (α)

Ln+1,F (x) = −

5

Fn+1 (α) [ˆ xF − (α + 1)(1 − ∂F )−1 ](∂F − 1)2 Ln,F (x) n+1 Fn+1 (α+1) = [ˆ xF (∂F − 1) + α + 1]Ln,F (x). n+1

The Spectral Theorem

We shall now define a natural inner product associated with the sequence {sn }n≥0 of Sheffer F -polynomials of the ∂F -delta operator Q(∂F ) relative to S. Definition 5.1. Let Q, S, {sn }n≥0 be as above. Let W be umbral operator: W : sn (x) → xn ( and linearly extented). We define the following bilinear form: (f (x), g(x))F := [(W f )(Q(∂F ))Sg(x)]x=0 ; f, g ∈ P. Proposition 5.2. [10] The bilinear form over reals defined above is a positive definite inner product such that: (sn (x), sk (x))F = Fn !δn,k . We shall call this scalar praduct the natural inner product associated with the sequence {sn }n≥0 of Sheffer F -polynomials. Unitary space (P, ( , )F ) can be completed to the unique Hilbert space H = P. Theorem 5.3. (Spectral Theorem) Let {sn }n≥0 be the sequence of Sheffer F -polynomials relative to the ∂F -shift invariant


763

invertible operator S for the ∂F -delta operator Q(∂F ) with ∂F -basic polynomial sequence {qn }n≥0 . Then there exists a unique operator AF : H → H of the form AF =

X uk + vˆk (x) k≥1

Fk−1 !

Q(∂F )k

with the following properties: (a) A is self adjoint; (b) The spectrum of A consists of n ∈ N and Asn = nsn for n ≥ 0; (c) Quantities uk and vˆk (x) are calculated according to d ′ −1 qk (x) uk = −[(log S) xˆF qk (x)]x=0 vˆF (x) = xˆF dx x=0 Proof: see [7].

6

The first elementary examples of some F-polynomials

(1) Here are the examples of Laguerre F -polynomials of order α = −1: L0,F (x) = 1 L1,F (x) = −x L2,F (x) = x2 − x L3,F (x) = −x3 + 4x2 − 2x L4,F (x) = x4 − 9x3 + 18x2 − 6x L5,F (x) = −x5 + 20x4 − 905x3 + 1280x2 − 30x L6,F (x) = x6 − 40x5 + 400x4 − 1200x3 + 1200x2 − 240x L7,F (x) = −x7 + 78x6 − 1560x5 + 10400x4 − 23400x3 + 18720x2 − − 3120x L8,F (x) = x8 − 147x7 + 5733x6 − 76440x5 + 382200x4 − 687960x3 + + 458640x2 − 65520x (2) Here are the examples of Laguerre F -polynomials of order α = 1: (1) L0,F (x) = 1 (1)

L1,F (x) = −x + 2

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E. Krot / Central European Journal of Mathematics 2(5) 2004 754–766 (1)

L2,F (x) = x2 − 3x + 3 (1)

L3,F (x) = −x3 + 8x2 − 12x + 8 (1)

L4,F (x) = x4 − 15x3 + 60x2 − 60x + 30 (1)

L5,F (x) = −x5 + 30x4 − 225x3 + 600x2 − 450x + 240 (1)

L6,F (x) = x6 − 56x5 + 840x4 − 4200x3 + 8400x2 − 5040x + 1680 (3) Here we give some examples of the Bernoullie’s F -polynomials of order 1: B0,F (x) = 1 B1,F (x) = x + 1 B2,F (x) = x2 + x +

1 2

B3,F (x) = x3 + 2x2 + x +

1 3

B4,F (x) = x4 + 3x3 + 3x2 + x + B5,F (x) = x5 + 5x4 +

15 3 x 2

1 5

+ 5x2 + x +

1 8

B6,F (x) = x6 + 8x5 + 20x4 + 20x3 + 8x2 + x + B7,F (x) = x7 + 13x6 + 52x5 + B8,F (x) = x8 + 21x7 +

273 6 x 2

260 4 x 3

+ 52x3 + 13x2 + x +

+ 364x5 + 364x4 +

B9,F (x) = x9 + 34x8 + 357x7 + 1547x6 + + 34x2 + x +

1 13

12376 5 x 5

273 3 x 2

1 21

+ 21x2 + x +

1 36

+ 1547x4 + 357x3 +

1 55

Remark 6.1. Let us observe that analogously to the ordinary case F -polynomials ,such as Abel, Laguerre or Bernoullie’s F -polynomials may have coefficients which are integer numbers (F -Abel, F -Laguerre) and non-integer rationals (F -Bernoulli). To see that recall for example the formula for Laguerre F -polynomials of order -1 (F -basic): n Fn X n−k k k k n x (−1) (n − 1)F Ln,F (x) = n k=1 Fk k


765

and the one for F -Laguerre of order α 6= −1 (F -Sheffer): X Fn ! α + n (α) (−x)k . Ln,F (x) = F k! n − k k≥0 Because binomial coefficients are integers the second formula gives us polynomials with integer coefficients. It is easy to verify that F -basic Laguerre polynomials do have this property too. Finally let p ∈ P while ak denote coefficient of this polynomial p at xk ,i.e. X p(x) = ak x k . k≥0

Consider now the Bernoullie’s F -polynomials of order 1. Because of the symmetry of n and some known divisibility properties of Fibonacci numbers [4, 1] for Bernoullie’s k F F -polynomial Bn,F (x) we have an−k = ak+1 n for k = 0, 1, .., 2 . Moreover from the formula for these polynomials we have that a0 =

1 Fn+1

.

Observe now that coefficients of Abel F -polynomials are integer numbers, so we may expect now that these polynomials enumerate some combinatorial objects like those of the now classical theory of binomial enumeration (see [11]).

Acknowledgment I would like to thank to Prof. A.K. Kwa´sniewski for his remarks and guidance.

References [1] B. Bondarienko: Generalized Pascal Triangles and Pyramids- Their Fractals , graphs and Applications, A reproduction by the Fibonacci Association 1993, Santa Clara University, Santa Clara, CA. [2] R.L. Graham, D.E. Knuth and O. Patashnik: Concrete mathematics.A Foundation for Computer Science, Addison-Wesley Publishing Company, Inc., Massachusetts, 1994. [3] C. Graves: “On the principles which regulate the interchange of symbols in certain symbolic equations“, Proc.Royal Irish Academy, Vol. 6, (1853-1857), pp. 144–152. [4] W.E. Hoggat,Jr: Fibonacci and Lucas numbers. A publication of The Fibonacci Association, University of Santa Clara, CA 95053. [5] D. Jarden: “Nullifying coefficiens“, Scripta Math., Vol. 19, (1953), pp. 239–241. [6] E. Krot: “ψ-extensions of q-Hermite and q-Laguerre Polynomials - properties and principal statements“, Czech. J. Phys., Vol. 51(12), (2001), pp. 1362–1367.

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[7] A.K. Kwa´sniewski: “Towards ψ-Extension of Rota’s Finite Operator Calculus“, Rep. Math. Phys., Vol. 47(305), (2001), pp. 305–342. [8] G. Markowsky: “Differential operators and the Theory of Binomial Enumeration“, Math.Anal.Appl., Vol. 63(145), (1978). [9] S. Pincherle and U. Amaldi: Le operazioni distributive e le loro applicazioni all analisi, N. Zanichelli, Bologna, 1901. [10] G.-C. Rota: Finite Operator Calculus, Academic Press, New York, 1975. [11] G.C. Rota and R. Mullin: “On the Foundations of cCombinatorial Theory, III: Theory of binominal Enumeration“, In: Graph Theory and its Applications, Academic Press, New York, 1970. [12] http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Fibonacci.html [13] A.K. Kwa´sniewski: “Information on Some Recent Applications of Umbral Extensions to Discrete Mathematics“, ArXiv:math.CO/0411145, Vol. 7, (2004), to be presented at ISRAMA Congress, Calcuta-India, December 2004

CEJM 2(5) 2005 767–792

Review article

Extended finite operator calculus - an example of algebraization of analysis Andrzej Krzysztof Kwa´sniewski1∗ , Ewa Borak2 1

Higher School of Mathematics and Applied Informatics, 15-021 Bialystok , ul.Kamienna 17, Poland 2 Institute of Computer Science, Bialystok University, 15-887 Bialystok, ul.Sosnowa 64, Poland

Received 15 December 2003; accepted 21 October 2004 Abstract: “A Calculus of Sequences” started in 1936 by Ward constitutes the general scheme for extensions of classical operator calculus of Rota - Mullin considered by many afterwards and after Ward. Because of the notation we shall call the Ward‘s calculus of sequences in its afterwards elaborated form - a ψ-calculus. The ψ-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota - Mullin or equivalently - of umbral calculus of Roman and Rota. At the same time this calculus is an example of the algebraization of the analysis - here restricted to the algebra of polynomials. Many of the results of ψ-calculus may be extended to Markowsky Q-umbral calculus where Q stands for a generalized difference operator, i.e. the one lowering the degree of any polynomial by one. This is a review article based on the recent first author contributions [1]. As the survey article it is supplemented by the short indicatory glossaries of notation and terms used by Ward [2], Viskov [7, 8] , Markowsky [12], Roman [28–32] on one side and the Rota-oriented notation on the other side [9–11, 1, 3, 4, 35] (see also [33]). c Central European Science Journals. All rights reserved.

Keywords: Extended umbral calculus, Graves-Heisenberg-Weyl algebra MSC (2000): 05A40 , 81S99

1

Introduction

We shall call the Wards calculus of sequences [2] in its afterwards last century elaborated form - a ψ-calculus because of the Viskov‘s efficient notation [3-8]-adopted from Boas ∗

E-mail: kwandr@pl

768

A.K. Kwa´sniewski, E. Borak / Central European Journal of Mathematics 2(5) 2005 767–792

and Buck . The efficiency of the Rota oriented language and our notation used has been already exemplified by easy proving of ψ-extended counterparts of all representation independent statements of ψ-calculus [2]. Here these are ψ-labelled representations of Graves-Heisenberg-Weyl (GHW) [1, 3, 16, 17] algebra of linear operators acting on the algebra P of polynomials. As a matter of fact ψ-calculus becomes in parts almost automatic extension of Rota - Mullin calculus [9] or equivalently - of umbral calculus of Roman and Rota [9, 10, 11]. The ψ-extension relies on the notion of ∂ψ -shift invariance of operators with ψ-derivatives ∂ψ staying for equivalence classes representatives of special differential operators lowering degree of polynomials by one [7, 8, 12]. Many of the results of ψ-calculus may be extended to Markowsky Q-umbral calculus [12] where Q stands for arbitrary generalized difference operator, i.e. the one lowering the degree of any polynomial by one. Q-umbral calculus [12] - as we call it - includes also those generalized difference operators, which are not series in ψ-derivative ∂ψ whatever an admissible ψ sequence would be (for - ”admissible” - see next section). The survey proposed here reviews the operator formulation of “A Calculus of Sequences” started in 1936 by Ward [2] with the indication of the decisive role the ψrepresentations of Graves-Heisenberg-Weyl (GHW) algebra account for formulation and derivation of principal statements of the ψ-extension of finite operator calculus of Rota and its extensions. Restating what was said above let us underline that all statements of standard finite operator calculus of Rota are valid also in the case of ψ-extension under the almost mnemonic , automatic replacement of {D, x ˆ, id} generators of GHW by their ψ-representation correspondents {∂ψ , xˆψ , id} - see definitions 2.1 and 2.5. Naturally any specification of admissible ψ - for example the famous one defining q-calculus - has its own characteristic properties not pertaining to the standard case of Rota calculus realization. Nevertheless the overall picture and system of statements depending only on GHW algebra is the same modulo some automatic replacements in formulas demonstrated in the sequel. The large part of that kind of job was already done in [1, 3, 35]. The aim of this presentation is to give a general picture ( see: Section 3) of the algebra of linear operators on polynomial algebra. The picture that emerges discloses the fact that any ψ-representation of finite operator calculus or equivalently - any ψ-representation of GHW algebra makes up an example of the algebraization of the analysis with generalized differential operators [12] acting on the algebra of polynomials. We shall delimit all our considerations to the algebra P of polynomials or sometimes to the algebra of formal series. Therefore the distinction between difference and differentiation operators disappears. All linear operators on P are both difference and differentiation operators if the degree of differentiation or difference operator is unlimited. If all this is extended to Markowsky Q-umbral calculus [12] then many of the results of ψ-calculus may be extended to Q-umbral calculus [12]. This is achieved under the almost automatic replacement of {D, xˆ, id} generators of GHW or their ψ-representation {∂ψ , xˆψ , id} by their Q-representation correspondents {Q, x ˆQ , id} - see definition 2.5.


769

The article is supplemented by the short indicatory glossaries of notation and terms used by Ward [1], Viskov [7], [8], Markowsky [12], Roman [28]-[31] on one side and the Rota-oriented [9]-[11] notation on the other side [3],[4, 35, 1].

2

Primary definitions, notation and general observations

In the following we shall consider the algebra P of polynomials P =F[x] over the field F of characteristic zero. All operators or functionals studied here are to be understood as linear operators on P . It shall be easy to see that they are always well defined. Throughout the note while saying “polynomial sequence {pn }∞ 0 ” we mean deg pn = n; n ≥ 0 and we adopt also the convention that deg pn < 0 iff pn ≡ 0. Consider ℑ - the family of functions‘ sequences (in conformity with Viskov [3, 7, 8] notation) such that: ℑ = {ψ; R ⊃ [a, b] ; q ∈ [a, b] ; ψ (q) : Z → F ; ψ0 (q) = 1 ; ψn (q) 6= 0; ψ−n (q) = 0; n ∈ N}. We shall call ψ = {ψn (q)}n≥0 ; ψn (q) 6= 0; n ≥ 0 and ψ0 (q) = 1 an admissible sequence. Let now nψ denotes [3, 4] nψ ≡ ψn−1 (q) ψn−1 (q) , n ≥ 0. Then (note that for admissible ψ, 0ψ = 0) nψ ! ≡ ψn−1 (q) ≡ nψ (n − 1)ψ (n − 2)ψ (n − 3)ψ ....2ψ 1ψ ;   n nkψ = nψ (n − 1)ψ .... (n − k + 1)ψ ,   ≡ k

k

nψ kψ !

and expψ {y} =

0ψ ! = 1 ∞ P

k=0

yk . kψ !

ψ

Definition 2.1. Let ψ be admissible. Let ∂ψ be the linear operator lowering degree of polynomials by one defined according to ∂ψ xn = nψ xn−1 ; n ≥ 0. Then ∂ψ is called the ψ-derivative. Remark 2.2. a) For any rational function R the corresponding factorial R (q n )! of the sequence R(q n ) is defined naturally [3, 4, 1] as it is defined for nψ sequence , i.e. : R(q n )! = R(q n )R(q n−1 )...R(q 1 ) The choice ψn (q)=[R (q n )!]−1 and R (x) = 1−x results in 1−q the well known q-factorial nq ! = nq (n − 1)q !; 1q ! = 0q ! = 1 while the ψ-derivative ∂ψ becomes now (nψ = nq ) the Jackson’s derivative [25, 26, 27, 2, 3] ∂q : (∂q ϕ) (x) = ϕ(x)−ϕ(qx) . (1−q)x b) Note also that if ψ = {ψn (q)}n≥0 and ϕ = {ϕn (q)}n≥0 are two admissible sequences then [∂ψ , ∂ϕ ]= 0 iff ψ = ϕ. Here [,] denotes the commutator of operators. Definition 2.3. Let E y (∂ψ ) ≡ expψ {y∂ψ } = translation operator.

∞ P

k=0

k y k ∂ψ . kψ !

E y (∂ψ ) is called the generalized

770


Note 2.4. [3, 4, 1] P E a (∂ψ ) f (x) ≡ f (x +ψ a) ; (x +ψ a)n ≡ E a (∂ψ ) xn ; E a (∂ψ ) f =

n≥0

an n ∂ f; nψ ! ψ

and in general (x +ψ a)n 6= (x +ψ a)n−1 (x +ψ a). Note also [1] that in general (1 +ψ (−1))2n+1 6= 0 ; n ≥ 0 though (1 +ψ (−1))2n = 0; n ≥ 1. Note 2.5. [1] expψ (x +ψ y) ≡ E x (∂ψ ) expψ {y} - while in general expψ {x + y} = 6 expψ {x} expψ {y}. Possible consequent use of the identity expψ (x +ψ y) ≡ expψ {x} expψ {y} is quite encouraging. It leads among others to “ψ-trigonometry” either ψ-elliptic or ψ-hyperbolic via introducing n ocosψ , sinψ [1], coshψ , sinhψ or in general ψ-hyperbolic functions of m-th (ψ)

order hj (α)

R∋α→

j∈Zm

defined according to [13]

(ψ) hj

k 1 X −kj 2π (α) = ω expψ ω α ; j ∈ Zm , ω = exp i . m k∈Z m m

where 1 < m ∈ N and Zm = {0, 1, ..., m − 1}. Definition 2.6. A polynomial sequence {pn }∞ o is of ψ -binomial type if it satisfies the recurrence   Xn E y (∂ψ ) pn (x) ≡ pn (x +ψ y) ≡   pk (x) pn−k (y) . k k≥0 ψ

Polynomial sequences of ψ-binomial type [3, 4, 1] are known to correspond in one-toone manner to special generalized differential operators Q, namely to those Q = Q (∂ψ ) which are ∂ψ -shift invariant operators [3, 4, 1]. We shall deal in this note mostly with this special case,i.e. with ψ-umbral calculus. However before to proceed let us supply a basic information referring to this general case of Q-umbral calculus. Definition 2.7. Let P = F[x]. Let Q be a linear map Q : P → P such that: ∀p∈P deg (Qp) = (deg p) − 1 (with the convention deg p = −1 means p = const = 0). Q is then called a generalized difference-tial operator [12] or Gel‘fond-Leontiev [7] operator. Right from the above definitions we infer that the following holds. Observation 2.8. Let Q be as in Definition 2.7. Let Qxn =

n P

bn,k xn−k where bn,1 6= 0

k=1

of course. Without loose of generality take b1,1 = 1. Then ∃ {qk }k≥2 ⊂ F and there exists admissible ψ such that X Q = ∂ψ + qk ∂ψk (1) k≥2


if and only if bn,k

  n =   bk,k ; k

n ≥ k ≥ 1; bn,1 6= 0; b1,1 = 1.

771

(2)

ψ

If {qk }k ≥ 2 and an admissible ψ exist then these are unique. Notation 2.9. In the case (2) is true we shall write : Q = Q (∂ψ ) because then and only then the generalized differential operator Q is a series in powers of ∂ψ . Remark 2.10. Note that operators of the (1) form constitute a group under superposition of formal power series (compare with the formula (S) in [13]). Of course not all generalized difference-tial operators satisfy (1) i.e. are series just only in corresponding ψ-derivative ∂ψ (see Proposition 3.1 ). For example [15] let Q = 12 Dˆ xD− 13 D 3 . Then Qxn = 12 n2 xn−1 − 1 3 n−3 nx so according to Observation 2.8 nψ = 12 n2 and there exists no admissible ψ such 3 that Q = Q (∂ψ ).Here xˆ denotes the operator of multiplication by x while nk is a special case of nψk for the choice nψ = n. Observation 2.11. >From theorem 3.1 in [12] we infer that generalized differential P operators give rise to subalgebras Q of linear maps (plus zero map of course) commuting with a given generalized difference-tial operator Q. The intersection of two different P P algebras Q1 and Q2 is just zero map added. The importance of the above Observation 2.11 as well as the definition below may be further fully appreciated in the context of the Theorem 2.20 and the Proposition 3.1 to come. Definition 2.12. Let {pn }n≥0 be the normal polynomial sequence [12] ,i.e. p0 (x) = 1 and pn (0) = 0 ; n ≥ 1. Then we call it the ψ-basic sequence of the generalized difference-tial operator Q if in addition Q pn = nψ pn−1 . In parallel we define a linear map xˆQ : P → P (n+1) such that xˆQ pn = (n+1) pn+1 ; n ≥ 0. We call the operator xˆQ the dual to Q operator. ψ

When Q = Q (∂ψ ) = ∂ψ we write for short: xˆQ(∂ψ ) ≡ xˆ∂ψ ≡ xˆψ (see: Definition 2.19). Of course [Q, xˆQ ]= id therefore {Q, xˆQ , id} provide us with a continuous family of generators of GHW in - as we call it - Q-representation of Graves-Heisenberg-Weyl algebra. In the following we shall restrict to special case of generalized differential operators Q, namely to those Q = Q (∂ψ ) which are ∂ψ -shift invariant operators [3, 4, 1] (see: Definition 2.14). At first let us start with appropriate ψ-Leibnitz rules for corresponding ψ-derivatives. ψ-Leibnitz rules: It is easy to see that the following hold for any formal series f and g: ˆ · (∂q g), where Qf ˆ (x) = f (qx); for ∂q : ∂q (f · g) = (∂q f ) · g + Qf ˆ ∂0 : ∂R (f · g)(z) = R q Q ˆ {(∂0 f )(z) · g(z) + f (0)(∂0 g)(z)} for ∂R = R q Q

772


ˆ xn−1 = nR xn−1 ; (nψ = nR = nR(q) = R (q n )) and finally where - note - R q Q for ∂ψ = n ˆ ψ ∂0 : ∂ψ (f · g)(z) = n ˆ ψ {(∂o f )(z) · g(z) + f (0)(∂0 g)(z)} where n ˆ ψ xn−1 = nψ xn−1 ; n ≥ 1. d Exampleo2.13. Let Q (∂ψ ) = Dˆ xD, where xˆf (x) = xf (x) and D = dx . Then ψ = n −1 ˆ ∂0 ≡ ∂R . Then ψ = [R (q n )!]−1 [(n2 )!] and Q = ∂ψ . Let Q (∂ψ ) R(q Q) and n≥0 n≥0

ˆ (x) = f (qx) and nψ = R(q n ). Q = ∂ψ ≡ ∂R . Here R(z) is any formal Laurent series; Qf ∂0 is q = 0 Jackson derivative which as a matter of fact - being a difference operator is the differential operator of infinite order at the same time: ∂0 =

∞ X

(−1)n+1

n=1

xn−1 dn . n! dxn

Naturally with the choice ψn (q) = [R (q n )!]−1 and R (x) = becomes the Jackson’s derivative [25, 26, 27, 2, 3] ∂q : (∂q ϕ) (x) =

(3) 1−x 1−q

the ψ-derivative ∂ψ

ˆ 1 − qQ ∂0 ϕ (x) . (1 − q)

The equivalent to (3) form of Bernoulli-Taylor expansion one may find [16] in Acta Eruditorum from November 1694 under the name “series univeralissima”. (Taylor‘s expansion was presented in his “Methodus incrementorum directa et inversa” in 1715 - edited in London). Definition 2.14. Let us denote by End(P ) the algebra of all linear operators acting on the algebra P of polynomials. Let X = {T ∈ End(P ); ∀ α ∈ F; [T, E α (∂ψ )] = 0}. ψ

P

Then ψ is a commutative subalgebra of End(P ) of F-linear operators. We shall call these operators T : ∂ψ -shift invariant operators. We are now in a position to define further basic objects of “ψ-umbral calculus” [3, 4, 1]. Definition 2.15. Let Q (∂ψ ) : P → P ; the linear operator Q (∂ψ ) is a ∂ψ -delta operator iff a) Q (∂ψ ) is ∂ψ - shift invariant; b) Q (∂ψ ) (id) = const 6= 0 where id(x)=x. The strictly related notion is that of the ∂ψ -basic polynomial sequence: Definition 2.16. Let Q (∂ψ ) : P → P ; be the ∂ψ -delta operator. A polynomial sequence {pn }n≥0 ; deg p n = n such that:


773

1) p0 (x) = 1; 2) pn (0) = 0; n > 0; 3) Q (∂ψ ) pn = nψ pn−1 ,∂ψ -delta operator Q (∂ψ )is called the ∂ψ -basic polynomial sequence of the ∂ψ -delta operator. Identification 2.17. It is easy to see that the following identification takes place: ∂ψ delta operator Q (∂ψ ) = ∂ψ -shift invariant generalized differential operator Q. Of course not every generalized differential operator might be considered to be such. Note 2.18. Let Φ (x; λ) =

P

n≥0

λn p nψ ! n

(x) denotes the ψ-exponential generating function

of the ∂ψ -basic polynomial sequence {pn }n≥0 of the ∂ψ -delta operator Q ≡ Q (∂ψ ) and let Φ (0; λ) = 1. Then QΦ (x; λ) = λΦ (x; λ) and Φ is the unique solution of this eigenvalue problem. If in addition (2.2) is satisfied then there exists such an admissible sequence ϕ that Φ (x; λ) = expϕ {λx} (see Example 3.1). The notation and naming established by Definitions 2.15 and 2.16 serve the target to preserve and to broaden simplicity of Rota‘s finite operator calculus also in its extended “ψ-umbral calculus” case [3, 4, 1]. As a matter of illustration of such notation efficiency let us quote after [3] the important Theorem 2.20 which might be proved using the fact that ∀ Q (∂ψ ) ∃! invertible S ∈ Σψ such that Q (∂ψ ) = ∂ψ S. ( For Theorem 2.20 see also Theorem 4.3. in [12], which holds for operators, introduced by the Definition 2.12). Let us define at first what follows. Definition 2.19. (compare with (17) in [8]) The Pincherle ψ-derivative is the linear map ’ : Σψ → Σψ ; T ’ = T xˆψ - xˆψ T ≡[T , xˆψ ] where the linear map xˆψ : P → P ; is defined in the basis {xn }n≥0 as follows xˆψ xn =

ψn+1 (q) (n + 1) n+1 (n + 1) n+1 x = x ; ψn (q) (n + 1)ψ

n ≥ 0.

Then the following theorem is true [3] Theorem 2.20. (ψ-Lagrange and ψ-Rodrigues formulas [34, 11, 12, 23, 3]) Let {pn (x)}∞ n=0 be ∂ψ -basic polynomial sequence of the ∂ψ -delta operator Q (∂ψ ). Let Q (∂ψ ) = ∂ψ S. Then for n > 0: (1) pn (x) = Q (∂ψ )’ S −n−1 xn ; n (2) pn (x) = S −n xn − nψ (S −n )’xn−1 ; n (3) pn (x) = nψ xˆψ S −n xn−1 ; n (4) pn (x) = nψ xˆψ (Q (∂ψ )’ )−1 pn−1 (x) (← Rodrigues ψ-formula ). For the proof one uses typical properties of the Pincherle ψ-derivative [3].Because ∂ψ ’ = id we arrive at the simple and crucial observation.

774


Observation 2.21. [3,35] The triples {∂ψ , x ˆψ , id} for any admissible ψ-constitute the set of generators of the ψlabelled representations of Graves-Heisenberg-Weyl (GHW) algebra [17, 18, 19, 35, 1]. Namely, as easily seen [∂ψ , xˆψ ] = id. (compare with Definition 2.12) Observation 2.22. In view of the Observation 2.21 the general Leibnitz rule in ψrepresentation of Graves-Heisenberg-Weyl algebra may be written (compare with 2.2.2 Proposition in [18]) as follows    Xnm m−k n−k ∂ψn xˆm = (4)     k! xˆψ ∂ψ . ψ k k k≥0 One derives the above ψ-Leibnitz rule from ψ-Heisenberg-Weyl exponential commutation rules exactly the same way as in {D, x ˆ, id} GHW representation - (compare with 2.2.1 Proposition in [18] ). ψ-Heisenberg-Weyl exponential commutation relations read: exp{t∂ψ } exp{aˆ xψ } = exp{at} exp{aˆ xψ } exp{t∂ψ }.

(5)

To this end let us introduce a pertinent ψ-multiplication ∗ψ of functions as specified below. Notation 2.23. x ∗ψ xn = xˆψ (xn ) =

(n+1) n+1 x ; n≥0 (n+1)ψ 1ψ (n+1)! n+1 xn ∗ψ x = xˆnψ (x) = (n+1) x ; n≥0 ψ! x ∗ψ α1 = x ∗ψ α = α1−1 ψ x and α1 ∗ψ x = n xψ )xn . ∀x, α ∈ F; f (x) ∗ψ x = f (ˆ

hence x ∗ψ 1 = 1−1 ψ x 6≡ x hence 1 ∗ψ x = x therefore α ∗ψ x = αx and

For k 6= n xn ∗ψ xk 6= xk ∗ψ xn as well as xn ∗ψ xk 6= xn+k - in general i.e. for arbitrary admissible ψ; compare this with (x +ψ a)n 6= (x +ψ a)n−1 (x +ψ a). In order to facilitate in the future formulation of observations accounted for on the basis of ψ-calculus representation of GHW algebra we shall use what follows. Definition 2.24. With Notation 2.5 adopted let us define the ∗ψ powers of x according to xn∗ψ ≡ x ∗ψ x(n−1)∗ψ = xˆψ (x(n−1)∗ψ ) = x ∗ψ x ∗ψ ... ∗ψ x = nn!ψ ! xn ; n ≥ 0. Note that xn∗ψ ∗ψ xk∗ψ = nn!ψ ! x(n+k)∗ψ 6= xk∗ψ ∗ψ xn∗ψ = kk!ψ ! x(n+k)∗ψ for k 6= n and x0∗ψ = 1. This noncommutative ψ-product ∗ψ is deviced so as to ensure the following observations. Observation 2.25. (a) ∂ψ xn∗ψ = nx(n−1)∗ψ ; n ≥ 0 (b) expψ [αx] ≡ exp{αˆ xψ }1


(c) (d) (e) (f )

775

exp[αx] ∗ψ (expψ {β xˆψ }1) = (expψ {[α + β]ˆ xψ })1 k n∗ψ k n∗ψ k ∂ψ (x ∗ψ x ) = (Dx ) ∗ψ x + x ∗ψ (∂ψ xn∗ψ ) hence ∂ψ (f ∗ψ g) = (Df ) ∗ψ g + f ∗ψ (∂ψ g) ; f, g - formal series f (ˆ xψ )g(ˆ xψ ) 1 = f (x) ∗ψ g˜(x) ; g˜(x) = g(ˆ xψ )1.

Now the consequences of Leibniz rule (e) for difference-ization of the product are easily P 1 pm (x); pm (x) = feasible. For example the Poisson ψ-process distribution πm (x) = N (λ,x) m≥0

1 is determined by

(λx)m pm (x) = ∗ψ expψ [−λx] (6) m! which is the unique solution (up to a constant factor) of the ∂ψ -difference equations systems ∂ψ pm (x) + λpm (x) = λpm−1 (x) m > 0 ; ∂ψ p0 (x) = −λp0 (x) (7) Naturally N(λ, x) = exp[λx] ∗ψ expψ [−λx]. As announced - the rules of ψ -product ∗ψ are accounted for on the basis of ψ-calculus representation of GHW algebra. Indeed,it is enough to consult Observation 2.25 and to introduce ψ-Pincherle derivation ∂ˆψ of series in powers of the symbol xˆψ as below. Then the correspondence between generic relative formulas turns out evident. xψ ) = [∂ψ , f (ˆ xψ )]. Then Observation 2.26. Let ∂ˆψ ≡ ∂ x∂ˆψ be defined according to ∂ˆψ f (ˆ ∂ˆψ xˆnψ = nˆ xψn−1 ; n ≥ 0 and ∂ˆψ xˆnψ 1 = ∂ψ xn∗ψ hence [∂ˆψ f (ˆ xψ )]1 = ∂ψ f (x) where f is a formal series in powers of xˆψ or equivalently in ∗ψ powers of x. As an example of application note how the solution of 7 is obtained from the obvious solution pm (ˆ xψ ) of the ∂ˆψ -Pincherle differential equation 8 formulated within G-H-W algebra generated by {∂ψ , xˆψ , id} ∂ˆψ pm (ˆ xψ ) + λpm (ˆ xψ ) = λpm−1 (ˆ xψ ) m > 0 ; ∂ψ p0 (ˆ xψ ) = −λp0 (ˆ xψ .)

(8)

Namely : due to Observation 2.25 (f) pm (x) = pm (ˆ xψ )1, where (λˆ xψ )m pm (ˆ xψ ) = expψ [−λˆ xψ ]. m!

3

(9)

The general picture of the algebra End(P ) from GHW algebra point of view

The general picture from the title above relates to the general picture of the algebra End(P ) of operators on P as in the following we shall consider the algebra P of polynomials P = F[x] over the field F of characteristic zero. With series of Propositions from [1,3,35,21] we shall draw an over view picture of the situation distinguished by possibility to develop further umbral calculus in its operator form for any polynomial sequences {pn }∞ 0 [12] instead of those of traditional binomial type only.

776


In 1901 it was proved [20] that every linear operator mapping P into P may be represented as infinite series in operators xˆ and D. In 1986 the authors of [21] supplied the explicit expression for such series in most general case of polynomials in one variable ( for many variables see: [22] ). Thus according to Proposition 1 from [21] one has: Proposition 3.1. Let Q be a linear operator that reduces by one the degree of each polynomial. Let {qn (ˆ x)}n≥0 be an arbitrary sequence of polynomials in the operator xˆ. P n ˆ Then T = qn (ˆ x)Q defines a linear operator that maps polynomials into polynomials. n≥0

Conversely, if Tˆ is linear operator that maps polynomials into polynomials then there exists a unique expansion of the form X Tˆ = qn (ˆ x)Qn . n≥0

It is also a rather matter of an easy exercise to prove the Proposition 2 from [21]: Proposition 3.2. Let Q be a linear operator that reduces by one the degree of each polynomial. Let {qn (ˆ x)}n≥0 be an arbitrary sequence of polynomials in the operator xˆ. Let a linear operator that maps polynomials into polynomials be given by P Tˆ = qn (ˆ x)Qn . n≥0 P n Let P (x; λ) = qn (x)λ denotes indicator of Tˆ. Then there exists a unique formal n≥0

series Φ (x; λ); Φ (0; λ) = 1 such that: QΦ (x; λ) = λΦ (x; λ) . −1 ˆ Then also P (x; λ) = Φ (x; λ) T Φ (x; λ). Example 3.3. Note that ∂ψ expψ {λx} = λ expψ {λx}; expψ [x] |x=0 = 1. (*) P Hence for indicator of Tˆ; Tˆ = qn (ˆ x)∂ψn we have: n≥0

ˆ expψ {λx }. (**) P (x; λ) = [expψ {λx }]−1 T After choosing ψn (q) = [nq !]−1 we get expψ {x } = expq {x }. In this connection note that 1 exp0 (x) = 1−x and exp(x) are mutual limit deformations for |x| < 1 due to: ∞ P exp0 (z)−1 1 = exp (z) ⇒ exp (z) = = z k ; |z| < 1 , i.e. o 0 z 1−z k=0

exp (x) ←− expq (x) = 1←q

∞ X n=0

xn 1 −→ . nq ! q→0 1 − x

1 Therefore corresponding specifications of (*) such as exp0 (λx) = 1−λx or exp(λx) lead to corresponding specifications of (**) for divided difference operator ∂0 and D operator including special cases from [21].

To be complete let us still introduce [3, 4] an important operator xˆQ(∂ψ ) dual to Q (∂ψ ).


777

Definition 3.4. (see Definition 2.12) Let {pn }n≥0 be the ∂ψ -basic polynomial sequence of the ∂ψ -delta operator Q (∂ψ ). A (n+1) linear map xˆQ(∂ψ ) : P → P ; xˆQ(∂ψ ) pn = (n+1) pn+1 ; n ≥ 0 is called the operator dual ψ to Q (∂ψ ). Comment 3.5. Dual in the above sense corresponds to adjoint in ψ-umbral calculus language of linear functionals’ umbral algebra (compare with Proposition 1.1.21 in [23] ). It is now obvious that the following holds. Proposition 3.6. Let {qn xˆQ(∂ψ ) }n≥0 be an arbitrary sequence of polynomials in the P operator xˆQ(∂ψ ) . Then T = qn xˆQ(∂ψ ) Q (∂ψ )n defines a linear operator that maps n≥0 polynomials into polynomials. Conversely, if T is linear operator that maps polynomials into polynomials then there exists a unique expansion of the form X T = qn xˆQ(∂ψ ) Q (∂ψ )n . (10) n≥0

Comment 3.7. The pair Q (∂ψ ) , xˆQ(∂ψ ) of dual operators is expected to play a role in the description of quantum-like processes apart from the q-case now vastly exploited [3, 4]. Naturally the Proposition 3.2 for Q (∂ψ ) and xˆQ(∂ψ ) dual operators is also valid. Summing up: we have the following picture for End(P ) - the algebra of all linear operators acting on the algebra P of polynomials. SP Q(P ) ≡ Q ⊂ End(P ) Q

and of course Q(P ) 6= End(P ) where the subfamily Q(P ) (with zero map) breaks up into P sum of subalgebras Q according to commutativity of these generalized difference-tial P operators Q (see Definition 2.7 and Observation 2.11). Also to each subalgebra ψ i.e. to each Q (∂ψ ) operator there corresponds its dual operator xˆQ(∂ψ ) xˆQ(∂ψ ) ∈ /

X

ψ

and both Q (∂ψ ) & xˆQ(∂ψ ) operators are sufficient to build up the whole algebra End(P ) according to unique representation given by (10) including the ∂ψ and xˆψ case. Summarising: for any admissible ψ we have the following general statement. General statement: End(P ) =[{∂ψ ,ˆ xψ }] = [{Q (∂ψ ) , xˆQ(∂ψ ) }] = [{Q , xˆQ }] i.e. the algebra End(P ) is generated by any dual pair {Q , xˆQ } including any dual pair {Q (∂ψ ) , xˆQ(∂ψ ) } or specifically by {∂ψ ,ˆ xψ } which in turn is determined by a choice of any admissible sequence ψ. As a matter of fact and in another words: we have bijective correspondences between different commutation classes of ∂ψ -shift invariant operators from End(P ), dif-

778


P ferent abelian subalgebras ψ , distinct ψ-representations of GHW algebra, different ψ-representations of the reduced incidence algebra R(L(S)) - isomorphic to the algebra Φψ of ψ-exponential formal power series [3] and finally - distinct ψ-umbral calculi [8, 12, 15, 24, 34, 3, 35]. These bijective correspondences may be naturally extended to encompass also Q-umbral calculi[12,1], Q-representations of GHW algebra [1] and abelian P subalgebras Q . (Recall: R(L(S)) is the reduced incidence algebra of L(S) where L(S)={A; A⊂S; |A| < ∞}; S is countable and (L(S); ⊆) is partially ordered set ordered by inclusion [11, 3] ). This is the way the Rota‘s devise has been carried into effect. The devise “much is the iteration of the few” [11] - much of the properties of literally all polynomial sequences - as well as GHW algebra representations - is the application of few basic principles of the ψ-umbral difference operator calculus [3, 35, 1]. ψ− Integration Remark : Recall : ∂o xn = xn−1 . ∂o is identical with divided difference operator. ∂o is identical with ˆ (x)f (qx). ∂ψ for ψ = {ψ (q)n }n≥0 ; ψ (q)n = 1 ; n ≥ 0 . Let Qf Recall also that there corresponds to the “∂q difference-ization” the q-integration [25, 26, 27] which is a right inverse operation to “q-difference-ization”[35, 1]. Namely F (z) :≡

Z q

∞ X ϕ (z) := (1 − q) z ϕ qk z qk

(11)

k=0

i.e. F (z) ≡

Z q

ϕ (z) = (1 − q) z

∞ X

!

ˆ k ϕ (z) = q Q k

(1 − q) z

k=0

Of course ∂q ◦

Z

= id

1 ˆ 1 − qQ

ϕ (z) . (12)

(13)

q

as

ˆ 1 − qQ 1 ∂0 (1 − q) zˆ = id. ˆ (1 − q) 1 − qQ

(14)

Naturally (14) might serve to define a right inverse operation to “q-difference-ization” ˆ 1−q Q (∂q ϕ) (x) = (1−q) ∂0 ϕ (x) and consequently the “q-integration “ as represented by (11) and (12). As it is well known the definite q-integral is an numerical approximation of the definite integral obtained in the q → 1 limit. Following the q-case example we introduce now an R-integration (consult Remark 2.2).   Z 1 1 xn = xˆ  xn = xn+1 ; n ≥ 0 (15) n+1 ) R (q ˆ R R qQ


Of course ∂R ◦

R

R

779

= id as  ˆ ∂o xˆ R qQ



1  = id. ˆ R qQ

(16)

Let us then finally introduce the analogous representation for ∂ψ difference-ization

Then

∂ψ = n ˆ ψ ∂o ; n ˆ ψ xn−1 = nψ xn−1 ; n ≥ 1.

(17)

1 1 x = xˆ xn = xn+1 ; n ≥ 0 n ˆ (n + 1) ψ ψ ψ

(18)

Z

n

and of course ∂ψ ◦

Z

= id

(19)

ψ

Closing Remark: The picture that emerges discloses the fact that any ψ-representation of finite operator calculus or equivalently - any ψ-representation of GHW algebra makes up an example of the algebraization of the analysis - naturally when constrained to the algebra of polynomials. We did restricted all our considerations to the algebra P of polynomials. Therefore the distinction in-between difference and differentiation operators disappears. All linear operators on P are both difference and differentiation operators if the degree P dk k d of differentiation or difference operator is unlimited. For example dx = ∆ where k! k≥1 d k P δn dn dk = dx x x=0 = (−1)k−1 (k − 1)! or ∆ = where δn = [∆xn ]x=0 = 1. Thus the n! dxn n≥1

difference and differential operators and equations are treated on the same footing. For new applications - due to the first author see [4,1,36-41]. Our goal here was to deliver the general scheme of ”ψ-umbral” algebraization of the analysis of general differential operators [12]. Most of the general features presented here are known to be pertinent to the Q representation of finite operator calculus (Viskov, Markowsky, Roman) where Q is any linear operator lowering degree of any polynomial by one . So it is most general example of the algebraization of the analysis for general differential operators [12].

4

Glossary

In order to facilitate the reader a simultaneous access to quoted references of classic Masters of umbral calculus - here now follow short indicatory glossaries of notation used by Ward [2], Viskov [7, 8], Markowsky [11], Roman [28]-[32] on one side and the Rota-oriented notation on the other side. See also [33].

780


Ward

Rota - oriented (this note)

[n] ; [n]!

nψ ; nψ !

basic binomial coefficient [n, r] =

[n]! [r]![n−r]!

ψ-binomial coefficient

n k ψ

D = Dx - the operator D

∂ψ - the ψ-derivative

D xn = [n] xn−1

∂ψ xn = nψ xn−1

(x + y)n

(x +ψ y)n

(x + y)n ≡

n P

[n, r] xn−r y r

(x +ψ y)n =

n P

k

≡

nψ kψ !

n xk y n−k k ψ

r=0

k=0

basic displacement symbol

generalized shift operator

E t; t ∈ Z

E y (∂ψ ) ≡ expψ {y∂ψ }; y ∈ F

Eϕ(x) = ϕ(x + 1)

E(∂ψ )ϕ(x) = ϕ(x +ψ 1)

E t ϕ(x) = ϕ x + t

E y (∂ψ )xn ≡ (x +ψ y)n


Ward


basic difference operator

ψ-difference delta operator

∆ = E − id

∆ψ = E y (∂ψ ) − id

∆ = ε(D) − id =

∞ P

n=0

Dn [n]!

− id

Roman


t; txn = nxn−1



htk |p(x)i = p(k) (0)

[∂ψk p(x)]|x=0

781

782


Roman


evaluation functional

generalized shift operator

ǫy (t) = exp {yt}

E y (∂ψ ) = expψ {y∂ψ }

htk |xn i = n!δn,k

hǫy (t)|p(x)i = p(y)

ǫy (t)xn =

P

k≥0

n k

xk y n−k

formal derivative

f ′ (t) ≡

d f (t) dt

[E y (∂ψ )pn (x)]|x=0 = pn (y)

E y (∂ψ )pn (x) =

P

k≥0

n p (x)pn−k (y) k ψ k

Pincherle derivative

[Q(∂ψ )]‘≡

d Q(∂ψ ) d∂ψ

f (t) compositional inverse of

Q−1 (∂ψ ) compositional inverse of

formal power series f (t)

formal power series Q(∂ψ )



Roman

θt ; θt xn = xn+1 ; n ≥ 0

θt t = xˆD

P

k≥0

sk (x) k t kψ !

=

xˆψ ; xˆψ xn =

n+1 xn+1 ; (n+1)ψ

n≥0

ˆ xˆψ ∂ψ = xˆD = N

P

k≥0

sk (x) k z kψ !

=

[g(f(z))]−1 exp {xf (t)}

s(q −1 (z)) expψ {xq −1 (z)}

{sn (x)}n≥0 - Sheffer sequence

q(t), s(t) indicators

for (g(t), f (t))

of Q(∂ψ ) and S∂ψ

g(t) sn (x) = qn (x) - sequence

sn (x) = S∂−1 qn (x) - ∂ψ - basic ψ

associated for f (t)

sequence of Q(∂ψ )

783

784


Roman


The expansion theorem:

The First Expansion Theorem

h(t) =

∞ P

k=0

hh(t)|pk (x)i f (t)k k!

T =

P

n≥0

[T pn (z)]|z=0 Q(∂ψ )n nψ

pn (x) - sequence associated for f (t) ∂ψ - basic polynomial sequence {pn }∞ 0

exp{yf(t)} =

∞ P

k=0

pk (y) k t k!

The Sheffer Identity:

sn (x + y) =

n P

k=0

n k

pn (y)sn−k (x)

expψ {xQ−1 (x)} =

P

k≥0

pk (y) k z k!

The Sheffer ψ-Binomial Theorem:

sn (x +ψ y) =

P

k≥0

n s (x)qn−k (y) k ψ k


Viskov


θψ - the ψ-derivative


θψ xn =

ψn−1 n−1 x ψn


Ap (p = {pn }∞ 0 )

Q

Ap pn = pn−1

Q pn = nψ pn−1

Bp (p = {pn }∞ 0 )

xˆQ

Bp pn = (n + 1) pn+1

Epy (p = {pn }∞ 0 )

Epy pn (x) =

n P

xˆQ pn =

n+1 p (n+1)ψ n+1

E y (∂ψ ) ≡ expψ {y∂ψ }

E y (∂ψ ) pn (x) =

pn−k (x)pk (y)

k=0

=

P

k≥0


785

786


Viskov


T − εp -operator:

E y - shift operator:

T Ap = Ap T

E y ϕ(x) = ϕ(x +ψ y)

T - ∂ψ -shift invariant operator:

∀y∈F T Epy = Epy T

∀α∈F [T, E α (∂ψ )] = 0

Q - δψ -operator:

Q(∂ψ ) - ∂ψ -delta-operator:

Q - ǫp -operator and

Q(∂ψ ) - ∂ψ -shift-invariant and

Qx = const 6= 0

Q(∂ψ )(id) = const 6= 0

{pn (x), n ≥ 0} - (Q, ψ)-basic

{pn }n≥0 -∂ψ -basic

polynomial sequence of the


δψ -operator Q

∂ψ -delta-operator Q(∂ψ )


=

Viskov


ψ-binomiality property

ψ-binomiality property

Ψy sn (x) =

E y (∂ψ )pn (x) =

n P

m=0

T =

ψn ψn−m sm (x)pn−m (y) ψn

P

ψn [V T pn (x)]Qn

=

P

k≥0

T =

n≥0

n≥0

T Ψy p(x) = P

n≥0

ψn sn (y)Qn ST p(x)

P


[T pn (z)]|z=0 Q(∂ψ )n nψ !

T p(x +ψ y) = P

k≥0

sk (y) Q(∂ψ )k ST p(x) kψ !

787

788


Markowsky

Rota - oriented

L - the differential operator

Q

L pn = pn−1

Q pn = nψ pn−1

M

xˆQ

n+1 p (n+1)ψ n+1

M pn = pn+1

xˆQ pn =

Ly

E y (Q) =

P

k≥0

E y (Q) pn (x) =

Ly pn (x) =

=

n P

k=0

n k

pk (x)pn−k (y)

pk (y) k Q kψ !

=

P

k≥0

n k ψ

pk (x)pn−k (y)

E a - shift-operator:

E y - ∂ψ -shift operator:

E a f (x) = f (x + a)

E y ϕ(x) = ϕ(x +ψ y)


Markowsky

Rota - oriented

G - shift-invariant operator:

T - ∂ψ -shift invariant operator:

EG = GE

∀α∈F [T, E(Q)] = 0

G - delta-operator:

L = L(Q) - Qψ -delta operator:

G - shift-invariant and

[L, Q] = 0 and

Gx = const 6= 0

L(id) = const 6= 0

DL (G)

G′ = [G(Q), x ˆQ ]

L - Pincherle derivative of G

Q - Pincherle derivative

DL (G) = [G, M]

789

790


Markowsky

Rota - oriented

{Q0 , Q1 , ...} - basic family

{pn }n≥0 -ψ-basic

for differential operator L


generalized difference operator Q

binomiality property

Q - ψ-binomiality property

Pn (x + y) =

E y (Q)pn (x) =

=

n P

i=0

n i

Pi (x)Pn−i (y)

=

P

k≥0


Acknowledgment The authors thank the Referee for suggestions , which have led us to improve the presentation of the paper. The authors express also their gratitude to Katarzyna Kwa´sniewska for preparation the LATEXversion of this contribution.

References [1] A.K. Kwa´sniewski: “On Simple Characterisations of Sheffer ψ-polynomials and Related Propositions of the Calculus of Sequences”, Bulletin de la Soc. des Sciences et des Letters de Lód´z 52 SERIE Reserchers sur les deformations, Vol. 36(45), (2002) (ArXiv: math.CO/0312397). [2] M. Ward: “A Calculus of Sequences”, Amer. J. Math., Vol. 58(255), (1936).


791

[3] A.K. Kwa´sniewski: “Towards ψ-extension of Finite Operator Calculus of Rota”, Rep. Math. Phys., Vol. 48(3), (2001) (ArXiv: math.CO/0402078 Feb 2004). [4] A.K. Kwa´sniewski: “On extended finite operator calculus of Rota and quantum groups”, Integral Transforms and Special Functions, Vol. 2(4), (2001). [5] R.P. Boas and R.C. Buck Jr.: “Polynomials Defined by Generating Relations”, Am. Math. Monthly, Vol. 63(626), (1959). [6] R.P. Boas and R.C. Buck Jr.: Springer, Berlin, 1964.

Polynomial Expansions of Analytic Functions,

[7] O.V. Viskov: “Operator characterization of generalized Appel polynomials”, Soviet Math. Dokl., Vol. 16(1521), (1975). [8] O.V. Viskov: “On the basis in the space of polynomials”, Soviet Math. Dokl., Vol. 19(250), (1978). [9] G.-C. Rota and R. Mullin: On the foundations of combinatorial theory, III. Theory of Binomial Enumeration in ”Graph Theory and Its Applications”, Academic Press, New York, 1970. [10] G.C. Rota, D. Kahaner and A. Odlyzko: “On the Foundations of combinatorial theory. VIII. Finite operator calculus”, J. Math. Anal. Appl., Vol. 42(684), (1973). [11] G.C. Rota: Finite Operator Calculus, Academic Press, New York, 1975. [12] G. Markowsky: “Differential operators and the Theory of Binomial Enumeration”, J. Math. Anal. Appl., Vol. 63(145), (1978). [13] A.K. Kwa´sniewski: “Higher Order Recurrences for Analytical Functions of Tchebysheff Type”, Advances in Applied Clifford Algebras, Vol. 9(41), (1999). [14] O.V. Viskov: “Noncommutative Approach to Classical Problems of Analysis”, Trudy Matiematicz‘eskovo Instituta AN SSSR, Vol. 177(21), (1986). [15] A. Di Bucchianico and D. Loeb: “A Simpler Characterization of Sheffer Polynomials, Studies in Applied Mathematics”, J. Math. Anal. Appl., Vol. 92(1), (1994). [16] N.Ya. Sonin: “Rjady Ivana Bernulli”, Izw. Akad. Nauk , Vol. 7(337), (1897). [17] C. Graves: “On the principles which regulate the interhange of symbols in certain symbolic equations”, Proc. Royal Irish Academy, Vol. 6(144), (1853-1857). [18] P. Feinsilver and R. Schott: Algebraic Structures and Operator Calculus, Kluwer Academic Publishers, New York, 1993. [19] O.V. Viskov: “Newton-Leibnitz Formula and the Taylor Expansion”, Integral Transforms and Special Functions, Vol. 1(2), (1997). [20] S. Pincherle and U. Amaldi: Le operazioni distributive e le loro applicazioni all‘analisi , N. Zanichelli, Bologna, 1901. [21] S.G. Kurbanov and V.M. Maximov: “Mutual Expansions of Differential operators and Divided Difference Operators”, Dokl. Akad. Nauk Uz. SSSR, Vol. 4(8), (1986). [22] A. Di Bucchianico and D. Loeb: Integral Transforms and Special Functions, Vol. 4(49), (1996). [23] P. Kirschenhofer: “Binomialfolgen, Shefferfolgen und Faktorfolgen in der q-Analysis”, Sitzunber. Abt. II Oster. Ackad. Wiss. Math. Naturw. Kl., Vol. 188(263), (1979).

792


[24] A. Di Bucchianico and D. Loeb: “Sequences of Binomial Type with persistent roots”, J. Math. Anal. Appl., Vol. 199(39), (1996). [25] F.H. Jackson: “q-difference equations”, Quart. J. Pure and Appl. Math., Vol. 41(193), (1910). [26] F.H. Jackson: “The q-integral analogous to Borels integral”, Messenger of Math., Vol. 47(57), (1917). [27] F.H. Jackson: “Basic Integration”, Quart. J. Math., Vol. 2(1), (1951). [28] S.M. Roman: “The Theory of Umbral Calculus. I”, J. Math. Anal. Appl., Vol. 87(58), (1982). [29] S.M. Roman: “The Theory of Umbral Calculus. II”, J. Math. Anal. Appl., Vol. 89(290), (1982). [30] S.M. Roman: “The Theory of Umbral Calculus. III”, J. Math. Anal. Appl., Vol. 95(528), (1983). [31] S.M. Roman: The umbral calculus, Academic Press, New York, 1984. [32] S.R. Roman: “More on the umbral calculus with emphasis on the q-umbral calculus”, J. Math. Anal. Appl., Vol. 107(222), (1985). [33] A.K. Kwasniewski and E. Gradzka: “Further remarks on ψ-extensions of finite operator calculus”, Rendiconti del Circolo Matematico di Palermo Serie II , Suppl., 69(117), (2002). [34] J.F. Steffensen: “The poweroid an extension of the mathematical notion of power”, Acta Mathematica, Vol. 73(333), (1941). [35] A.K. Kwa´sniewski: “Main theorems of extended finite operator calculus”, Integral Transforms and Special Functions, Vol. 14(499), (2003). [36] A.K.Kwa´sniewski: “The logarithmic Fib-binomial formula”, Advan. Stud. Contemp. Math. , Vol. 9(1), (2004) (19-26 ArXiv: math.CO/0406258 13 June 2004). [37] A.K. Kwa´sniewski: “On ψ-basic Bernoulli-Ward polynomials”, Bull. Soc. Sci. Lett. Lodz, in print (ArXiv: math.CO/0405577 30 May 2004). [38] A.K. Kwa´sniewski: “ψ-Appell polynomials‘ solutions of the -difference calculus nonhomogeneous equation”, Bull. Soc. Sci. Lett. Lodz, in print (ArXiv: math.CO/0405578 30 May 2004). [39] A.K. Kwa´sniewski: “On ψ-umbral difference Bernoulli-Taylor formula with Cauchy type remainder”, Bull. Soc. Sci. Lett. Lodz, in print (ArXiv: math.GM/0312401 December 2003). [40] A.K. Kwa´sniewski: “First contact remarks on umbra difference calculus references streams” , Bull. Soc. Sci. Lett. Lodz, in print (ArXiv: math.CO/0403139 v1 8 March 2004). [41] A.K. Kwa´sniewski: “On extended umbral calculus, oscillator-like algebras and Generalized Clifford Algebra”, Advances in Applied Clifford Algebras, Vol. 11(2), (2001), pp. 267–279 (ArXiv: math.QA/0401083 January 2004).

CEJM 2(5) 2005 793–800

On Martin Bordemann’s proof of the existence of projectively equivariant quantizations Pierre B. A. Lecomte∗ Department of Mathematics, University of Liege, B-4000 Liege ,Belgium

Received 15 December 2003; accepted 17 April 2004 Abstract: The paper explains the notion of projectively equivariant quantization. It gives a sketch of Martin Bordemann’s proof of the existence of projectively equivariant quantization on arbitrary manifolds. c Central European Science Journals. All rights reserved.

Keywords: Quantization, projective structure, differential operators, smooth manifolds MSC (2000): 53C05, 53D55

1

Introduction

The definition of the notion of projectively equivariant quantization has been introduced in [7]. The idea was to extend results concerning flat projective manifolds obtained by V. Ovsienko and myself to arbitrary smooth manifolds . In the middle of 2002, M. Bordemann wrote a nice paper [2] where he showed that such quantizations exist on every smooth manifold. The purpose of this note is to explain what the quantizations for which the notion has been induced are and to explain the main steps of the proof of M. Bordemann. This was the content of my talk at the meeting in Krynica, in May 2003. I would like to express my best gratitude to the organisers of the meeting for their warm invitation and for the opportunity of giving my talk. Throughout this note, by a smooth manifold we mean a connected, Hausdorf and second countable manifold of class C ∞ . Its dimension will always be assumed to be at least two. ∗


794

2

P. Lecomte / Central European Journal of Mathematics 2(5) 2005 793–800

Projectively equivariant quantizations

2.1 Differential operators on IRm The idea of projectively equivariant quantization has arisen slowly, trying to understand the behavior of a differential operator A : f 7→

X X

i1 ...ir

A

0≤r≤k i1 ,...,ir

∂r f ∂xi1 · · · xis

(1)

under a change of local coordinates. At least in a formal way, such an operator is completely encoded by the polynomial PA : ξ 7→

X X

Ai1 ...ir ξi1 · · · ξir

(2)

0≤r≤k i1 ,...,ir

where ξ = (ξ1 , . . . , ξm ) ∈ IRm∗ , with coefficients depending smoothly on the variables xj : the map A 7→ PA is a linear bijection from the space D(IRm ) of differential operators in the space S(IRm ) of these polynomials. However, it does not keep track of the behavior of the operator under arbitrary change of coordinates: the terms of higher order in A and of higher degree in PA have the same behavior but that of the other terms is indeed extremely different because, in general, the terms of order r of the operator affect the terms at each order less than r, and only these terms, while in the polynomial, the modifications induced by a term stay within the homogeneous part of the same degree of the polynomial. On the other hand, both A and PA change in the same way when affected by an affine change of coordinates. In other words, both D(IRm ) and S(IRm ) are modules of the group of diffeomorphisms of IRm and, as such, they seem to be non isomorphic while they could be isomorphic under the action of some nontrivial subgroups. Observe that the space D(IRm ) is filtered, by the order, and that S(IRm ) is graded, by the degree. The latter is isomorphic to the graded module associated to the former. Derivating the action of the flows of vector fields, we see that D(IRm ) and S(IRm ) are also representations of the Lie algebra V ect(IRm ) of the vector fields of IRm acting by Lie derivatives, as well as of its Lie subalgebras. In particular, they are isomorphic as representations of the Lie algebra af fm of vector fields the coefficients of which are polynomials of order at most 1 in the xj ’s. Between af fm and V ect(IRm ), there was room for more interesting Lie subalgebras of which D(IRm ) and S(IRm ) could be isomorphic representations. One of them is the projective embedding slm+1 of sl(m + 1, IR). It is the linear span of the set j

j

{∂i , x ∂i , x (

m X k=1

xk ∂k )|i, j ∈ {1, ..., m}}


795

of vector fields of IRm . It is a Lie subalgebra and it is isomorphic to the Lie algebra sl(m + 1, IR). The subalgebra slm+1 may be viewed as the set of local expressions of the fundamental vector fields associated with the natural action of SL(m, IR) on the m-dimensional real projective space, in any of its canonical chart. The following statement is a special case of a result of [9]. Theorem 2.1. There exists a unique linear slm+1 -equivariant bijection Q : S(IRm ) → D(IRm ) that is filtered ant that preserves the principal symbol. The principal symbol σA of a differential operator A of order k is the image of A under the projection from the space of all differential operators onto its associated graded space - on IRm , it is just the sum of the terms of PA of highest degree. The above property 2.1 has been used as a tool in the study of the space of differential operators on an arbitrary manifolds (see for instance [9] and also [10]). For instance, it gives a very simple proof of the fact that the modules D(M) and S(M) are not isomorphic under the action of the Lie algebra V ect(M) of all vector fields of the manifold M. From this point of view, it is interesting to note that slm+1 is a maximal subalgebra of the Lie algebra V ect∗ (IRm ) of vector fields of IRm having polynomial coefficients. In a sense, Theorem 2.1 is thus optimal. The maximal finite dimensional subalgebras of V ect∗ (IRm ) have been classified in [11] and Theorem 2.1 has been extended in various ways (see for example [1],[8], [5] and the references given in these papers). Tentative efforts have been made to write the map Q of Theorem in an explicit coordinate free form. Solutions have been obtained for operators of order 2 and 3 (see for instance [3], [4]). In the expressions obtained, partial derivatives were replaced by covariant derivation and terms in the curvature of the derivation appears in the formulas.

2.2 Quantizations The elements of S(IRm ) may be viewed as the ”classical observables” of some mechanical system having IRm as configuration space and T ∗ IRm as phase space, the coordinates xj of IRm corresponding to position, while the components ξj of ξ correspond to impulsions. Therefore, a map such as Q of Theorem 2.1 is the beginning of a sort of quantization procedure: it assigns to each classical observable some differential operator. With that in mind, it is natural to deal with operators whose arguments and values are densities instead of functions (as we implicitly assumed up to here), specifically densities of weight 1/2. We will denote by Dλ,µ (M) the space of differential operators on a smooth manifold M working from the space Fλ (M) of (fields of) densities of weight λ of M into the space Fµ (M) and by Sδ (M) the space of smooth functions on T ∗ M which are polynomial on the fibers and which take values in the space of densities of weight δ = µ − λ. Both are representation of the Lie algebra of vector fields V ect(M) of M and the latter is again the graded module associated with the filtration of the former by the order. By a

796


quantization, we mean here a linear map Q : Sδ (M) → Dλ,µ (M) which is filtered and which preserves the principal symbol (this means that it induces the identity on the associated graded spaces; in particular, it is a bijection). We denote by Qλ,µ (M) the space of quantizations of M, for given weights λ and µ.

2.3 Projective structures The naive idea of a projective structure is that of the set of straight lines. On a given manifold M, there is in general no notion of line. Of course, one could replace lines by geodesics but then, there is in general no canonical notion of geodesic because M does not carry any canonical linear connection. Therefore, a projective structure on a manifold is understood as an equivalence class among the set of linear connections of M (with vanishing torsion) for the relation: ”to have the same set of geometrical geodesics”. As linear connections are characterized by their covariant derivations, we can use these instead. A classical result of Weyl states that two covariant derivations define the same projective structure if and only if they differ by a symmetric tensor of type ( 21 ) of the form (X, Y ) 7→ ω(X)Y + ω(Y )X. A diffeomorphism preserves a projective structure if it transforms any covariant derivation of the structure into another one and a vector field is said to be an infinitesimal automorphism of the structure if its flow preserves the structure. On IRm , the infinitesimal automorphism of the projective structure associated with the canonical flat connection are precisely the elements of slm+1 . More on projective structures could be found for instance in [6].

2.4 Projectively equivariant quantization The above considerations lead me, in [7], to propose a generalization of the notion of quantization. The idea was to restore V ect(M)-equivariance, on arbitrary smooth manifolds M, by adding a new argument: a covariant derivation ∇, the slm+1 -equivariance being replaced by the fact that the dependance on ∇ is only through its projective equivalence class. More precisely, a projectively equivariant quantization, a qpe in short, is a map Q : CM → Qλ,µ (M) (where CM denotes the space of covariant derivation with vanishing torsion of M) which is natural and which is such that Q∇ = Q∇′ if ∇ and ∇′ define the same projective structure of M. In [7], by natural, I meant commuting with all Lie derivatives with respect to vector fields. In [2], M. Bordemann used the more proper sense, that of the well known theory of natural bundles due to Nijenhuis.


3

797

Existence of qpe

I will sketch now the proof of the existence of qpe given by M. Bordemann. First, I recall some elementary facts.

3.1 The canonical map CM → Qλ,µ (M) There is a canonical way to assign to each ∇ ∈ CM a quantization τ∇ . It is a folk construction, but I would like to refer to it as the Lichnerowicz’s map as A. Lichnerowicz has heavily used it in the theory of star-products. It is easy to understand. First one iterates the covariant derivation. For any density f of weight λ, ∇p f is a symmetric p-covariant tensor field valued in the densities of weight λ and is defined inductively: ∇1 f = ∇f and ∇p f is the symmetrization of the tensor field (X1 , ..., Xp ) 7→ (∇X1 (∇p−1 f ))(X2 , ..., Xp ). Then, given a homogeneous polynomial P of degree p valued in the δ-densities, one defines τ∇ (P ) as being the differential operator τ∇ (P ) : f 7→< P, ∇p f > The map τ , extended by linearity to the whole Sδ (M) is clearly a natural map from CM into Qλ,µ (M). Of course, it is not invariant under changes of the covariant derivation within the same projective structure. We will denote it τ M if we want to stress its dependence to M. M. Bordemann calls it the ”standard ordering”.

3.2 The ideas The idea of M. Bordemann is to replace M by some manifold M, fibered over M, to ˜ of covariant derivations, P → P˜ of polynomials and construct natural lifts ∇ → ∇ ϕ → ϕ˜ of densities over M in such a way that τ M (P˜ )(ϕ) ˜ is itself the lift of some density (Q∇ (P ))(ϕ). The lift of ∇ should depend only on the projective class of ∇ instead of ∇ itself. Moreover, that of P depends on the lifted covariant derivation and turns out to be also projectively independent. The key point is to choose M and the lift appropriately. I will try below to explain what it means precisely.

3.3 The bundle M We choose once for all a nonzero weight λ0 of density. We denote by Mλ0 - M for short - the set of positive λ0 -densities of M. It is a subbundle of the bundle Fλ0 (M) of λ0 densities of M. It is a principal bundle with structure group IR+ , the group of positive real numbers, the action of r on a density α being just the product rα. This action is generated by one fundamental vector field on M, namely the Euler’s field ξ that generates the homotheties on Fλ0 (M).

798


The bundle Fλ (M) of λ-densities of M is associated with M and the action (r, s) 7→ r s of IR+ on IR. The (fields of) densities of weight λ of M, namely the sections ϕ ∈ Γ(Fλ (M)) of Fλ (M), are thus in a natural one-to-one onto correspondence with the real valued functions on M which are equivariant for that action, or equivalently, which ˜ is the lift of some density, we are such that ξ.f = − λλ0 f . In order to ensure that τ M (P˜ )(ϕ) will impose on the various lifts an appropriate behaviour under the action of ξ. Without going into details, let us say that we will ask for the following properties: ˜ =0 • Lξ ∇ • Lξ P˜ = − λδ0 • div∇˜ P˜ = 0 (The letter L denotes Lie derivative; recall that P ∈ Sδ (M), where δ = µ − λ; div∇ denotes the divergence operator associated to the covariant derivation ∇.) The last condition is less directly related to the vector field ξ. We will see later its role in a more accurate fashion. λ/λ0

3.4 Lifting the covariant derivations ˜ for which Lξ ∇ ˜ = 0 are uniquely M. Bordemann has shown that all natural lifts ∇ 7→ ∇ determined by three real numbers u, v, w and by the conditions ˜ X h Y h = (∇X Y )h + λ0 Rica (X, Y ) + uRics (X, Y )ξ • ∇ 2 ˜ Xh E = ∇ ˜ ξ X h = vX h • ∇ ˜ ξ ξ = wξ • ∇ For any X ∈ V ect(M), X h denote the horizontal lift of X on M. (Observe that the covariant derivation ∇ of M induces in a natural way a covariant derivation on M. This is due to the fact that the bundle Fλ0 (M) is associated with the principal bundle of linear frames of M.) Moreover, Ric denotes the Ricci curvature of ∇, the exponents a and s referring to its skewsymmetric and symmetric parts respectively. It is worth noticing that, at least locally, M has local frames consisting of ξ and of horizontal lifts of vector fields of M. ˜ is projectively invariant if and only The second important fact is that the lift ∇ 7→ ∇ if λ0 m + 1 1 u=− &v=w= 2 m−1 λ0 (m + 1) where m is the dimension of M.

3.5 Lifting tensor fields As easily seen, any tensor field T ∈ Sδk (M) such that Lξ T = − λδ0 T has a unique decomposition k X h T = Pk−i ξi i=0

where Pjh is the horizontal lift of some Pj ∈ Sδj (M).


799

The condition div∇˜ T = 0 will now determine all of them completely in terms of Pk . Indeed, that condition reads Pk−1 = −θ1 div∇ Pk .. . Pk−(i+1) = −θi+1 (div∇ Pk−i − 2λ0 < r, Pk−i+1 >) .. . P0 = −θk (div∇ P1 − 2λ0 < r, P2 >) where λ0 = (m + 1)λ0 , θj =

λ0 j[m − (m + 1)δ + 2k − j]

and

1 Rics (X, Y ) m−1 This shows that the P ′s are completely determined by Pk provided that none of the denominators in the above equation vanishes. In other word, we shall assume that δ is different from m+j , j ∈ IN. m+1 Under that assumption, in particular if δ = 0, it follows that [2] the above equations define a linear bijection between Sδ (M) and the space r(X, Y ) =

Sδ,λ0 (M) = {T ∈ Sδ (M)|Lξ T = −

δ T }. λ0

References [1] F. Boniver, H. Hansoul, P. Mathonet and N. Poncin: “Equivariant symbol calculus for differential operators acting on forms”, Lett. Math. Phys., Vol. 62(3), (2002), pp. 219–232. [2] M. Bordemann: “Sur l’existence d’une prescription d’ordre naturelle projectivement invariante” (arXiv:math.DG/0208171v1 22Aug2002). [3] S. Bouarroudj: “Projectively equivariant quantization map”, Lett. Math. Phys., Vol. 51(4), (2000), pp. 265–274. [4] S. Bouarroudj: “Formula for the projectively invariant quantization on degree three”, C. R. Acad. Sci. Paris Sér. I Math., Vol. 333(4), (2001), pp. 34–346. [5] C. Duval, P. Lecomte and V. Ovsienko: “Conformally equivariant quantization: existence and uniqueness”, Ann. Inst. Fourier (Grenoble), Vol. 49(6), (1999), pp. 1999–2029. [6] S. Kobayashi: Transformation groups in differential geometry, Springer, Berlin, 1972. [7] P. Lecomte: “Towards projectively equivariant Quantization. Noncommutative geometry and string theory”, Meada at al. (Eds.): Progress in theoretical physics, Vol. 144, (2001), pp. 125–132.

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[8] P. Lecomte: “On the cohomology of sl(m + 1, IR) acting on differential Operators and sl(m + 1, IR)-equivariant symbols”, Indaga. Math., NS, Vol. 11(1), (2000), pp. 95–114. [9] P. Lecomte and V. Ovsienko: “Projectively equivariant Symbol Calculus”, Letters in Math. Phys., Vol. 49, (1999), pp. 173–196, . [10] P. Lecomte and V. Ovsienko: “Cohomology of the Vector Fields Lie Algebra and Modules of differential Operators on a smooth Manifold”, Comp. Math., Vol. 124, (2000), pp. 95–110. [11] P. Mathonet and F. Boniver: “Maximal subalgebras of vector fields for equivariant quantizations”, J. Math. Phys., Vol. 42(2), (2001), pp. 582–589.

CEJM 2(5) 2005 801–810

The natural functions on the cotangent bundle of higher order vector tangent bundles over fibered manifolds Jan Kurek1∗ , Wlodzimierz M. Mikulski2† 1 Institute of Mathematics, Maria Curie Sklodowska University, Plac Marii Curie Sklodowskiej 1, Lublin, Poland 2 Institute of Mathematics, Jagiellonian University, Reymonta 4, Krak´ ow, Poland

Received 15 December 2003; accepted 30 September 2004 Abstract: For natural numbers r, s, q, m, n with s ≥ r ≤ q we determine all natural functions g : T ∗ (J (r,s,q) (Y, R1,1 )0 )∗ → R for any fibered manifold Y with m-dimensional base and ndimensional fibers. For natural numbers r, s, m, n with s ≥ r we determine all natural functions g : T ∗ (J (r,s) (Y, R)0 )∗ → R for any Y as above. c Central European Science Journals. All rights reserved.

Keywords: Bundle functor, natural function MSC (2000): 58A20, 53A55

Introduction In [2], I. Koláˇr introduced the concept of natural functions and described all natural functions g : T ∗ F M → R for some large class of natural bundles F M over m-manifolds M. In [6], the second author classified all natural functions g : T ∗ (J r (M, R)0 )∗ → R for any m-dimensional manifold M. In the present paper we extend the result of [6] on fibered manifolds instead of manifolds. More precisely, for natural numbers r, s, q, m, n with s ≥ r ≤ q we describe all ∗ †


802

J. Kurek, W.M. Mikulski / Central European Journal of Mathematics 2(5) 2005 801–810

natural functions g : T ∗ (J (r,s,q) (Y, R1,1 )0 )∗ → R for any (m, n)-dimensional fibered manifold Y . For natural numbers r, s, m, n with s ≥ r we determine all natural functions g : T ∗ (J (r,s) (Y, R)0 )∗ → R for any Y as above. Some type natural geometric objects on some natural bundles GY play very importrant roles in the differential geometry. For example, natural affinors on GY can be used to study torsions of generalized connections, [5], [1]. Natural affinors on GY form a module over the ring of natural functions GY → R. Then to classify all natural affinors on GY it is necessary to describe all natural functions GY → R. So, the results of the present paper can be considered as the first step to classify all natural affinors on GY = T ∗ (J (r,s,q)(Y, R1,1 )0 )∗ and GY = T ∗ (J (r,s) (Y, R)0 )∗ . All manifolds and maps are assumed to be infinitely differentiable.

1

Natural functions on T ∗ T (r)

For a comfort we cite below the main result of [6]. Let r, m be natural numbers. For every manifold M we have the r-th order vector tangent bundle T (r) M = (J r (M, R))∗ over M. Every map f : M → N induces a vector bundle map T (r) f : T (r) M → T (r) N over (r) f such that < T (r) f (ω), jfr (x) γ >=< ω, jxr (γ ◦ f ) >, ω ∈ Tx M, γ : N → R, γ ◦ f (x) = 0, x ∈ M. The correspondence T (r) : Mf → VB is a vector bundle functor in the sense of [3], where Mf is the category of manifolds and their maps and VB is the category of vector bundles and their homomorphisms. Let Mfm be the category of m-dimensional manifolds and their embeddings. A (r) natural function g on T ∗ T|Mfn is a system of functions gM : T ∗ T (r) M → R for any m-manifold M satisfying gM = gN ◦ T ∗ T (r) ϕ for any embedding ϕ : M → N between m-manifolds. ∗ (r) Example 1.1. [6] Let t = 1, ..., r. For every m-manifold M we define λ M→ M : T T R by λ ◦ π)(a), q(a) > , M (a) :=< (A

where q : T ∗ T (r) M → T (r) M is the cotangent bundle projection, A : (T (r) M)∗ → (T (r) M)∗ is a fibre bundle morphism over idM given by A (jxr γ) := jxr (γ t ), γ : M → R, γ(x) = 0, x ∈ M , γ t is the t-th power of γ and π : T ∗ T (r) M → (T (r) M)∗ is a fibre bundle morphism over idM given by π(a) := a|Vq(a) T (r) M =T ˜ x(r) M, a ∈ (T ∗ T (r) )x M, x ∈ M . (r)

∗ The system λ = {λ M } is a natural function on T T|Mfn .


803

(r)

Theorem 1.2. [6] All natural functions g on T ∗ T|Mfm are of the form gM = H ◦ (λ M , ..., λM ) ,

where H ∈ C ∞ (Rr ) is a smooth function of r variables.

2

Natural functions on T ∗ T (r,s,q) and the main result

In this section we present the main result of the present paper. Let r, s, q, m, n be natural numbers such that s ≥ r ≤ q. The construction of T (r) : Mf → VB can be generalized as follows, see [4] or [7]. Let Y → M and Z → N be fibered manifolds. We recall that two fibered maps f, g : (r,s,q) (r,s,q) Y → Z with base maps f , g : M → N determine the same (r, s, q)-jet jy f = jy g r r s s q q at y ∈ Yx , x ∈ M, if jy f = jy g , jy (f |Yx ) = jy (g|Yx ) and jx f = jx g . The space of all (r, s, q)-jets of Y into Z is denoted by J (r,s,q)(Y, Z). The composition of fibered maps induces the composition of (r, s, q)-jets, [3], p. 126. Let R1,1 = R × R be the trivial bundle over R. The space J (r,s,q) (Y, R1,1)0 , 0 ∈ R2 , has an induced structure of a vector bundle over Y . Every fibered map f : Y → Z, (r,s,q) (r,s,q) (r,s,q) f (y) = z, induces a linear map λ(jy f ) : Jz (Z, R1,1)0 → Jy (Y, R1,1 )0 by means of the jet composition. If we denote by T (r,s,q) Y the dual vector bundle of J (r,s,q) (Y, R1,1)0 (r,s,q) and define T (r,s,q) f : T (r,s,q)Y → T (r,s,q) Z by using the dual maps to λ(jy f ), we obtain (r,s,q) a vector bundle functor T : F M → VB, where F M is the category of fibered manifolds and their fibered maps. The bundle functor T (r,s,q) is called the (r, s, q)-order vector tangent bundle. Let F Mm,n be the category of fibered manifolds with m-dimensional bases and n(r,s,q) dimensional fibers and their fibered embeddings. A natural function g on T ∗ T|FMm,n is a system of functions gY : T ∗ T (r,s,q)Y → R for any F Mm,n -object Y satisfying gY = gZ ◦ T ∗ T (r,s,q)ϕ for any F Mm,n -morphism ϕ : Y → Z between F Mm,n -objects. Example 2.1. Let i = 1, ..., q. For every F Mm,n -object Y we define µ : T ∗ T (r,s,q)Y → Y R by µ ◦ π)(a), q˜(a) > , Y (a) :=< (B where q˜ : T ∗ T (r,s,q)Y → T (r,s,q) Y is the cotangent projection, B : (T (r,s,q)Y )∗ → (T (r,s,q)Y )∗ is a fibre bundle morphism over idY given by B (jy(r,s,q) γ) := jy(r,s,q) ((γ 1 )i , 0), γ = (γ 1 , γ 2 ) : Y → R1,1 , γ(y) = 0, y ∈ Y , and π : T ∗ T (r,s,q)Y → (T (r,s,q) Y )∗ is a fibre bundle morphism over idY given by π(a) := a|Vq˜(a) T (r,s,q)Y =T ˜ y(r,s,q) Y, a ∈ (T ∗ T (r,s,q))y Y, y ∈ Y . (r,s,q)

∗ The system µ = {µ Y } is a natural function on T T|FMm,n .

804


Example 2.2. Let j = 1, ..., s. For every F Mm,n -object Y we define νY<j> : T ∗ T (r,s,q)Y → R by νY<j> (a) :=< (C <j> ◦ π)(a), q˜(a) > , where q˜ is as in Example 2, C <j> : (T (r,s,q)Y )∗ → (T (r,s,q) Y )∗ is a fibre bundle morphism over idY given by C <j> (jy(r,s,q) γ) := jy(r,s,q) (0, (γ 2 )j ), γ = (γ 1 , γ 2 ) : Y → R1,1, γ(y) = 0, y ∈ Y , and π is as in Example 2.1. (r,s,q) The system ν <j> = {νY<j> } is a natural function on T ∗ T|FMm,n . Example 2.3. Let k ∈ N and l ∈ N ∪ {0} be numbers such that k + l ≤ r. For every F Mm,n -object Y we define σY : T ∗ T (r,s,q)Y → R by σY (a) :=< (D ◦ π)(a), q˜(a) > , where q˜ is as in Example 2, D : (T (r,s,q) Y )∗ → (T (r,s,q)Y )∗ is a fibre bundle morphism over idY given by D (jy(r,s,q)γ) := jy(r,s,q)(0, (γ 1 )k (γ 2 )l ), γ = (γ 1 , γ 2 ) : Y → R1,1 , γ(y) = 0, y ∈ Y , and π is as in Example 2.1. (r,s,q) The system σ = {σY} is a natural function on T ∗ T|FMm,n . The main result of the present paper is the following classification theorem. Theorem 2.4. Let r, s, q, m, n be natural numbers such that s ≥ r ≤ q. All natural (r,s,q) functions g on T ∗ T|FMm,n are <j> gY = H ◦ (µ , σY ) Y , νY

for some unique H ∈ C ∞ (RN ), where N = q + s + 12 (r + 1)r. The proof of Theorem 2.4 will occupy Sections 3 and 4. From now on r, s, q, m, n are numbers as in Theorem 2.4 and q˜ and π are maps as in Example 2.1. The usual coordinates on the standard F Mm,n -object Rm,n , the trivial bundle Rm × Rn over Rm , are denoted by x1 , ..., xm , y 1, ..., y n and the canonical vector fields induced by x1 , ..., xm , y1 , ..., y n on Rm × Rn are denoted by ∂1 , ..., ∂m , ∂˜1 , ..., ∂ñ . For any projectable vector field X on a F Mm,n -object Y the flow lift of X to T (r,s,q)Y is denoted by T (r,s,q) X.

3

Preparations to the proof of Theorem 2.4

In this section we prove some lemmas.


805

(r,s,q)

Lemma 3.1. Let g be a natural function on T ∗ T|FMm,n . Suppose that gRm,n (a) = 0 for (r,s,q)

all a ∈ (T ∗ T (r,s,q))0 Rm,n with π(a) = j0

(x1 , y 1). Then g = 0.

Proof. The lemma is a simple consequence of the naturality of g with respect to F Mm,n (r,s,q) 1 morphisms and the fact that j0 (x , y 1) has dense orbit (with respect to F Mm,n (r,s,q) morphisms) in J0 (Rm,n , R1,1 )0 . (r,s,q)

Lemma 3.2. Let g be a natural function on T ∗ T|FMm,n . Suppose gRm,n (a) = 0 for (r,s,q)

all a ∈ (T ∗ T (r,s,q))0 Rm,n with π(a) = j0 (x1 , y 1), < a, T (r,s,q) ∂i (˜ q (a)) >= 0 and < (r,s,q) ˜ a, T ∂j (˜ q (a)) >= 0 for i = 1, ..., m − 1 and j = 1, ..., n − 1. Then g = 0. (r,s,q)

Proof. Consider a ∈ (T ∗ T (r,s,q))0 Rm,n with π(a) = j0 (x1 , y 1). Using Lemma 3.1 it is sufficient to show that gRm,n (a) = 0. Because of the density argument we can assume that < a, T (r,s,q)∂ñ (˜ q (a)) >6= 0. ∗ m n (r,s,q) Define Θ ∈ T0 (R × R ) by < Θ, Z(0) >=< a, T Z(˜ q(a)) > for all constant m n (r,s,q) ˜ vector fields Z on R × R . Since < a, T ∂n (˜ q (a)) >6= 0 and n ≥ 2, there is a (r,s,q) 1 m,n m,n linear F Mm,n -morphism ψ : R →R preserving j0 (x , y 1) and sending Θ into αd0 xm + βd0 y n for some α, β ∈ R. Then a = T ∗ T (r,s,q)ψ(a) satisfies the conditions π(a) = (r,s,q) 1 j0 (x , y 1 ), < a, T (r,s,q)∂i (˜ q (a)) >= 0 and < a, T (r,s,q)∂˜j (˜ q (a)) >= 0 for i = 1, ..., m − 1 and j = 1, ..., n − 1. Then by the assumption of the lemma we have gRm,n (a) = 0. Then by the naturality of g with respect to ψ we have gRm,n (a) = 0. It is clear that T (r,s,q) ((y 1)s ∂˜1 ) and T (r,s,q) ((x1 )q ∂1 ) are vertical over 0 ∈ Rm × Rn . (r,s,q) m,n (r,s,q) 1 Lemma 3.3. Set {y ∈ T0 R | < T (r,s,q)((y 1)s ∂˜1 )(y), j0 (x , y 1) >6= 0 , (r,s,q) (r,s,q) < T (r,s,q)((x1 )q ∂1 )(y), j0 (x1 , y 1) >6= 0} is dense in T0 Rm,n , where obviously (r,s,q) Vy T (r,s,q) Rm,n =T ˜ 0 Rm,n .

Proof. Let ϕt be the flow of (y 1 )s ∂˜1 near 0. For any y ∈ T0

(r,s,q)

(r,s,q)

< T (r,s,q)((y 1 )s ∂˜1 )(y), j0

Rm,n we have

d (r,s,q) (r,s,q) 1 (x1 , y 1 ) > = < |t=0 T0 ϕt (y), j0 (x , y 1 ) > dt d (r,s,q) 1 < T (r,s,q) ϕt (y), j0 (x , y 1 ) > |t=0 = dt d (r,s,q) = < y, j0 ((x1 , y 1) ◦ ϕt ) > |t=0 dt (r,s,q) ∂ = < y, j0 ( ((x1 , y 1) ◦ ϕt )t=0 ) > ∂t (r,s,q) = < y, j0 (0, (y 1)s ) > . (r,s,q)

Similarly T (r,s,q)((x1 )q ∂1 )(y), j0 These facts imply our lemma.

(r,s,q)

(x1 , y 1) >=< y, j0

(1)

((x1 )q , 0) >.

(r,s,q)

Lemma 3.4. Let g be a natural function on T ∗ T|FMm,n . Suppose gRm,n (a) = 0 for (r,s,q)

all a ∈ (T ∗ T (r,s,q))0 Rm,n with π(a) = j0

(x1 , y 1), < a, T (r,s,q) ∂i (˜ q (a)) >= 0 and
= 0 for i = 1, ..., m and j = 1, ..., n. Then g = 0. (r,s,q)

Proof. Consider a point a ∈ (T ∗ T (r) )0 Rn satisfying the conditions π(a) = j0 (x1 , y 1 ), < a, T (r,s,q)∂i (˜ q(a)) >= 0 and < a, T (r,s,q) ∂˜j (˜ q(a)) >= 0 for i = 1, ..., m − 1 and j = 1, ..., n − 1. By Lemma 3.2, it is sufficient to show that gRm,n (a) = 0. By Lemma 3.3 we can additionally assume that 1 , α1 1 (r,s,q) 1 < T (r,s,q)((y 1 )q ∂˜1 )(˜ q(a)), j0 (x , y 1) > = α2 (r,s,q)

< T (r,s,q)((x1 )q ∂1 )(˜ q(a)), j0

(x1 , y 1) > =

(2)

for some α1 , α2 ∈ R. We will proced by several steps. Step 1. (F Mm,n -morphism ϕ1 and element a1 ) Denote < a, T (r,s,q) ∂ñ (˜ q (a)) >= β1 . Since j0s (∂ñ − α2 β1 (y 1 )s ∂˜1 ) = j0s (∂ñ ), there exists an F Mm,n -morphism ϕ1 : Rm,n → Rm,n , ϕ1 (0) = 0, such that: (r,s,q)

(r,s,q)

j0 (ϕ1 ) = j0 (id) , 1 s˜ ˜ germ0 (T ϕ1 ◦ (∂n − α2 β1 (y ) ∂1 )) = germ0 (∂ñ ◦ ϕ1 ) , germ0 (T ϕ1 ◦ ∂i ) = germ0 (∂i ◦ ϕ1 ) , germ0 (T ϕ1 ◦ ∂˜j ) = germ0 (∂˜j ◦ ϕ1 )

(3)

for i = 1, ..., m and j = 2, ..., n − 1, cf. [8]. Let a1 = T ∗ T (r,s,q)ϕ1 (a) ∈ (T ∗ T (r,s,q) )0 Rm,n . We see that (r,s,q) ˜ < a1 , T (r,s,q)∂ñ (˜ q (a1 )) > = < a, T ∗ T (r,s,q)ϕ−1 ∂n (˜ q (a1 ))) > 1 (T (r,s,q) ˜ = < a, T ∂n (˜ q(a)) − α2 β1 T (r,s,q) ((y 1)s ∂˜1 )(˜ q(a)) > 1 = β1 − α2 β1 =0. α2

(4)

(r,s,q)

It is clear that π(a1 ) = j0 (x1 , y 1), q˜(a) = q˜(a1 ), < a1 , T (r,s,q)∂i (˜ q (a1 )) >= 0 and (r,s,q) ˜ < a1 , T ∂j (˜ q (a1 )) >= 0 for i = 1, ..., m − 1 and j = 2, ..., n − 1. Step 2. (F Mm,n -morphism ϕ2 and element a2 ) Denote < a1 , T (r,s,q)∂˜1 (˜ q (a1 )) >= β2 . Since j0s (∂˜1 − α2 β2 (y 1 )s ∂˜1 ) = j0s (∂˜1 ), there exists an F Mm,n -morphism ϕ2 : Rm,n → Rm,n , ϕ2 (0) = 0, such that: (r,s,q)

(r,s,q)

j0 (ϕ2 ) = j0 (id) , 1 s˜ ˜ germ0 (T ϕ2 ◦ (∂1 − α2 β2 (y ) ∂1 )) = germ0 (∂˜1 ◦ ϕ2 ) , germ0 (T ϕ2 ◦ ∂i ) = germ0 (∂i ◦ ϕ2 ) , germ0 (T ϕ2 ◦ ∂˜j ) = germ0 (∂˜j ◦ ϕ2 ) for i = 1, ..., m and j = 2, ..., n. Let a2 = T ∗ T (r,s,q)ϕ1 (a1 ) ∈ (T ∗ T (r,s,q))0 Rm,n .

(5)


807

We see that (r,s,q) ˜ < a2 , T (r,s,q)∂˜1 (˜ q (a2 )) > = < a1 , T ∗ T (r,s,q) ϕ−1 ∂1 (˜ q (a2 ))) > 2 (T (r,s,q) ˜ = < a1 , T ∂1 (˜ q (a1 )) − α2 β2 T (r,s,q) ((y 1)s ∂˜1 )(˜ q (a1 )) > 1 = β2 − α2 β2 =0. (6) α2 (r,s,q)

It is clear that π(a2 ) = j0 (x1 , y 1), q˜(a1 ) = q˜(a2 ), < a2 , T (r,s,q)∂i (˜ q (a2 )) >= 0 and (r,s,q) ˜ < a2 , T ∂j (˜ q (a2 )) >= 0 for i = 1, ..., m − 1 and j = 2, ..., n. Step 3. (F Mm,n -morphism ϕ3 and element a3 ) Denote < a2 , T (r,s,q)∂m (˜ q (a2 )) >= β3 . Since j0q (∂m − α1 β3 (x1 )q ∂1 ) = j0q (∂m ), there exists an F Mm,n -morphism ϕ3 : Rm,n → Rm,n , ϕ3 (0) = 0, such that: (r,s,q)

(r,s,q)

j0 (ϕ3 ) = j0 (id) , 1 q germ0 (T ϕ3 ◦ (∂m − α1 β3 (x ) ∂1 )) = germ0 (∂m ◦ ϕ3 ) , germ0 (T ϕ3 ◦ ∂i ) = germ0 (∂i ◦ ϕ3 ) , germ0 (T ϕ3 ◦ ∂˜j ) = germ0 (∂˜j ◦ ϕ3 )

(7)

for i = 2, ..., m − 1 and j = 1, ..., n. Let a3 = T ∗ T (r,s,q)ϕ3 (a2 ) ∈ (T ∗ T (r,s,q))0 Rm,n . We see that (r,s,q) < a3 , T (r,s,q)∂m (˜ q (a3 )) > = < a2 , T ∗ T (r,s,q)ϕ−1 ∂m (˜ q(a3 ))) > 3 (T (r,s,q) = < a2 , T ∂m (˜ q (a2 )) − α1 β3 T (r,s,q)((x1 )q ∂1 )(˜ q (a2 )) > 1 =0. (8) = β3 − α1 β3 α1 (r,s,q)

It is clear that π(a3 ) = j0 (x1 , y 1), q˜(a2 ) = q˜(a3 ), < a3 , T (r,s,q)∂i (˜ q (a3 )) >= 0 and (r,s,q) ˜ < a3 , T ∂j (˜ q (a3 )) >= 0 for i = 2, ..., m − 1 and j = 1, ..., n. Step 4. (F Mm,n -morphism ϕ4 and element a4 ) Denote < a3 , T (r,s,q)∂1 (˜ q (a3 )) >= β4 . Since j0q (∂1 − α1 β4 (x1 )q ∂1 ) = j0q (∂1 ), there exists an F Mm,n -morphism ϕ4 : Rm,n → Rm,n , ϕ4 (0) = 0, such that: (r,s,q)

(r,s,q)

j0 (ϕ4 ) = j0 (id) , 1 q germ0 (T ϕ4 ◦ (∂1 − α1 β4 (x ) ∂1 )) = germ0 (∂1 ◦ ϕ4 ) , germ0 (T ϕ4 ◦ ∂i ) = germ0 (∂i ◦ ϕ4 ) , germ0 (T ϕ4 ◦ ∂˜j ) = germ0 (∂˜j ◦ ϕ4 )

(9)

for i = 2, ..., m and j = 1, ..., n. Let a4 = T ∗ T (r,s,q)ϕ4 (a3 ) ∈ (T ∗ T (r,s,q))0 Rm,n . We see that (r,s,q) < a4 , T (r,s,q) ∂1 (˜ q (a4 )) > = < a3 , T ∗ T (r,s,q)ϕ−1 ∂1 (˜ q (a4 ))) > 4 (T (r,s,q) = < a3 , T ∂1 (˜ q(a3 )) − α1 β4 T (r,s,q)((x1 )q ∂1 )(˜ q(a3 )) > 1 = β4 − α1 β4 =0. (10) α1

808

J. Kurek, W.M. Mikulski / Central European Journal of Mathematics 2(5) 2005 801–810 (r,s,q)

It is clear that π(a4 ) = j0 (x1 , y 1), q˜(a3 ) = q˜(a4 ), < a4 , T (r,s,q)∂i (˜ q (a4 )) >= 0 and (r,s,q) ˜ < a4 , T ∂j (˜ q (a4 )) >= 0 for i = 2, ..., m and j = 1, ..., n. Step 5. (End of the proof) By the assumption of the lemma we have gRm,n (a4 ) = 0. Then by the naturality of g with respect to the composition ϕ4 ◦ ϕ3 ◦ ϕ2 ◦ ϕ1 we get gRm,n (a) = 0. This ends the proof of Lemma 3.4. (r,s,q)

Lemma 3.5. Let g be a natural function on T ∗ T|FMm,n . Suppose that gRm,n (a) = 0 for every a ∈ (T ∗ T (r) )0 Rn satisfying the following equalities (11) (r,s,q)

π(a) = j0 (x1 , y 1 ), < a, T (r,s,q) ∂i (˜ q (a)) > (r,s,q) ˜ < a, T ∂j (˜ q (a)) > (r,s,q) α < q˜(a), j0 (x , 0) > (r,s,q) < q˜(a), j0 (0, xβ y γ ) > (r,s,q) < q˜(a), j0 (0, y σ ) >

= = = = =

0 0 0 0 0

, , , , (11)

where i = 1, ..., m, j = 1, ..., n, α ∈ (N ∪ {0})m , 1 ≤ |α| ≤ q, α2 + ... + αn = 0, β ∈ (N ∪{0})m , γ ∈ (N ∪{0})n , 1 ≤ |β| + |γ| ≤ r, |β| ≥ 1, β2 + ...+ βm + γ2 + ...+ βn = 0, σ ∈ (N ∪ {0})n , 1 ≤ |σ| ≤ s, σ2 + ... + σn = 0. Then g = 0. (r,s,q)

Proof. Let us consider a point a ∈ (T ∗ T (r) )0 Rn such that π(a) = j0 (x1 , y 1 ), < a, T (r,s,q)∂i (˜ q (a)) >= 0 and < a, T (r,s,q) ∂˜j (˜ q(a)) >= 0 for i = 1, ..., m and j = 1, ..., n. By Lemma 3.4 it is sufficient to show that gRm,n (a) = 0. Let ct := (x1 , tx2 , ..., txm , y 1, ty 2 , ..., ty n ) : Rm,n → Rm,n , t 6= 0. It is easy to see that there exists ao such that (11) and T ∗ T (r,s,q)ct (a) → ao as t → 0. By the assumption of the lemma we have gRm,n (ao ) = 0. Then using the naturality of g with respect to ct we deduce that gRm,n (a) = gRm,n (ao ) = 0.

4

Proof of Theorem 2.4

Proof. We are now in position to prove Theorem 2.4. (r,s,q) Let g be a natural function on T ∗ T|FMm,n . 1

Define H : Rq × Rs × R 2 r(r+1) → R, H(ξ) = gRm,n (aξ ) , 1

3 where ξ = ((ξi1)qi=1 , (ξj2)sj=1 , (ξ(k,l) )(k,l)∈Pr ) ∈ Rq × Rs × R 2 r(r+1) , Pr = {(k, l) | k ∈ N , l ∈ N ∪ {0} , k + l ≤ r}. Here aξ ∈ (T ∗ T (r,s,q))0 Rm,n is an unique element such that the following two conditions (I) and (II) hold. (I) We have (r,s,q)

π(aξ ) = j0

(x1 , y 1) ,


< aξ , T (r,s,q)∂i (˜ q(aξ )) > (r,s,q) ˜ < aξ , T ∂j (˜ q(aξ )) > (r,s,q) α < q˜(aξ ), j0 (x , 0) > (r,s,q) < q˜(aξ ), j0 (0, xβ y γ ) > (r,s,q) < q˜(aξ ), j0 (0, y σ ) >

= = = = =

0 0 0 0 0

, , , , ,

809

(12)

where i = 1, ..., m, j = 1, ..., n, α ∈ (N ∪ {0})m , 1 ≤ |α| ≤ q, α2 + ... + αn = 0, β ∈ (N ∪ {0})m , γ ∈ (N ∪ {0})n , 1 ≤ |β| + |γ| ≤ r, |β| ≥ 1, β2 + ... + βm + γ2 + ... + βn = 0, σ ∈ (N ∪ {0})n , 1 ≤ |σ| ≤ s, σ2 + ... + σn = 0. (II) We have (r,s,q)

< q˜(aξ ), j0 ((x1 )i , 0) > = ξi1 , (r,s,q) < q˜(aξ ), j0 (0, (y 1)j ) > = ξj2 , (r,s,q) 3 < q˜(aξ ), j0 (0, (x1 )k (y 1 )l ) > = ξ(k,l) ,

(13)

where i = 1, ..., q, j = 1, ..., s and (k, l) ∈ Pr = {(k, l) | k ∈ N , l ∈ N ∪ {0} , k + l ≤ r}. It is clear that H is smooth. Moreover, we see that q <j> s gRm,n (aξ ) = H((µ Rm,n (aξ ))i=1 , (νRm,n (aξ ))j=1 , (σRm,n (aξ ))(k,l)∈Pr ) , 1

3 where ξ = ((ξi1)qi=1 , (ξj2)sj=1 , (ξ(k,l) )(k,l)∈Pr ) ∈ Rq × Rs × R 2 r(r+1) . Then by Lemma 3.5 we <j> have gY = H ◦ (µ , σY ) for any fibered manifold Y of dimension (m, n). Y , νY Such H is unique because the image of the fiber (T ∗ T (r,s,q))0 Rm,n by the system q <j> s q+s+ 21 r(r+1) ((µ . The proof of Theorem 2.4 is Rm,n )i=1 , (νRm,n )j=1 , (σRm,n )(k,l)∈Pr ) is R

complete.

5

Natural functions on T ∗ T (r,s)

We can solve similar problem with T (r,s) = (J (r,s) (., R)0 )∗ : Mf → VB instead of T (r,s,q) as follows. Let Y → M be a fibered manifold and Q be a manifold. Two maps f, g : Y → Q (r,s) (r,s) determine the same (r, s)-jet jy f = jy g at y ∈ Yx , x ∈ M, if jyr f = jyr g and jys (f |Yx ) = jyr (g|Yx). The space of all (r, s)-jets of Y into Q is denoted by J (r,s) (Y, Q), see [3]. The space T r,s∗Y = J (r,s) (Y, R)0 has an induced structure of a vector bundle over Y , Every F M-morphism h : Z → Y , h(z) = y, induces a linear map λ(h)y,z : Tyr,x∗ Y → Tzr,s∗Z, jyr,s f → jzr,s (f ◦ h). If we denote by T (r,s) Y the dual bundle of T r,s∗Y and define T (r,s) h : T (r,s) Z → T (r,s) Y by using the dual maps to λ(h)y,z , we obtain a vector bundle functor T (r,s) on F M, see [7]. (r,s)

A natural function g on T ∗ T|FMm,n is a system of functions gY : T ∗ T (r,s) Y → R for any F Mm,n -object Y satisfying gY = gZ ◦T ∗ T (r,s) ϕ for any F Mm,n -morphism ϕ : Y → Z between F Mm,n -objects.

810


Example 5.1. Let j = 1, ..., s. For every F Mm,n -object Y we define ν˜Y<j> : T ∗ T (r,s) Y → R by ν˜Y<j> (a) :=< (C˜ <j> ◦ π)(a), q˜(a) > , where q˜ : T ∗ T (r,s) Y → T (r,s) Y is the cotangent bundle projection, C˜ <j> : (T (r,s) Y )∗ → (T (r,s) Y )∗ is a fibre bundle morphism over idY given by C˜ <j> (jy(r,s) γ) := jy(r,s) (γ j ), γ : Y → R, γ(y) = 0, y ∈ Y , and π : T ∗ T (r,s) Y → (T (r,s) )∗ is a fiber bundle morphism over idY given by π(a) := a|Vq˜(a) T (r,s) Y =T ˜ y(r,s) Y, a ∈ (T ∗ T (r,s) )y Y, y ∈ Y . (r,s)

The system ν˜<j> = {˜ νY<j> } is a natural function on T ∗ T|FMm,n . Theorem 5.2. Let r, s, m, n be natural numbers such that s ≥ r. All natural functions g (r,s) on T ∗ T|FMm,n are gY = H ◦ (˜ νY<j> ) for some unique H ∈ C ∞ (Rs ). Proof. We use quite similar arguments to the proof of Theorem 2.4.

References [1] M. Doupovec and J. Kurek: “Torsions of connections on higher order cotangent bundles”, Czechoslovak Math. J., Vol 53(4), (2003), pp. 949–962. [2] I. Koláˇr: “On cotangent bundles of some natural bundles”, Supl. Rendiconti Circolo Math. Palermo, Vol. 37(II), (1994), pp. 115–120. [3] I. Koláˇr, P. Michor and J. Slovák: Natural operations in differential geometry, Springer-Verlag, 1993. [4] I. Koláˇr and W.M. Mikulski: “Contact elements on fibered manifolds”, Czechoslovak Math. J., Vol. 53(4), (2003), pp. 1017–1030. [5] I. Koláˇr and M. Modugno: “Torsions of connections on some natural bundles”, Differential Geom. Appl., Vol. 2, (1992), pp. 1–16. [6] W.M. Mikulski: “Natural functions on T ∗ T (r) and T ∗ T r∗ ”, Arch. Math. Brno, Vol. 31(1), (1995), pp. 1–7. [7] W.M. Mikulski: “Natural affinors on (J r,s,q (R1,1)0 )∗ ”, Comment. Math. Univ. Carolinae, Vol. 42(4), (2001), pp. 655–663. [8] A. Zajtz, “On the order of natural operators and liftings”, Ann. Polon. Math., Vol. 49, (1988), pp. 169–178.

CEJM 2(5) 2005 811–825

Lie algebraic characterization of manifolds∗ Janusz Grabowski1† , Norbert Poncin2‡ 1

Institute of Mathematics, Polish Academy of Sciences, ´ Sniadeckich 8, P.O.Box 21, 00-956 Warsaw, Poland 2 Mathematics Laboratory, University of Luxembourg, avenue de la Fa¨ıencerie, 162 A, L-1511 Luxembourg City, Grand-Duchy of Luxembourg

Received 15 December 2003; accepted 12 November 2004 Abstract: Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds. c Central European Science Journals. All rights reserved.

Keywords: Algebraic characterization; smooth, real-analytic, and automorphisms; Lie algebras; differential operators; principal symbols MSC (2000): 17B63, 13N10, 16S32, 17B40, 17B65, 53D17

1

Stein

manifolds;

Algebraic characterizations of manifold structures

Algebraic characterizations of topological spaces and manifolds can be traced back to the work of I. Gel’fand and A. Kolmogoroff [13] in which compact topological spaces K are characterized by the algebras A = C(K) of continuous functions on them. In particular, points p of these spaces are identified with maximal ideals p∗ in these algebras consisting of functions vanishing at p. This identification easily implies that isomorphisms of the algebras C(K1 ) and C(K2 ) are induced by homeomorphisms between K1 and K2 , so that ∗

The research of Janusz Grabowski supported by the Polish Ministry of Scientific Research and Information Technology under the grant No. 2 P03A 020 24, that of Norbert Poncin by grant C.U.L./02/010. † E-mail: [email protected] ‡ E-mail: [email protected]

812

J. Grabowski, N. Poncin / Central European Journal of Mathematics 2(5) 2005 811–825

the algebraic structure of C(K) characterizes K uniquely up to homeomorphism. Here C(K) may consist of complex or real functions as well. All above can be carried over when we replace K with a compact smooth manifold M and C(K) with the algebra A = C ∞ (M) of all real smooth functions on M. In this case algebraic isomorphisms between C ∞ (M1 ) and C ∞ (M2 ) are induced by diffeomorphisms φ : M2 → M1 , i.e. they are of the form φ∗ (f ) = f ◦ φ. If our manifold is non-compact then no longer maximal ideals must be of the form p∗ . However, there is still an algebraic characterization of ideals p∗ as one-codimensional (or just maximal finitecodimensional) ones, so that the algebraic structure of C ∞ (M) still characterizes M, for M being Hausdorff and second countable. A similar result is true in the real-analytic (respectively the holomorphic) case, i.e. when we assume that M is a real-analytic (respectively a Stein) manifold and that A is the algebra of all real-analytic (respectively all holomorphic) functions on M (see [14]). As it was recently pointed out to us by Alan Weinstein, the assumption that M is second countable is crucial for the standard proofs that one-codimensional ideals are of the form p∗ . This remark resulted in alternative proofs [27, 17], which are valid without additional assumptions about manifolds. There are other algebraic structures canonically associated with a smooth manifold M, for example the Lie algebra X (M) of all smooth vector fields on M or the associative (or Lie) algebra D(M) of linear differential operators acting on C ∞ (M). Characterization of manifolds by associated Lie algebras is a topic initiated in 1954 with the well-known paper [32] of L.E. Pursell and M.E. Shanks that appeared in the Proceedings of the American Mathematical Society. The main theorem of this article states that, if the Lie algebras Xc (Mi ) of smooth compactly supported vector fields of two smooth manifolds Mi (i ∈ {1, 2}) are isomorphic Lie algebras, then the underlying manifolds are diffeomorphic, and—of course—vice versa. The central idea of the proof of this result is the following. If M is a smooth variety and p a point of M, denote by I(Xc (M)) the set of maximal ideals of Xc (M) and by p∞ the maximal ideal p∞ = {X ∈ Xc (M) : X is flat at p}. The map p ∈ M → p∞ ∈ I(Xc (M)) being a bijection and the property ”maximal ideal” being an algebra-isomorphism invariant, the correspondence ∞ p∞ 1 ∈ I(Xc (M1 )) → p2 ∈ I(Xc (M2 ))

↑

↓

p1 ∈ M1

p2 ∈ M2

is a bijection and even a diffeomorphism. Similar upshots exist in the analytic cases. Note that here ”flat” means zero and that the Lie algebra of all R-analytic vector fields of an R-analytic compact connected manifold is simple, so has no proper ideals. Hence, the above maximal ideals in particular, and ideals in general, are of no use in these cases. Maximal finite-codimensional subalgebras


813

turned out to be an efficient substitute for maximal ideals. This idea appeared around 1975 in several papers, e.g. in [2], and is the basis of the general algebraic framework developed in [14] and containing the smooth, the R-analytic and the holomorphic cases. Here, if M(X• (M)) is the set of all maximal finite-codimensional subalgebras of X• (M), where subscript • means smooth, R-analytic or holomorphic, the fundamental bijection is p ∈ M → p0 ∈ M(X• (M)) with p0 = {X ∈ X• (M) : (Xf )(p) = 0, ∀f ∈ C • (M)}. This method works well for a large class of the Lie algebras of vector fields which are simultaneously modules over the algebra of functions A. A pure algebraic framework in this direction has been developed by S. M. Skryabin [34] who has proven a very general ”algebraic Pursell-Shanks theorem”: Theorem 1.1. If Li is a Lie subalgebra in the Lie algebra DerAi of derivations of a commutative associative unital algebra Ai over a field of characteristic 0 which is simultaneously an Ai-submodule of DerAi and is non-singular in the sense that Li (Ai) = Ai , i = 1, 2, then any Lie algebra isomorphism Φ : L1 → L2 is induced by an associative algebra isomorphism φ∗ : A2 → A1 , i.e. Φ(X)(f ) = (φ∗ )−1 (X(φ∗ (f ))). Note that Skryabin’s proof does not refer to the structure of maximal ideals in A but uses the A-module structure on L. Other types of Lie algebras of vector fields (see e.g. [22]) have also been considered but the corresponding methods have been developed for each case separately. Let us mention the Lie algebras of vector fields preserving a given submanifold [21], a given (generalized) foliation [15], a symplectic or contact form [28], the Lie algebras of Hamiltonian vector fields or Poisson brackets of functions on a symplectic manifold [3] and Jacobi brackets in general [16], Lie algebras of vector fields on orbit spaces and G-manifolds [1], Lie algebras of vector fields on affine and toric varieties [20, 5, 33], Lie algebroids [12], and many others. In our work [18], we have examined the Lie algebra D(M) of all linear differential operators on the space C ∞ (M) of smooth functions of M, its Lie subalgebra D 1 (M) of all first-order differential operators and the Poisson algebra S(M) = P ol(T ∗ M) of all polynomial functions on the cotangent bundle, the symbols of the operators in D(M). We have obtained in each case Pursell-Shanks type results in a purely algebraic way. Furthermore, we have provided an explicit description of all the automorphisms of any of these Lie algebras. In this notes we depict this last paper assuming a philosophical and pedagogic point of view, we prove a general algebraic Pursell-Shanks type result, and, with the help of some topology, we extend our smooth Pursell-Shanks type result from [18] to the real-analytic and holomorphic cases.

814

2


Lie algebras of differential operators

2.1 Abstract definitions The goal being a work on the algebraic level, we must define some general algebra, modelled on D(M), call it a quantum Poisson algebra. A quantum Poisson algebra is an associative filtered algebra D = ∪i D i (D i = {0}, for i < 0) with unit 1, D i · D j ⊂ D i+j , such that the canonical Lie bracket verifies [D i , D j ] ⊂ D i+j−1 .

(1)

Note that A = D 0 is an associative commutative subalgebra of D and that K, the underlying field, is naturally imbedded in A. Similarly we heave the algebra S(M) = P ol(T ∗M) (or S(M) = Γ(ST M)) of smooth functions on T ∗ M that are polynomial along the fibers (respectively, of symmetric contravariant tensor fields of M), classical counterpart of D(M), on the algebraic level and define a classical Poisson algebra as a commutative associative graded algebra S = ⊕i Si (Si = {0}, for i < 0) with unit 1, Si Sj ⊂ Si+j , which is equipped with a Poisson bracket {., .} such that {Si , Sj } ⊂ Si+j−1. Here A = S0 is obviously an associative and Lie-commutative subalgebra of S. Let us point out that quantum Poisson algebras canonically induce classical Poisson algebras. Indeed, starting from D = ∪i D i , we get a graded vector space when setting Si = D i /D i−1 . If the degree deg(D) of an arbitrary non-zero D ∈ D is given by the lowest filter that contains D and if cli denotes the class in the quotient Si , we define the principal symbol σ(D) of D by σ(D) = cldeg(D) (D) and the symbol σi (D) of order i ≥ deg(D) by    0, if i > deg(D), σi (D) = cli (D) =   σ(D), if i = deg(D).

Now the commutative associative multiplication and the Poisson bracket are obtained as follows: σ(D1 )σ(D2 ) = σdeg(D1 )+deg(D2 ) (D1 · D2 ) (D1 , D2 ∈ D) and {σ(D1 ), σ(D2 )} = σdeg(D1 )+deg(D2 )−1 ([D1 , D2 ]) (D1 , D2 ∈ D).


815

Remark that the commutativity of the associative multiplication is a direct consequence of Equation (1) and that Leibniz’s rule simply passes from [., .] to {., .}. So ”dequantization” is actually a passage from non-commutativity to commutativity, the trace of non-commutativity on the classical level being the Poisson bracket. Note also that the quantum Poisson algebra D and the induced classical limit S of D have the same basic algebra A. The principal symbol map σ : D → S has the following important property:    σ([D1 , D2 ]) {σ(D1 ), σ(D2 )} = . or   0 There is a canonical quantum Poisson algebra associated with any unital associative commutative algebra A, namely the algebra D(A) of linear differential operators on A. Note that this algebraic approach to differential operators goes back to some ideas of Grothendieck and that it was extensively developed by A. M. Vinogradov. The filtration D i (A) is defined inductively: D 0 (A) = {mf : f ∈ A}, where mf (g) = f · g (so D 0 (A) is canonically isomorphic with A), and D i+1 (A) = {D ∈ Hom(A) : [D, mf ] ∈ D i(A), for all f ∈ A}, where [., .] is the commutator. It can be seen that, in the fundamental example A = C ∞ (M), we get D = D(A) = D(M), S = S(M) = P ol(T ∗M) = Γ(ST M) and the above algebraically defined Poisson bracket coincides with the canonical Poisson bracket on S, i.e. the standard symplectic bracket in the first interpretation of S, and the symmetric Schouten bracket in the second. The situation is completely analogous in real-analytic and holomorphic cases. Note only that in the holomorphic case the role of T ∗ M is played by a complex vector bundle ∗ T(1,0) M over M whose sections are holomorphic 1-forms of the type (1, 0) and the Poisson ∗ structure on T(1,0) M is represented by a holomorphic bivector field being a combination of wedge products of vector fields of type (1, 0).

2.2 Key-idea Remember that the first objective is to establish Pursell-Shanks type results, that is— roughly speaking—to deduce a geometric conclusion from algebraic information. As in previous papers on this topic, functions should play a central role. So our initial concern is to obtain an algebraic characterization of ”functions”, i.e., in the general algebraic context of an arbitrary quantum Poisson algebra D, of A ⊂ D, and more generally of all filters D i ⊂ D. It is clear that A ⊂ Nil(D)

n

}| { z := {D ∈ D : ∀∆ ∈ D, ∃n ∈ N :[D, [D, . . . [D, ∆]]] = 0}

816


and that D i+1 ⊂ {D ∈ D : [D, A] ⊂ D i} (i ≥ −1). Our conjecture is that functions (respectively, (i + 1)th order ”differential operators”) are the only locally nilpotent operators, i.e. the sole operators whose repeated adjoint action upon any operator ends up by zero (respectively, the only operators for which the commutator with all functions is of order i). It turns out that both guesses are confirmed if we show that for any D ∈ D, n

z }| { ∀f ∈ A, ∃n ∈ N :[D, [D, . . . [D, f ]]] = 0 =⇒ D ∈ A. We call this property, which states that if the repeated adjoint action of an operator upon any function ends up by zero then this operator is a function, the distinguishing property of the Lie bracket. At this point it is natural to ask if any bracket is distinguishing and—in the negative— if the commutator bracket of D(M) is. Obviously, the algebra of all linear differential operators acting on the polynomials in a variable x ∈ R is a simple example of a nondistinguishing Lie algebra. It suffices to consider the operator d/dx. The answer to the second question is positive. We refrain from describing here the technical constructive proof given in [18]. We will present further another topological proof, which can be adapted to real-analytic and holomorphic cases.

2.3 An algebraic Pursell-Shanks type result We aspire to give an algebraic proof of the theorem stating that two manifolds M1 and M2 are diffeomorphic if the Lie algebras D(M1 ) and D(M2 ) are isomorphic. So let us consider a Lie algebra isomorphism Φ : D1 → D2 between two quantum Poisson algebras D1 and D2 . In the following, we discuss two necessary assumptions. 2.3.1 Distinguishing property The next proposition is a first step towards our aim. If D1 , D2 are distinguishing quantum Poisson algebras then Φ respects the filtration. The proof is by induction on the ”order of differentiation” and uses the above algebraic characterizations of functions and filters, hence the distinguishing character of D1 and D2 . For instance, Φ(A1 ) = Φ(Nil(D1 )) = Nil(D2 ) = A2 (so that in particular Φ(D10 ) ⊂ D20 ). 2.3.2 Non-singularity property Since Φ(A1 ) = A2 , the Lie algebra isomorphism Φ restricts to a vector space isomorphism between A1 and A2 . If this restriction respected the associative multiplication it would


817

be an associative algebra isomorphism, which—as well known—would in the geometric context, Di = D(Mi ) (i ∈ {1, 2}), be induced by a diffeomorphism between M1 and M2 . So the question arises if we are able to deduce from the Lie algebra structure any information regarding the associative algebra structure and in particular the left and right multiplications ℓf : D ∋ D → f · D ∈ D and rf : D ∋ D → D · f ∈ D by a function f ∈ A. Observe first that ℓf and rf commute with the adjoint action by functions, i.e. are members of the centralizer of ad A in the Lie algebra End(D) of endomorphisms of D, which is the Lie subalgebra C(D) = {Ψ ∈ End(D) : Ψ ◦ ad A = ad A ◦ Ψ}. On the other hand, it is not possible to extract from the Lie bracket more information than ℓf − rf = ad f, (2) where the right hand side is of course a lowering member of the centralizer, i.e. a mapping in the centralizer which lowers the order of differential operators. Thus the centralizer might be the brain wave. In particular it should, in view of (2), be possible to describe it as the algebra of those endomorphisms Ψ of D that respect the filtration and are of the form Ψ = ℓf + Ψ1 , where f = Ψ(1) and Ψ1 ∈ C(D) is lowering. When trying to prove this conjecture, we realize that it holds if [D 1 , A] = A, in the sense that any function is a finite sum of brackets. In the geometric context this means that [X (M), C ∞ (M)] = X (M) (C ∞ (M)) = C ∞ (M), a non-singularity assumption that appears in many papers of this type, e.g. [14, 34], and is of course verified. Hence, a second proposition: If D is a non-singular and distinguishing quantum Poisson algebra then any Ψ ∈ C(D) respects the filtration and Ψ = ℓΨ(1) + Ψ1 , Ψ1 ∈ C(D) being lowering. 2.3.3 Isomorphisms Having in view to use the centralizer to show that Φ respects the associative multiplication, we must visibly read the Lie algebra isomorphism Φ : D1 → D2 as Lie algebra isomorphism Φ∗ : End(D1 ) → End(D2 ). We only need set Φ∗ (Ψ) = Φ ◦ Ψ ◦ Φ−1 , Ψ ∈ End(D1 ). As Φ∗ (C(D1 )) = C(D2 ), it follows from the above depicted structure of the centralizer that for any f, g ∈ A1 , Φ∗ (ℓf )(Φ(g)) = (Φ∗ (ℓf )) (1) · Φ(g), i.e. Φ(f · g) = Φ(f · Φ−1 (1)) · Φ(g).

(3)

Remark that ζ = Φ−1 (1) ∈ Z(D1 ), where Z(D1 ) denotes the center of the Lie algebra D1 . In view of (3), Φ(f · g) = Φ(g · f ) = Φ(f ) · Φ(g · ζ) = Φ(f ) · Φ(ζ · g) = Φ(ζ 2 ) · Φ(f ) · Φ(g),

(4)

for all f, g ∈ A1 . This in turn implies that Φ(ζ 2 ) is invertible, say Φ(ζ 2 ) = κ−1 , since the image of Φ, so A2 , is contained in the ideal generated by Φ(ζ 2 ). If we put A(f ) = κ−1 · Φ(f ), then, due to (4), A(f · g) = A(f ) · A(g), so A is an associative algebra isomorphism. Thus we get the following algebraic Pursell-Shanks type theorem.

818


Theorem 2.1. Let Di (i ∈ {1, 2}) be non-singular and distinguishing quantum Poisson algebras. Then every Lie algebra isomorphism Φ : D1 → D2 respects the filtration and its restriction Φ|A1 to A1 has the form Φ|A1 = κA, where κ ∈ A2 is invertible and central in D2 and A : A1 → A2 is an associative algebra isomorphism. Remark. The central elements in D(M) are just constants. This immediately follows from the well-known corresponding property of the symplectic Poisson bracket on S = S(M) = P ol(T ∗ M). If this symplectic property holds good for the classical limit S of D, we say that D is symplectic. We have assumed this property to obtain a version of Theorem 2.1 in [18]. Now we have proven that this assumption is superfluous.

2.4 Isomorphisms of the Lie algebras of differential operators Let now M be a finite-dimensional paracompact and second countable smooth (respectively real-analytic, Stein) manifold, let A = A(M) be the commutative associative algebra of all real smooth (respectively real-analytic, holomorphic) functions on M, let D = D(M) = D(A(M)) be the corresponding algebra of differential operators, and let S = S(M) be the classical limit of D(M). Lemma 2.2. The quantum Poisson algebra D(M) is distinguishing and non-singular. Proof. Let us work first in the smooth case. Let D be a linear differential operator on A(M) such that for every f ∈ A(M) there is n for which (adD )n (f ) = 0, where adD (D ′ ) = [D, D ′] is the commutator in the algebra of differential operators. It suffices to show that D ∈ A(M). The algebra A(M) admits a complete metric, which makes it into a topological algebra such that all linear differential operators are continuous (see section 6 of the book [23]). It is then easy to see that Ker((adD )n ) = {f ∈ A(M) : (adD )n (f ) = 0}, n = 1, 2, . . ., is a S closed subspace of A(M). By assumption, n Ker((adD )n ) = A(M), so Ker((adD )n0 ) = A(M) for a certain n0 according to the Baire property of the topology on A(M). Passing n0 (f ) = 0 for all f ∈ A(M), where XD is the now to principal symbols, we can write XD ∗ Hamiltonian vector field on T M of the principal symbol σ(D) of D with respect to the canonical Poisson bracket on T ∗ M. Here we regard A(M) as canonically embedded in n0 the algebra of polynomial functions on T ∗ M. Hence the 0-order operator (adg )n0 (XD ), n0 which is the multiplication by n0 !(XD (g)) , vanishes on A(M) for all g ∈ A(M), so XD (g) = {σ(D), g} = 0 for all g ∈ A(M). This in turn implies that σ(D) ∈ A(M), so D ∈ A(M). The proof in the holomorphic case is completely analogous (for the topology we refer to section 8 of [23]), so let us pass to the real-analytic case. Now, the natural topology is not completely metrizable and we cannot apply the above procedure. However, there ˜ of M, i.e. a Stein manifold M ˜ of complex dimension equal is a Stein neighbourhood M to the dimension of M, containing M as a real-analytic closed submanifold, whose germ


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along M is unique [19], so that D can be complexified to a linear holomorphic differential ˜ The complexified operator has clearly the analogous property: for every operator D. ˜ there is n for which (ad ˜ )n (f ) = 0, so D, ˜ thus D, is a holomorphic function f on M D multiplication by a function. The non-singularity of D(M) follows directly from Proposition 3.5 of [14]. The above Lemma shows that we can apply Theorem 2.1 to D(M) in all, i.e. smooth, real-analytic, and holomorphic, cases. We obtain in the same way—mutatis mutandis— Pursell-Shanks type results for the Lie algebras S(M) and D 1 (M): Theorem 2.3. The Lie algebras D(M1 ) and D(M2 ) (respectively S(M1 ) and S(M2 ), or D 1 (M1 ) and D 1 (M2 )) of all differential operators (respectively all symmetric contravariant tensors, or all differential operators of order 1) on two smooth (respectively real-analytic, holomorphic) manifolds M1 and M2 are isomorphic if and only if the manifolds M1 and M2 are smoothly (respectively bianalytically, biholomorphically) diffeomorphic.

2.5 Automorphisms of the Lie algebras of differential operators In view of the above Pursell-Shanks type results, the study of the Lie algebra isomorphisms, e.g. between D(M1 ) and D(M2 ), can be reduced to the examination of the corresponding Lie algebra automorphisms, say Φ ∈ Aut(D(M), [., .]). The standard idea in this kind of problems is the simplification of the considered arbitrary automorphism Φ via multiplication by automorphisms identified a priori. Here, the automorphism A = κ−1 Φ ∈ Aut(C ∞ (M), ·) induced by Φ (see Theorem 2.1) can canonically be extended to an automorphism A∗ ∈ Aut(D(M), [., .]). It suffices to set A∗ (D) = A ◦ D ◦ A−1 , for any D ∈ D(M). Then, evidently, Φ1 = A−1 ∗ ◦ Φ ∈ Aut(D(M), [., .]) and Φ1|C ∞ (M ) = κ id, where id is the identity map of C ∞ (M). Some notations are necessary. In the following, we use the canonical splitting D(M) = ∞ C (M) ⊕ Dc (M), where Dc (M) is the Lie algebra of differential operators vanishing on constants. Moreover, we denote by •0 and •c the projections onto C ∞ (M) = D 0 (M) and Dc (M) respectively. 2.5.1 Automorphisms of D 1 (M) The formerly explained reduction to the problem of the determination of all automorphisms Φ1 , which coincide with κ id on functions, is still valid. Furthermore, in the case of the Lie algebra D 1 (M) the preceding splitting reads D 1 (M) = C ∞ (M) ⊕ Dc1 (M) = C ∞ (M) ⊕ X (M). It follows from the automorphism property that (Φ1 )c|X (M ) = id and (Φ1 )0|X (M ) ∈ Z 1 (X (M), C ∞ (M)),

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where Z 1 (X (M), C ∞ (M)) is the space of 1-cocycles of the Lie algebra of vector fields canonically represented upon functions by Lie derivatives. We know that these cocycles are locally given by (Φ1 )0|X (M ) = λ div + df, (5) where λ ∈ R and f is a smooth function. When trying to globally define the right hand side of Equation (5), we naturally substitute a closed 1-form ω ∈ Ω1 (M) ∩ ker d to the exact 1-form df . In order to globalize the divergence div, note the following. If M is oriented by a volume Ω, we have divΩ = div−Ω . So the divergence can be defined with respect to | Ω |. This pseudo-volume may be viewed as a pair {Ω, −Ω} and exists on any manifold, orientable or not. Alternatively, we may interpret | Ω | as a nowhere vanishing tensor density of weight 1, i.e. as a section of the vector bundle IF1 (T M) of 1-densities, which is everywhere non-zero. This bundle being of rank 1, the existence of such a section is equivalent to the triviality of the bundle. However, the fiber bundle IF1 (T M) is known to be trivial for any manifold M. Thus, a nowhere vanishing tensor 1-density ρ0 always exists and the divergence can be defined with respect to this ρ0 . For a more rigorous approach the reader is referred to [18]. Eventually, the above cocycles globally read (Φ1 )0|X (M ) = λ divρ0 + ω. Let us fix a divergence div. For any f + X ∈ D 1 (M) = C ∞ (M) ⊕ X (M), we then obtain Φ1 (f + X) = κf + λ div X + ω(X) + X. Since the initial arbitrary automorphism Φ ∈ Aut(D 1 (M), [., .]) has been decomposed as Φ = A∗ ◦ Φ1 , with A∗ induced by a diffeomorphism ϕ ∈ Dif f (M) and denoted ϕ∗ below, we finally have the theorem: Theorem 2.4. A linear map Φ : D 1 (M) → D 1 (M) is an automorphism of the Lie algebra D 1 (M) = C ∞ (M) ⊕ X (M) of linear first-order differential operators on C ∞ (M) if and only if it can be written in the form Φ(f + X) = (κf + λ div X + ω(X)) ◦ ϕ−1 + ϕ∗ (X), where ϕ is a diffeomorphism of M, λ, κ are constants (κ 6= 0), ω is a closed 1-form on M, and ϕ∗ is defined by (ϕ∗ (X))(f ) = (X(f ◦ ϕ)) ◦ ϕ−1 . All the objects ϕ, λ, κ, ω are uniquely determined by Φ. 2.5.2 Automorphisms of D(M) As previously, we need only seek the automorphisms Φ1 ∈ Aut(D(M), [., .]), such that Φ1 (f ) = κf , f ∈ C ∞ (M). Such an automorphism visibly restricts to a similar automorphism of the Lie algebra D 1 (M). Hence, Φ1 (f + X) = κf + λ div X + ω(X) + X.

(6)


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Since ω ∈ Z 1 (X (M), C ∞ (M)), it is reasonable to think that ω might be extended to ω ∈ Z 1 (D(M), D(M)) = Der(D(M)), where Der(D(M)) is the Lie algebra of all derivations of (D(M), [., .]). If in addition this derivation ω were lowering, it would generate an automorphism eω , which could possibly be used to cancel the term ω(X) in Equation (6). But ω is locally exact, ω|U = dfU (U: open subset of M, fU ∈ C ∞ (U)). So it suffices to ensure that the inner derivations associated to the functions fU glue together. Lastly, ω(D)|U = [D|U , fU ], D ∈ D(M), and Φ2 = Φ1 ◦ e−κ

−1 ω

∈ Aut(D(M), [., .]) actually verifies

Φ2 (f + X) = κf + λ div X + X.

(7)

It is interesting to note that the automorphism eω is, for ω = df (f ∈ C ∞ (M)), simply the inner automorphism eω : D(M) ∋ D → ef · D · e−f ∈ D(M). An analogous extension div of the cocycle div is at least not canonical. At this stage a new idea has to be injected. It is easily checked that every quantum ˜ of S(M), Φ(σ(D)) ˜ automorphism Φ of D(M) induces a classical automorphism Φ = σ(Φ(D)). Of course, the converse is not accurate. Moreover, any automorphism of S(M) restricts to an automorphism of S0 (M) ⊕ S1 (M) ≃ D 1 (M). So it would be natural to try to benefit from this ”algebra hierarchy” and compute, having already obtained the automorphisms of D 1 (M), those of S(M). This approach however turns out to be rather merely elegant than really necessary. We do not employ it here. The automorphism property shows that Φ2|Di(M ) = κ1−i id + ψi ,

(8)

where i ∈ N and ψi ∈ HomR (D i (M), D i−1 (M)). This is equivalent to saying that ˜ 2|S (M ) = κ1−i id. Φ i Such a classical automorphism Uκ : Si (M) ∋ P → κ1−i P ∈ Si (M) really exists. Indeed, the degree deg : Si (M) ∋ P → (i − 1)P ∈ Si (M) is known to be a derivation of S(M) and, for κ > 0, the automorphism of S(M) generated by −log(κ) · deg is precisely Uκ . Nevertheless, it is hard to imagine that this classical automorphism Uκ is induced by a quantum automorphism, since no graduation exists on the quantum level. Therefore, the first guess is ? κ = 1. In order to validate or invalidate this supposition, we project the automorphism property, combined with the newly discovered structure (8) of Φ2 , onto C ∞ (M). It is worth stressing that all information is, here as well as above, enclosed in the 0-order terms. Our exploitation of this projection is based on formal calculus. In the main, this 1 n symbolism consists in the substitution of monomials ξ1α . . . ξnα in the components of a 1 n linear form ξ ∈ (Rn )∗ to the derivatives ∂xα1 . . . ∂xαn f of a function f .

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This method is known in Mechanics as the normal ordering or canonical symbolization/quantization. Its systematic use in Differential Geometry is originated in papers by M. Flato and A. Lichnerowicz [10] as well as M. De Wilde and P. Lecomte [8], dealing with the Chevalley-Eilenberg cohomology of the Lie algebra of vector fields associated with the Lie derivative of differential forms. This polynomial modus operandi matured during the last twenty years and developed into a powerful computing technique, successfully applied in numerous works (see e.g. [25] or [29]). A by now standard application of the normal ordering leads to a system of equations in specially the above constants λ and κ. When solving the system, we get two possibilities, (λ, κ) = (0, 1) and (λ, κ) = (1, −1). This outcome surprisingly cancels the conjecture κ = 1. 2.5.2.1 (λ, κ) = (0, 1) Equation (7), which gives Φ2 on D 1 (M), suggests that Φ2 could coincide with id on the whole algebra D(M). In fact our automorphism equations show that computations reduce to the determination of some intertwining operators between the X (M)-modules of kth-order linear differential operators mapping differential p-forms into functions. These equivariant operators have been obtained in [31] and [4]. They allow to conclude that we actually have Φ2 = id. The paper [31], following works by P. Lecomte, P. Mathonet, and E. Tousset [25], H. Gargoubi and V. Ovsienko [11], P. Cohen, Yu. Manin, and D. Zagier [6], C. Duval and V. Ovsienko [9], gives the classification of the preceding modules. Additionally, it provides the complete description of the above-mentioned intertwining operators, thus answering a question by P. Lecomte whether some homotopy operator—which locally coincides with the Koszul differential [24] and is equivariant if restricted to low-order differential operators—is intertwining for all orders of differentiation. A small dimensional hypothesis in [31], which was believed to be inherent in the used canonical symbolization technique, was the starting point of [4]. Here, the authors prove the existence and the uniqueness of a projectively equivariant symbol map (in the sense of P. Lecomte and V. Ovsienko [26]) for the spaces of differential operators transforming p-forms into functions, the underlying manifold being endowed with a flat projective structure. The substitution of this equivariant symbol to the previously used canonical symbol allowed to get rid of the dimensional assumption and unexpected intertwining operators were discovered in the few supplementary dimensions. 2.5.2.2 (λ, κ) = (1, −1) Computations being rather technical, we confine ourselves to mention that our quest for automorphisms, by means of the canonical symbolization, leads to a symbol that reminds of the opposite of the conjugation operator. Thus, this operator might be an


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astonishing automorphism. Remember that for an oriented manifold M with volume Ω, the conjugate D ∗ ∈ D(M) of a differential operator D ∈ D(M) is defined by Z

D(f ) · g | Ω |=

M

Z

f · D ∗ (g) | Ω |,

M

for any compactly supported f, g ∈ C ∞ (M). Since (D ◦ ∆)∗ = ∆∗ ◦ D ∗ , D, ∆ ∈ D(M), the operator C := −∗ verifies C(D ◦ ∆) = −C(∆) ◦ C(D) and is thus an automorphism of D(M). Formal calculus allows to show that this automorphism exists for any manifold (orientable or not). At last, the computations can again be reduced to the formerly described intertwining operators and we get Φ2 = C. Hence the conclusion: Theorem 2.5. A linear map Φ : D(M) → D(M) is an automorphism of the Lie algebra D(M) of linear differential operators on C ∞ (M) if and only if it can be written in the form Φ = ϕ∗ ◦ C a ◦ eω , where ϕ is a diffeomorphism of M, a = 0, 1, C 0 = id and C 1 = C, and ω is a closed 1-form on M. All the objects ϕ, a, ω are uniquely determined by Φ. 2.5.3 Automorphisms of S(M) The study of the automorphisms of S(M) is similar to the preceding one regarding D(M) and even simpler, in view of the existence of the degree-automorphism Uκ . We obtain the following upshot. Theorem 2.6. A linear map Φ : S(M) → S(M) is an automorphism of the Lie algebra S(M) of polynomial functions on T ∗ M with respect to the canonical symplectic bracket if and only if it can be written in the form Φ(P ) = Uκ (P ) ◦ ϕ∗ ◦ Exp(ω v ), where κ is a non-zero constant, Uκ (P ) = κ1−i P for P ∈ Si (M), ϕ∗ is the phase lift of a diffeomorphism ϕ of M, and Exp(ω v ) is the vertical symplectic diffeomorphism of T ∗ M, which is nothing but the translation by a closed 1-form ω on M. All the objects κ, ϕ, ω are uniquely determined by Φ. It is interesting to compare this result with those of [3] and [16].

Acknowledgment The authors are grateful to Peter Michor for providing helpful comments on topologies in spaces of differentiable maps.

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References [1] K. Abe: “Pursell-Shanks type theorem for orbit spaces and G-manifolds”, Publ. Res. Inst. Math. Sci., Vol. 18, (1982), pp. 265–282. [2] I. Amemiya: “Lie algebra of vector fields and complex structure”, J. Math. Soc. Japan, Vol. 27, (1975), pp. 545–549. [3] C.J. Atkin and J. Grabowski: “Homomorphisms of the Lie algebras associated with a symplectic manifold”, Compos. Math., Vol. 76, (1990), pp. 315–348. [4] F. Boniver, S. Hansoul, P. Mathonet and N. Poncin: “Equivariant symbol calculus for differential operators acting on forms”, Lett. Math. Phys., Vol. 62, (2002), pp. 219–232. [5] A. Campillo, J. Grabowski and G. M¨ uller G: “Derivation algebras of toric varieties”, Comp. Math., Vol. 116, (1999), pp. 119–132. [6] P. Cohen, Yu Manin and D. Zagier: “Automorphic pseudodifferential operators, Algebraic Aspects of Integrable Systems”, Progr. Nonlinear Differential Equations Appl., Vol. 26, (1997), pp. 17–47. [7] M. De Wilde and P. Lecomte: “Some Characterizations of Differential Operators on Vector Bundles”, E.B. Christoffel, P. Butzer and F. Feher (Eds.), Brikh˝auser Verlag, Basel, 1981, pp. 543–549. [8] M. De Wilde and P. Lecomte: “Cohomology of the Lie Algebra of Smooth Vector Fields of a Manifold, associated to the Lie Derivative of Smooth Forms”, J. Math. pures et appl., Vol. 62, (1983), pp. 197–214. [9] C. Duval and V. Ovsienko: “Space of second order linear differential operators as a module over the Lie algebra of vector fields”, Adv. in Math., Vol. 132(2), (1997), pp. 316–333. [10] M. Flato and A. Lichnerowicz: “Cohomologie des représentations definies par la derivation de Lie et à valeurs dans les formes, de l’algèbre de Lie des champs de vecteurs d’une variété différentiable. Premiers espaces de cohomologie. Applications”, C. R. Acad. Sci., Sér. A, Vol. 291, (1980), pp. 331–335. [11] H. Gargoubi and V. Ovsienko: “Space of linear differential operators on the real line as a module over the Lie algebra of vector fields”, Internat. Math. Res. Notices, Vol. 5, (1996), pp. 235–251. [12] K. Grabowska and J. Grabowski: “The Lie algebra of a Lie algebroid”, In: J. Kubarski et al. (Eds.): Lie Algebroids and Related Topics in Differential Geometry, Vol. 54, Banach Center Publications, Warszawa, 2001, pp. 43–50. [13] I. Gel’fand and A. Kolmogoroff: “On rings of continuous functions on topological spaces”, C. R. (Dokl.) Acad. Sci. URSS, Vol. 22, (1939), pp. 11–15. [14] J. Grabowski: “Isomorphisms and ideals of the Lie algebras of vector fields”, Invent. math., Vol. 50, (1978), pp. 13–33. [15] J. Grabowski: “Lie algebras of vector fields and generalized foliations”, Publ. Matem., Vol. 37, (1993), pp. 359–367. [16] J. Grabowski: “Isomorphisms of Poisson and Jacobi brackets”, In: J. Grabowski and P. Urba´ nski (Eds.): Poisson Geometry, Vol. 51, Banach Center Publications, Warszawa, 2000, pp. 79–85.


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[17] J. Grabowski: “Isomorphisms of algebras of smooth functions revisited”, Archiv Math., to appear, electronic version at http://arXiv.org/abs/math.DG/0310295. [18] J. Grabowski and N. Poncin: “Automorphisms of quantum and classical Poisson algebras”, Comp. Math.London Math. Soc., Vol. 140, (2004), pp. 511–527 (http://arXiv.org/abs/math.RA/0211175 v1). [19] H. Grauert: “On Levi’s problem and the embedding of real analytic manifolds”, Ann. Math., Vol. 68, (1958), pp. 460–472. [20] H. Hauser and G. M¨ uller: “Affine varieties and Lie algebras of vector fields”, Manusc. Math., Vol. 80, (1993), pp. 309–337. [21] A. Koriyama: “On Lie algebras of vector fields with invariant submanifolds”, Nagoya Math. J., Vol. 55, (1974), pp. 91–110. [22] A. Koriyama, Y. Maeda and H. Omori: “On Lie algebras of vector fields”, Trans. Amer. Math. Soc., Vol. 226, (1977), pp. 89–117. [23] A. Kriegl and P.W. Michor: “The Convenient Setting of Global Analysis”, Math. Surv. Monog., Vol. 53, American Mathematical Society, (1997). [24] P. Lecomte: “On some sequence of graded Lie algebras associated to manifolds”, Ann. Glob. Anal. Geom., Vol. 12, (1994), pp. 183–192 [25] P. Lecomte, P. Mathonet and E. Tousset: “Comparison of some modules of the Lie algebra of vector fields”, Indag. Math., Vol. 7(4), (1996), pp. 461–471. [26] P. Lecomte and V. Ovsienko: “Projectively equivariant symbol calculus”, Lett. Math. Phys., Vol. 49, (1999), pp. 173–196. [27] J. Mrˇcun: On isomorphisms of algebras of smooth functions, electronic version at http://arXiv.org/abs/math.DG/0309179. [28] H. Omori: “Infinite dimensional Lie transformation groups”, Lect. Notes in Math., Vol. 427, (1976). [29] N. Poncin: “Cohomologie de l’algèbre de Lie des opérateurs différentiels sur une variété, à coefficients dans les fonctions”, C.R.A.S. Paris, Vol. 328, Serie I, (1999), pp. 789–794. [30] N. Poncin: “Equivariant Operators between some Modules of the Lie Algebra of Vector Fields”, Comm. in Alg., Vol. 32(7), (2004), pp. 2559–2572. [31] N. Poncin: Equivariant Operators between some Modules of the Lie Algebra of Vector Fields, preprint, Centre Universitaire de Luxembourg, (http://arXiv.org/abs/math.RT/0205297). [32] M.E. Shanks and L.E. Pursell: “The Lie algebra of a smooth manifold”, Proc. Amer. Math. Soc., Vol. 5, (1954), pp. 468–472. [33] T. Siebert: “Lie algebras of derivations and affine differential geometr over fields of characteristic 0”, Mat. Ann., Vol. 305, (1996), pp. 271–286. [34] S.M. Skryabin: The regular Lie rings of derivations of commutative rings, preprint WINITI 4403-W87 (1987). [35] H. Whitney: “Differentiable manifolds”, Ann. Math., Vol. 37, (1936), pp. 645–680.

CEJM 2(5) 2005 826–839

On higher order geometry on anchored vector bundles∗ Paul Popescu† Department of Applied Mathematics, University of Craiova, 13, Al.I. Cuza st., Craiova, 1100, Romania

Received 15 December 2003; accepted 17 September 2004 Abstract: Some geometric objects of higher order concerning extensions, semi-sprays, connections and Lagrange metrics are constructed using an anchored vector bundle. c Central European Science Journals. All rights reserved.

Keywords: Anchored vector bundle, affine bundle, almost Lie structure, r-prolongation, rconnection, lagrangian of order r, r-semi-spray MSC (2000): 14R25, 44A15, 53C07, 53B15, 22A30, 70S05

Introduction An anchored vector bundle (AVB) (or a relative tangent space, see [12, 15]) is a couple (θ, D), where θ = (E, p, M) is a vector bundle and D : θ → τ M is a vector bundle morphism called an anchor (an arrow, or a tangent map). The vector bundle τ M = (T M, p, M) is the tangent bundle of M. For example, if θ is a vector subbundle of τ M and i : θ → τ M is the inclusion morphism, then (θ, i) is an AVB. A bracket (or a Lie map) on an AVB (θ, D) is a map [·, ·] : Γ(θ) × Γ(θ) → Γ(θ) which enjoys the properties that it is bilinear over IR, is skew symmetric and [X, f Y ] = (DX)(f )Y + f [X, Y ], (∀)X, Y ∈ Γ(θ) and f ∈ F (M). An almost Lie structure (ALS) is a triple (θ, D, [·, ·]). Linear and non-linear connections associated with AVB’s and ALS’s are studied for the first time in [12, 13], where they are called R-connections. The linear R-connections are known as Wong connections [18] and they were recently considered in [5]. ∗ †

The paper was partially supported by a 2005 MEC-CNCSIS grant. E-mail: Paul [email protected]

P. Popescu / Central European Journal of Mathematics 2(5) 2005 826–839

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An algebroid is an ALS (θ, D, [·, ·]) which enjoys the property that [DX, DY ] = D[X, Y ], (∀)X, Y ∈ Γ(θ), where the first bracket is the Lie bracket on X (M). A Lie algebroid is an algebroid (θ, D, [·, ·]) which has a null Jacobiator, i.e. P J (X, Y, Z) ≡ [[X, Y ], Z] = 0, (∀)X, Y, Z ∈ Γ(θ).. The linear R-connections defined cicl.

using Lie algebroids are used in [4] (where they are called A-connections). A theory of higher order spaces and geometric structures is considered in [6]-[8]. We use the point of view of these papers concerning higher order extensions, and also the local calculus, which is used in the mainly part of the basic constructions. We consider below some cases where the constructions in our paper apply. 1. Let M be a smooth manifold. Any subbundle θ ⊂ τ M defines an anchored vector bundle. 2. A Lie algebroid. 3. Let M be a smooth manifold. Let M be a smooth manifold and {·, ·} : F (M) × F (M) → F (M) be a Poisson bracket. It is well-known that the association df → Xf (the hamiltonian vector field which correspond to a real function f ∈ F (M) is given by Xf (g) = {f, g}) extends to an anchor D : τ ∗ M → τ M. The bracket of functions defines also by extension a bracket of real forms. A Lie algebroid is defined in this way on τ ∗ M. 4. Let M be a smooth manifold. A two-vector field P ∈ Λ2 (M) define an anchor D : τ ∗ M → τ M. In particular, if ω comes from the inverse of a symplectic form on M, then a (non-singular) Poisson bracket follows (thus 3. applies). 5. Let M be a smooth manifold. The canonical symplectic form on T ∗ M defines an anchor D : τ ∗ T ∗ M → τ T ∗ M (thus 4. applies). The first section contains the basic constructions of the prolongations of order r ≥ 2 of an AVB and the other vector bundles associated with these prolongations. Connections of higher order (r-connections) are defined in the second section. A construction of an rconnection without using the dual coefficients (for example in [6]) is given by Proposition 2.1. A semi-spray of higher order (an r-semi-spray) is defined in Section 3. We prove that every r-semi-spray define an r-connections (Theorem 3.1). In Section 4 we define the r-lagrangians and we prove that every regular r-lagrangian define an r-semi-spray (Proposition 4.1).

1

The r-prolongations of an anchored vector bundle

Let (θ, D) be an anchored vector bundle. We define θ(0) = M, i.e. the trivial vector bundle over M, and θ(1) = θ = (E, π, M) = (E (1) , π (1) , E (0) ). Assume that {gγβ (xi )} is the cocycle (structural functions) of E. Thus, on the intersection of two domains of two adapted charts, the local coordinates on E (1) change according to the rules x¯i = x¯i (xj ), y¯(1)α = gβαy (1)β .

828


We define the affine bundle θ(2) = (E (2) , π (2) , E (1) ) using, for the coordinates on the fibers, the change rules ∂ y¯(1)α 2¯ y (2)α = 2gβα y (2)β + y (1)β Dβi . ∂xi Assuming that the affine bundles θ(0) , θ(1) ,. . ., θ(r) = (E (r) , π (r) , E (r−1) ) has been defined, we define the affine bundle θ(r+1) = (E (r+1) , π (r+1) , E (r) ) according to the formula: ∂ y¯(r)α ¯(r)α (2)β ∂ y + 2y +···+ ∂xi ∂y (1)β ¯(r)α ¯(r)α (r+1)β ∂ y (r)β ∂ y +ry + (r + 1)y ∂y (r−1)β ∂y (r)β

(r + 1)¯ y (r+1)α = y (1)β Dβi

for the change rule of the coordinates on the fibers of E (r+1) . It is easy to see that ∂ y¯(r)α = gβα , (∀)r ≥ 1. More synthetic, denoting as ∂y (r)β Γ(r) = y (1)β Dβi

∂ ∂ ∂ + 2y (2)β (1)β + · · · + ry (r)β (r−1)β , r ≥ 0, i ∂x ∂y ∂y

then (r + 1)¯ y (r+1)α = Γ(r) (¯ y (r)α ) + (r + 1)y (r+1)β gβα . Notice that the change rule of the coordinates on the manifold E (r) is: y¯(1)α = gβαy (1)β , 2¯ y (2)α = y (1)β Dβi ...

∂ y¯(1)α ∂ y¯(1)α (2)β + 2 y , ∂xi ∂y (1)β

∂ y¯(r−1)α ¯(r−1)α (2)β ∂ y + 2y +···+ ∂xi ∂y (1)β ∂ y¯(r−1)α ∂ y¯(r−1)α +(r − 1)y (r−1)β (r−2)β + ry (r)β (r−1)β ∂y ∂y

r¯ y (r)α = y (1)β Dβi

and Γ(r) is a local vector field, not a global one, its domain being the local domain of the local coordinates on E (r) . According to [17], we say that Γ(r) is a vector pseudo-field. Proposition 1.1. The bundles θ(r) = (E (r) , π (r) , E (r−1) ), r ≥ 1, are affine bundles. Proof. We proceed by induction over r. For r = 1, θ(1) is a vector bundle, thus it is a central affine bundle. Let us assume that the property is true for ≤ r and prove that it is true for r + 1. i (1)α (r)α (r+1)α Consider some new coordinates (¯ x , y¯ , . . . , y¯ , y¯ ) which change according to the rules i

i

x¯ = x¯ (¯ xj ), y¯(1)α = gβα y¯(1)β ,


(2)α

2¯ y

(1)β

= y¯

... (r)α

r y¯

(1)β

= y¯

(1)α ¯ i ∂y Dβ ∂xi

+

(r−1)α ¯ i ∂y Dβ ∂xi

829

(1)α ¯ (2)β ∂ y 2¯ y , ∂y (1)β

+

(r−1)α ¯ (2)β ∂ y 2¯ y ∂ y¯(1)β

(r−1)α

+···+ (r−1)α

∂ y¯ ∂ y¯ + r¯ y (r)β (r−1)β , (r−2)β ∂ y¯ ∂ y¯ (r)α (r)α ∂ y¯ ∂ y¯ = y¯(1)β Dβi + 2¯ y (2)β (1)β + · · · + i ∂x ∂ y¯ (r)α (r)α ¯ ¯ (r)β ∂ y (r+1)β ∂ y +r¯ y + (r + 1)¯ y ∂ y¯(r−1)β ∂ y¯(r)β +(r − 1)¯ y (r−1)β

y (r+1)α (r + 1)¯

(r+1)β

(r)β

¯ (r) (¯ thus y¯ = g¯γβ (¯ xi )¯ y (r+1)γ + Γ y ). We have to express the link between the coordii (1)α (r)α (r+1)α nates (¯ x , y¯ , . . . , y¯ , y¯ ) and (xi , y (1)α , . . . , y (r)α , y (r+1)α ) respectively on the intersection of their domains. The link between the couples of coordinates (xi , y (1)α , . . . , y (r)α ) i (1)α (r)α and (¯ x , y¯ , . . . , y¯ ) respectively has the form x¯i = x¯i (xj ), (1)α

= g¯β y (1)β ,

(2)α

= y (1)β Dβi

y¯

α

(1)α

y 2¯

... (r)α

2¯ y

=y

(1)β

(1)α

∂ y¯ ¯ (2)β ∂ y + 2y , ∂xi ∂y (1)β

(r−1)α ¯ i ∂y Dβ ∂xi

+

(r−1)α ¯ (2)β ∂ y 2y ∂ y¯(1)β

(r−1)α

+(r − 1)y (r−1)β

+···+ (r−1)α

∂ y¯ ∂ y¯ + ry (r)β (r−1)β . (r−2)β ∂ y¯ ∂ y¯

¯ (r) the local forms of the given vector pseudo-field on E on the We denote by Γ(r) and Γ domains of the coordinates (xi , y (1)α , . . . , y (r)α ) and (¯ xi , y¯(1)α , . . . , y¯(r)α ) respectively. We have (r+1)β (r)β ¯ (r) (¯ (r + 1)¯ y = (r + 1)¯ gγβ (¯ xi )¯ y (r+1)γ + Γ y )= (r)β ¯ (r) (¯ (r + 1)¯ gγβ (¯ xi )(gδγ (xi )y (r+1)δ + Γ(r) (¯ y (r)γ )) + Γ y )= ¯ (r) (¯ (r + 1)¯ g αδ y (r+1)δ + (r + 1)¯ gγβ Γ(r) (¯ y (r)γ ) + Γ y (r)β ). (r)β ¯ (r) (¯ y ) = (r + 1)¯ gγβ Γ(r) (¯ y (r)γ )+ But (r + 1)¯ gγβ Γ(r) (¯ y (r)γ ) + Γ (r)β ∂ y¯ (r)β (r)β Γ(r) (¯ y ) − (r + 1)Γ(r) (¯ y γ ) (r)γ = Γ(r) (¯ y ). ∂ y¯ We have proved that π (r+1)

(r + 1)¯ y (r+1)β = (r + 1)¯ g αδ y (r+1)β + Γ(r) (¯ y (r)β ), thus E (r+1) → E (r) is an affine bundle. The local adapted coordinates on E (r) , r ≥ 0, used above, are called adapted. An important particular case is obtained when θ = τ M, the tangent bundle of M (i.e. E = T M) and the anchor D = 1T M is the identity. In this case (τ M)(n) = T (n) M is the total space of the acceleration bundle of order k, studied for example in [6, 7, 8].

830


Notice that for every r ≥ 1, using local coordinates which are adapted, the natural local basis in τ E (r) change according to the rules: ∂ ∂ x¯j ∂ ∂ y¯(1)β ∂ ∂ y¯(r)β ∂ = + + · · · + , ∂xi ∂xi ∂ x¯j ∂xi ∂ y¯(1)β ∂xi ∂ y¯(r)β ∂ y¯(1)β ∂ ∂ y¯(r)β ∂ ∂ = + · · · + , ∂y (1)α ∂y (1)α ∂ y¯(1)β ∂y (1)α ∂ y¯(r)β .. . ∂ ∂ y¯(r)β ∂ = . ∂y (r)α ∂y (r)α ∂ y¯(r)β

(1)

On the intersection of two domains of coordinates one has: ¯ (r) = Γ(r) − Γ(r) (¯ Γ y (r)α )

∂ . ∂ y¯(r)α

(2)

Indeed, ∂ ∂ ∂ + 2y (2)β (1)β + · · · + ry (r)β (r−1)β = i ∂x ∂y ∂y j (1)β ∂ x¯ ∂ ∂ y¯ ∂ ∂ y¯(r)β ∂ (1)β i y Dβ + +···+ + ∂xi ∂ x¯j ∂xi ∂ y¯(1)β ∂xi ∂ y¯(r)β (1)β ∂ y¯ ∂ ∂ y¯(r)β ∂ (2)β + · · · + (1)α (r)β + · · · + 2y ∂y (1)α ∂ y¯(1)β ∂y ∂ y¯ (r−1)β ∂ y¯ ∂ ∂ y¯(r)β ∂ ∂ (r−1)β + · · · + (r−1)α (r)β + ry (r)α gαβ (r)β = +(r − 1)y (r−1)α r−(1)β ∂y ∂ y¯ ∂y ∂ y¯ ∂ y¯ ∂ ∂ ∂ ¯ βi ∂ + 2¯ y¯(1)β D y (2)β (1)β + · · · + r¯ y (r)β (r−1)β + Γ(r) (¯ y (r)α ) (r)α = i ∂ x¯ ∂ y¯ ∂ y¯ ∂ y¯ ∂ ¯ (r) + Γ(r) (¯ Γ y (r)α ) (r)α . ∂ y¯

Γ(r) = y (1)β Dβi

We consider, on the same intersection domains U ∩ U¯ of local adapted charts on E (r) , the local matrix: 

MU¯ ∩U

∂ y¯(1)α0 ∂y (1)β0 ∂ y¯(2)α1 ∂y (1)β0 .. .

0 ··· 0    ∂ y¯(2)α1  ··· 0  ∂y (2)β1 =  .. ..  . .  (r+1)αr  ∂ y¯(r+1)αr ∂ y¯(r+1)αr ∂ y¯ · · · (1)β (2)β ∂y 0 ∂y 1 ∂y (r+1)βr



     .    

(3)

Lemma 1.2. The equality ∂ y¯(r+1)α ∂ y¯(r)α = , ∂y (u)β ∂y (u−1)β holds for 2 ≤ u ≤ r + 1.

(4)


831

Proof. We use the induction over r ≥ 1. For r = 1 the only possibility for u is u = 2, ∂ y¯(1)α ∂ y¯(2)α when = = gβα . ∂y (1)β ∂y (2)β We assume the equality (4) is true for ≤ r − 1, thus ∂ y¯(r−1)α ∂ y¯(r)α = , 2 ≤ u ≤ r. ∂y (u)β ∂y (u−1)β

(5)

∂ y¯(r+1)α ∂ ¯(r)α ¯(r)α (1)β i ∂ y (2)β ∂ y = (y D + 2y +···+ β ∂y (u)β ∂y (u)β ∂xi ∂y(1)β ¯(r)α ∂ ∂ ∂ y¯(r)α (r+1)β ∂ y (1)β i ∂ (2)β (r+1)β (r + 1)y ) = (y Dβ i + 2y + · · ·+ (r + 1)y ) + ∂y (r)β ∂x ∂y (1)β ∂y (r)β ∂y (u)β ∂ ∂ ∂ ∂ y¯(r−1)α ∂ y¯(r)α (u−1+1) (u−1)β = (y (1)β Dβi i +2y (2)β (1)β +· · ·+(r +1)y (r+1)β (r)β ) + ∂y ∂x ∂y ∂y ∂y (u−1)β ∂ y¯(r−1)α ∂ y¯(r)α ∂ ∂ y¯(r−1)α ¯(r−1)α ¯(r−1)α (2)β ∂ y (r)β ∂ y (u−1) (u−2)β + (u−1)β = (u−1)β (y (1)β Dβi +2y +· · ·+ry ) ∂y ∂y ∂y ∂xi ∂y (1)β ∂y (r)β ∂ y¯(r)α ∂ y¯(r)α ∂ y¯(r)α ∂ y¯(r+1)α ∂ y¯(r)α + (u−1)β = r (u−1)β + (u−1)β . For u = r + 1, = = gβα , thus the equ(r+1)β (u)β ∂y ∂y ∂y ∂y ∂y ality (4) holds. For 2 ≤ u ≤ r we have: (r + 1)



It follows that MU¯ ∩U¯

∂ y¯(1)α0 ∂y (1)β0 ∂ y¯(2)α1 ∂y (1)β0 .. .

0 ··· 0    ∂ y¯(2)α1  ··· 0  ∂y (2)β1 =  .. ..  . .  (r+1)αr  ∂ y¯(r+1)αr ∂ y¯(r+1)αr ∂ y¯ · · · (1)β (2)β ∂y 0 ∂y 1 ∂y (r+1)βr   ∂ y¯(1)α0 0 ··· 0   ∂y (1)β0    ∂ y¯(2)α1 ∂ y¯(1)α1   0   ∂y (1)β0 ∂y (1)β1 · · ·  = .   .. .. ..   . . .   (r)αr   ∂ y¯(r+1)αr ∂ y¯(r)αr ∂ y¯ ··· ∂y (1)β0 ∂y (1)β1 ∂y (r)βr



     =    

Proposition 1.3. The local matrices {MU ∩U¯ } verify the cocycle conditions MU¯ ∩U¯ · MU¯ ∩U = MU¯ ∩U , thus there is a vector bundle ξ (r) = (F (r) , q (r) , E (r) ) over the base E (r) which has as structural functions these local matrices. For every r ≥ 1 the local correspondence (xi , y (1)α1 , . . . , y (r−1)αr−1 , y (r)αr , Y (0)β0 , Y (1)β1 , . . . , Y (r−1)βr−1 , Y (r)βr )→ → (xi , y (1)α1 , . . . , y (r−1)αr−1 , Y (0)β0 , Y (1)β1 , . . . , Y (r−1)βr−1 ) is an π (r) -epimorphism p(r) : ξ (r) → ξ (r−1) of vector bundles. The vector subbundle (r) ker p(r) ⊂ ξ (r) is isomorphic with the vector subbundle ker π∗ ⊂ τ E (r) , called here

832


the partial r-vertical bundle of the anchored vector bundle (θ, D). We call it partial in order to distinct from a total r-vertical bundle, defined as follows. The compositions Π(r) = π (1) ◦ π (2) ◦ · · · ◦ π (r) define a fibered manifold (E (r) , Π(r) , M) over the base M, (r) for every r ≥ 1. We say that the vector subbundle ker Π∗ ⊂ τ E (r) is the total r(r) (u) (u+1) vertical bundle.. We can also consider, for u = 1, ◦ · · · ◦ π (r) r, Πu = π ◦ π (r) (r) (r) (r) and the vector subbundles ker Πr ⊂ ker Πr−1 ⊂ · · · ⊂ ker Π1 = ker Π∗ ⊂ ∗

∗

∗

τ E (r) (they are involutive subbundles of τ E (r) ). Analogously to the partial r-vertical bundle, these vector bundles can be also obtained as a subbundle of ξ (r) , since the vector (r) subbundle ker Pu ⊂ ξ (r) , for 1 ≤ u ≤ r, defined by the vector bundle epimorphism (r) Pu = p(u) ◦ · · · ◦p(r) : E (r) → E (1) = E, is obviously isomorphic with the total r-vertical (r) (r) (r) not. bundle ker Πu ; we denote it as V θu . Notice that V θ1 = V θ(r) is the total r-vertical ∗ bundle. On a domain U of a local adapted chart on E (r) , there are some adapted local sections on ξ (r) , denoted as {s(0)α0 , s(1)α1 , . . . , s(r)αr }α0 ,...,αr =1,m , which are a local base of the module of sections of Γ(ξ (r) ) restricted to U. Let U¯ be an other domain such that U ∩ U¯ 6= ∅. Then, using the matrices (3), on the intersection U ∩ U¯ one has the relations: ∂ y¯(1)β ∂ y¯(2)β ∂ y¯(r+1)β s ¯ + s ¯ + · · · + s¯(r)β , (0)β (1)β ∂y (1)α ∂y (1)α ∂y (1)α ∂ y¯(1)β ∂ y¯(r)β s(1)α = s ¯ + · · · + s¯(r)β , (1)β ∂y (1)α ∂y (1)α .. . ∂ y¯(r)β s¯(r)β . s(r)α = ∂y (r)α Notice that, using lemma 1.2, the following equality holds:    (1)β ∂ y¯(1)β ∂ y¯(2)β ∂ y¯(r+1)β ∂ y¯ 0 · · · 0 · · ·  ∂y (1)α   ∂y (1)α ∂y (1)α ∂y (1)α    (2)β  ∂ y¯(2)β ∂ y¯(1)β   ∂ y¯ ∂ y¯(r+1)β    ···  ∂y (1)α ∂y (1)α · · · 0   0 ∂y (1)α ∂y (2)α  =  .. .. ..   .. .. ..    . . . . . .     ∂ y¯(r+1)β ∂ y¯(r)β ∂ y¯(r)β   ∂ y¯(r+1)β · · · 0 0 · · · ∂y (1)α ∂y (1)α ∂y (r)α ∂y (r+1)α s(0)α =

(6)

t

      .    

If we denote as {ω (0)α0 , ω (1)α1 , . . . , ω (r)αr } the dual base of local (real) linear forms on ξ (r) , which are local sections of the dual ξ (r)∗ , then on the intersection U ∩ U¯ one has the relations: ∂ y¯(1)β (0)β ω ¯ (0)α = ω , ∂y (1)α ∂ y¯(2)β (0)β ∂ y¯(2)β (1)β ω ¯ (1)α = ω + (2)α ω , (7) ∂y (1)α ∂y .. . ∂ y¯(r+1)β (0)β ∂ y¯(r+1)β (1)β ∂ y¯(r+1)β (r−1)β ω ¯ (r)α = ω + ω + · · · + ω . ∂y (1)α ∂y (2)α ∂y (r+1)α


833

Obviously, for 1 ≤ u ≤ r, the set of local sections {s(u)αu , s(u+1)αu+1 , . . ., s(r)αr } (r) (r) is a local base of the sections of ker Pu = V θu and the set of local vector fields ∂ ∂ ∂ (r) (r) ∼ { (u)α , (u+1)α , . . . , (r)α } is a local base of the sections of ker Πu = V θu . ∂y ∂y ∂y ∗ Taking account into the formulas (6) and (7), the following result follows: Proposition 1.4. a) The local definitions J(s(0)α ) = s(1)α , J(s(1)α ) = s(2)α , . . . , J(s(r−1)α ) = s(r)α , J(s(r)α ) = 0, (∀)α = 1, m give a global endomorphism on ξ (r) , called the r-tangent structure. b) The local definitions J ∗ (ω (0)α ) = 0, J ∗ (ω (1)α ) = ω (0)α , . . . , J ∗ (ω (r)α ) = ω (r−1)α , (∀)α = 1, m give a global endomorphism on ξ (r)∗ , called the r-cotangent structure. Notice that: (r) Restricted to the r-vertical vector subbundle ker P1 ⊂ ξ (r) , J induces an endomorphism (denoted also by J) of V θ(r) . (r) (r) (r) (r) (r) (r) (r) J(V θ1 ) = V θ2 , J(V θ2 ) = V θ3 ,. . ., J(V θr−1 ) = V θr , J(V θr ) = 0E (r) (the null vector bundle over E (r) ). ∗ ∗ ∗ r ◦ · · · ◦ J }∗ = 0. J r =J | ◦ J {z | ◦ J ◦{z· · · ◦ J}= 0 and also (J ) =J r+1

times

r+1

times

As tensors (1-covariant and 1-contravariant) on ξ (r) , respectively on ξ (r)∗ , the endomorphisms J and J ∗ have the forms: J = s(1)α ⊗ ω (0)α + s(2)α ⊗ ω (1)α + · · · + s(r)α ⊗ ω (r−1)α , ∗

J =ω

(0)α

⊗ s(1)α + ω

(1)α

⊗ s(2)α + · · · + ω (r)

(r−1)α

⊗ s(r)α .

(8) (9)

r

Proposition 1.5. The local section in ker P1 = V θ(r) defined by Γ = y (1)α s(1)α + 2y (2)α s(2)α + · · · + ry (r)α s(r)α is a global one, called the Liouville section. Proof. We have, on the intersection of the domains of two charts: y (1)α s(1)α + 2y (2)α s(2)α + · · · + ry (r)α s(r)α = (1)β ∂ y¯ ∂ y¯(2)β ∂ y¯(r)β (1)α y s¯ + (1)α s¯(2)α + · · · + (1)α s¯(r)α + (1)α (1)α ∂y ∂y ∂y (2)β (3)β ∂ y¯ ∂ y¯ ∂ y¯(r)β (2)α +2y s¯(2)α + (2)α s¯(3)α + · · · + (2)α s¯(r)α + · · · ∂y (2)α ∂y ∂y (2)β (r)β (1)β (2)β ∂ y ¯ ∂ y ¯ ∂ y ¯ ∂ y ¯ +ry (r)α (r)α s¯(r)α = y (1)α (1)α s¯(1)α + y (1)α (1)α + 2y (2)α (2)α s¯(2)α +· · · ∂y ∂y ∂y ∂y (r)β (r)β (r)β ∂ y ¯ ∂ y ¯ ∂ y ¯ + y (1)α (1)α + 2y (2)α (2)α + · · · + ry (r)α (r)α s¯(r)α = ∂y ∂y ∂y (1)α (2)α (r)α y¯ s¯(1)α + 2¯ y s¯(2)α + · · · + r¯ y s¯(r)α .

834

P. Popescu / Central European Journal of Mathematics 2(5) 2005 826–839 r

Using the isomorphism of the r-vertical bundles, it follows a global section Γ0 = ∂ ∂ ∂ (r) (r) ∼ y (1)α (1)α + 2y (2)α (2)α + · · · + ry (r)α (r)α on ker Πu = V θu . ∂y ∂y ∂y ∗

2

Connections of higher-order

An r-connection on θ is equivalently given by a left splitting of the inclusion i : V θ(r) → ξ (r) of the total r-vertical subbundle (i.e. a vector bundle map C : ξ (r) → V θ(r) , called the r-connection map) such that C ◦ i = 1V θ(r) ) or a vector subbundle Hθ(r) of the vector bundle ξ (r) (called the r-horizontal bundle), such that ξ (r) = V θ(r) ⊕ Hθ(r) (Whitney sum). The link between the r-connection map C and the r-horizontal vector bundle Hθ(r) is Hθ(r) = ker C. In local coordinates, the r-connection map C has the form: C(s(0)α ) =N βα s(1)β + · · · + N βα s(r)β , C(s(1)α ) = s(1)α , . . . , C(s(r)α ) = s(r)α . 1

r

The local functions N βα , . . . N 1

β α

r

are called the local coefficients of the r-connection.

It follows that the local sections {t(0)α }α=1,m defined by t(0)α = s(0)α − N βα s(1)β − · · · − N βα s(r)β 1

r

are a local base of the sections of Hθ(r) . On the intersection of two local domains on E (r) , these local sections change according to the rule t¯(0)α = gαβ t(0)β . Notice that for every r-connection on θ, the r-horizontal vector bundle Hθ(r) is isomorphic with each of the following vector bundles: the quotient vector bundle ξ (r) /V θ(r) and the induced vector bundle (Π(r) )∗ θ, where Π(r) : E (r) → M is the canonical projection. The local sections t(1)α = J(t(0)α ) = s(1)α − N βα s(2)β − · · · − N 1

r−1

β α s(r)β , . . .

. . . , t(r)α = J(t(r−1)α ) = s(1)α are total r-vertical sections and the set {t(0)α0 , t(1)α1 , . . . , t(r)αr }α0 ,...,αr =1,m , is a local base of the module of sections of Γ(ξ (r) ). Let us denote by {ρ(0)α0 , ρ(1)α1 , . . . , ρ(r)αr } the local dual base. This base has a special form: ρ(r)α = ω (r)α + M αβ ω (r−1)β + · · · + M (1)

(r−1)α

ρ

∗

(r)α

= J (ρ

)=ω

(r−1)

(r−1)α

+M (1)

α (1)β + βω

α (r−2)β βω

α (0)β M βω , (r)

+ ···+ M

(r−1)

α (0)β , βω


835

.. . ρ(1)α = J ∗ (ρ(2)α ) = ω (1)α + M αβ ω (0)β , (1)

(1)α

ρ

∗

(1)α

= J (ρ

)=ω

(0)α

.

The local functions M , . . . M are called the (local) dual coefficients of the r-connection. (1)

(2)

The link between the coefficients and the dual coefficients is given by the following formulae: M

β α

=N

M

β α

= N βα + N

(1) (2)

(1) (2)

β α

, (1)

γ α

M (1)

β γ

(10)

,

.. . M (r)

β α

= N βα + N (r)

(r−1)

γ α

M (1)

β γ

+ ···+ N

(1)

γ α

M

(r−1)

β γ

,

which follows from the condition that the bases are dual each to other. It follows that in order to give the coefficients of an r-connection is equivalently to give the dual coefficients. Since the calculus with the coefficients and dual coefficients is quite complicated, we prefer to consider the following construction of an r-connection. We denote as V ∗ ξ (r) the conormal bundle of the vertical vector subbundle V ξ (r) ⊂ ξ (r) , i.e. the subbundle of ξ (r)∗ which consists of real linear forms in ξ (r)∗ which annihilate on V ξ (r) .We say also that V ξ (r) is the annihilator of V ∗ ξ (r) . Generally, for any vector subbundle, the conormal and the annihilator can be considered in the same manner. Proposition 2.1. Let µ(r) ⊂ ξ (r)∗ be a vector subbundle, such that the restrictions of J ∗: J∗

J∗

J∗

J∗

J∗

µ(r) → J ∗ (µ(r) ) → (J ∗ )2 (µ(r) ) → · · · → (J ∗ )r−1 (µ(r) ) → (J ∗ )r (µ(r) ) = V ∗ ξ (r) are isomorphisms. Then the annihilator of the vector subbundle µ(r) ⊕ J ∗ (µ(r) ) ⊕ (J ∗ )2 (µ(r) ) ⊕ · · · ⊕ (J ∗ )r−1 (µ(r) ) ⊂ ξ (r)∗ is the horizontal vector subbundle Hξ (r) of an r-connection of θ. Conversely, if Hξ (r) is the horizontal vector subbundle Hξ (r) of an r-connection of θ and µ(r) ⊂ ξ (r)∗ is the annihilator of the vector subbundle Hξ (r) ⊕ J(Hξ (r) ) ⊕ · · · ⊕ J (r−1) (Hξ (r) ) ⊂ ξ (r) , then the construction above gives the r-connection defined by Hξ (r) . Since the vector bundle V ∗ ξ (r) is isomorphic with the vector bundle (Π(r) )∗ θ, it follows that µ(r) is also isomorphic with this vector bundle.

3

Semi-sprays of higher-order

Let S : E (r) → E (r+1) be an affine section. It has the local form (xi , y (1)α , . . . , y (r)α ) → (xi , y (1)α1 , . . . , y (r)αr , S β (xi , y (1)α1 , . . . , y (r)αr )). The local vector

836


field on E (r) defined by X = y (1)α Dαi

∂ ∂ ∂ + 2y (2)α (1)α + · · · + ry (r)α (r−1)α + (r + i ∂x ∂y ∂y

∂ ∂ = Γ(r) + (r + 1)S α (r)α is a global vector field on E (r) , since on the inter(r)α ∂y ∂y ¯ (r) + (r + 1)S¯α s¯(r)α = Γ(r) − Γ(r) (¯ section of two domains on E (r) one has: Γ y (r)α )¯ s(r)α + 1 (r + 1)(gβαS β + Γ(r) (¯ y (r)α ))¯ s(r)α = Γ(r) + (r + 1)S β s(r)β . Let us notice that, on the r+1 intersection of two domains on E one has: 1 S¯β = gβα S β + Γ(r) (¯ y (r)α . (11) r+1

1)S α

We say that X is an r-semi-spray on θ. We prove now that any r-semi-spray can define an r-connection. Theorem 3.1. If X is an r-semi-spray on θ, then there is an r-connection determined by X. Proof. We are going to use proposition 2.1. Let X = Γ(r) + S β s(r)β . Consider the local sections on V ∗ ξ (r) defined by the formula ∂S α (r−1)β ∂S α ∂S α (0)β (r−2)β ω + ω + · · · + ω = (r)β (r−1)β (1)β ∂y ∂y ∂y ∂ ∂ ∂ (r)α (0)β (1)β (r−1)β ω + ω ⊗ (1)β + ω ⊗ (2)β + · · · + ω ⊗ (r)β (S α ) . ∂y ∂y ∂y

Ωα = ω (r)α +

We have β β ∂S β (r−1)γ α ∂S (r−2)γ α ∂S ω + g ω + · · · + g ω (1)γ + β β ∂y (r)γ ∂y (r−1)γ ∂y (2)γ ∂S β +gβα (1)γ ω (0)γ = gβα ω (r)β + J gβα S β , ∂y

gβα Ωβ = gβαω (r)β + gβα

where J = ω (0)γ ⊗

∂ ∂ ∂ + ω (1)γ ⊗ (2)γ + · · · + ω (r−1)γ ⊗ (r)γ . (1)γ ∂y ∂y ∂y

Let us consider the intersection domains of two coordinates systems on E (r) . Using formulas (7) it follows that ω ¯ (r)β = gβα ω (r)β + J (¯ y (r+1)α ) and from (8), it follows that J =J . Using also formula (11), we have: gβα Ωβ

1 α (r) (r)α =ω ¯ − J (¯ y ) + J S¯ + Γ (¯ y ) = r+1 1 (r)α α (r) (r)α (r+1)α ω ¯ + J S¯ + J Γ (¯ y ) − y¯ = r+1 ¯ α + J gβα y (r+1)β = Ω ¯ α, Ω (r)β

(r+1)α

thus ¯ α. gβαΩβ = Ω


837

It follows that the local sections {Ωα }α=1,m define a vector subbundle µ(r) ⊂ ξ (r)∗ . It is obvious that µ(r) satisfies the hypothesis of the first part of Proposition 2.1, thus the conclusion of the Theorem follows. Notice that the coefficients which appear in the local definition of Ωα are in fact the dual coefficients of the r-connection. They give, by formula (10), all the coefficients of the r-connection.

4

Lagrangians of higher-order

A lagrangian of order r ≥ 1 on an anchored vector bundle (θ, D) is a continuous function L : E (r) → IR which is smooth on E (r)# = E (r) \s0 (M), where s0 : M → E (r) is the (global) section defined by s0 (xi ) = (xi , 0, . . . , 0) (called the null section). The lagrangian 2 ∂ L is regular if the vertical Hessian hαβ = (r)α (r)β of L is non-degenerate. In ∂y y 1≤α,β≤m this case the vertical Hessian defines a (pseudo)metric structure on the fibers of the vertical bundle V E (or V E˜ if the lagrangian is admissible). In this case we denote by hαβ = (hαβ )−1 . Proposition 4.1. Let (θ, D, [·, ·]θ ) be an almost Lie structure. There is an r-semi-spray defined by a regular lagrangian L on θ. Proof. Let U0 be domain of an adapted chart on E and U be the corresponding domain on E (r) . We define ∂L ∂L 1 (1)γ ∂L δ 1 αβ (r) α h Γ − (r−1)β + y B , (12) S = r+1 ∂y (r)β ∂y 2 ∂y (1)δ γβ δ where [sα , sβ ]θ = Bαβ s)δ is the local form of the bracket on θ. Let U0′ ⊂ M and U ⊂ E (r) be the domains of other adapted coordinates, on the intersection of the domains. Let ′ sα = gαα s¯α′ , α = 1, m be the change rule of the local sections of θ on U.. Then ′

′

′

′

′

′

α α α gbβ gγγ Bβα′ γ ′ = Dβi gγ,i − Dγi gβ,i + gαα Bβγ .

(13)

(see [13]). According to the formulas (1) and using Lemma 1.2, we have ∂ ∂y (r−1)α and

′

′

′

∂ y¯(r−1)α ∂ ∂ y¯(r)α ∂ ∂ ∂ y¯(2)α ∂ α′ = + = g + ′ ′ ′ α (r−1)α (r−1)α (r−1)α (r)α (r−1)α (1)α ∂y ∂ y¯ ∂y ∂ y¯ ∂ y¯ ∂y ∂ y¯(r)α′ ′

∂ ∂ y¯(r)α ∂ ∂ α′ = . ′ = gα (r)α (r)α (r)α ∂y ∂y ∂ y¯ ∂ y¯(r)α′ Using also formula (2), we have: ∂L ∂L 1 ∂L δ (r) Γ − (r−1)β + y (1)γ (1)δ Bγβ = (r)β ∂y ∂y 2 ∂y

838


Γ

(r)

Γ

(r)

′ gββ

∂L ∂ y¯(r)β ′

′ gββ

−

′ gββ

∂L ∂ y¯(r−1)β ′

′

∂ y¯(2)β ∂ 1 (1)γ ∂L α′ α − y g B = ′ + (1)β (r)β ∂y ∂ y¯ 2 ∂ y¯(1)α′ α γβ

′ ∂L ∂L ∂L ∂ y¯(2)β ∂ β ′ (r) β′ − gβ + ′ + gβ Γ ′ ′ − (r)β (r)β (r−1)β (1)β ∂ y¯ ∂ y¯ ∂ y¯ ∂y ∂ y¯(r)β ′

1 ∂L ′ α + y (1)γ (1)α′ gαα Bγβ = 2 ∂ y¯ ∂L ∂ ∂L β′ (1)γ i β ′ (r) (r)α′ ¯ y Dγ gβ,i (r)β ′ + gβ Γ + Γ(¯ y ) α′ − ∂ y¯ ∂ y¯ ∂ y¯(r)β ′ −gββ

′

∂L ∂ y¯(r−1)β ′

−

∂L 1 (1)γ i β ′ ′ y Dα gγ + y (1)γ Dβi gγβ + 2 ∂ y¯(r)β ′

1 ∂L ′ ′ ′ α′ α′ + y (1)γ (1)α′ gγγ gββ Bγα′ β ′ − Dγi gβ,i + Dβi gγ,i = 2 ∂ y¯ ′ gββ

∂L ∂L 1 (1)γ ′ ∂L ∂2L (r) α′ (r)γ ′ ¯ Γ − (r−1)β ′ + y¯ B ′ ′ + Γ(¯ y ) γ′ β′ . ∂ y¯(r)β ′ ∂ y¯ 2 ∂ y¯(1)α′ γ β ∂ y¯ ∂ y¯

Thus ′ gαα S α

1 α′ αβ ∂L ∂L 1 (1)γ ∂L δ (r) = g h Γ − (r−1)β + y B = r+1 α ∂y (r)β ∂y 2 ∂y (1)δ γβ 1 α′ β ′ ¯ (r) ∂L ∂L 1 (1)γ ′ ∂L α′ h Γ − + y ¯ B ′β′ + ′ ′ ′ γ r+1 ∂ y¯(r)β ∂ y¯(r−1)β 2 ∂ y¯(1)α 1 ′ ′ ′ + Γ(¯ y (r)γ )hγ ′ β ′ = S¯α + Γ(r) (¯ y α ). r+1

The relation gαα S α = S¯α + ′

spray.

′

1 ′ Γ(r) (¯ y α ) shows that formulas (12) define an r-semir+1

References [1] A. Bejancu: Vectorial Finsler connections and theory of Finsler subspaces, Seminar on Geometry and Topology, Timi¸soara, 1986. [2] M.B. Boyom: Anchored vector bundles and algebroids, arXiv:math.DG/0208012. [3] I. Bucataru: “Horizontal lift in the higher order geometry”, Publ. Math. Debrecen, Vol. 56(1-2), (2000), pp. 21–32. [4] R.L. Fernandes: “Lie algebroids, holonomy and characteristic classes”, Adv. in Math., Vol. 70, (2002), pp. 119–179 (arXiv:math-DG 0007132). [5] Frans Cantrijn and Bavo Langerock: “Generalised Connections over a Vector Bundle Map”, Diff. Geom. Appl., Vol. 18, (2003), pp. 295–317 (arXiv: math.DG/0201274).


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[6] R. Miron: The Geometry of Higher Order Lagrange Spaces. Applications to Mechanics and Physics, Kluwer, Dordrecht, FTPH no 82, 1997. [7] R. Miron and Gh. Atanasiu: “Compendium on the higher order Lagrange spaces”, Tensor, N.S., Vol. 53 (1993), pp. 39–57. [8] R. Miron and Gh. Atanasiu: “Differential geometry of the k-osculator bundle”, Rev.Roum.Math.Pures Appl., Vol. 41(3-4), (1996), pp. 205–236. [9] R. Miron and M. Anastasiei: Vector bundles. Lagrange spaces. Applications to the theory of relativity, Ed. Academiei, Bucure¸sti, 1987. [10] R. Miron and M. Anastasiei: The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Acad. Publ., 1994. [11] M. Popescu: “Connections on Finsler bundles” (The second international workshop on diff.geom.and appl., 25-28 septembrie 1995, Constan¸ta), An. St. Univ. Ovidius Constan¸ta, Seria mat., Vol. III(2), (1995), pp. 97–101. [12] P. Popescu: “On the geometry of relative tangent spaces”, Rev. Roum. Math. Pures Appl., Vol. 37(8), (1992), pp. 727–733. [13] P. Popescu: “Almost Lie structures, derivations and R-curvature on relative tangent spaces”, Rev. Roum. Math. Pures Appl., Vol. 37(8), (1992), pp. 779–789. [14] P. Popescu: On quasi-connections on fibered manifolds, New Developements in Diff. Geom., Vol. 350, Kluwer Academic Publ., 1996, pp. 343–352. [15] P. Popescu: “Categories of modules with differentials”, Journal of Algebra, Vol. 185, (1996), pp. 50–73. [16] M. Popescu and P. Popescu: “Geometric objects defined by almost Lie structures”, In: J.Kubarski, P. Urbanski and R. Wolak (Eds.): Lie Algebroids and Related Topics in Differential Geometry, Vol. 54, Banach Center Publ., 2001, pp. 217–233. [17] P. Popescu and M. Popescu: “A general background of higher order geometry and induced objects on subspaces”, Balkan Journal of Differential Geometry and its Applications, Vol. 7(1), (2002), pp. 79–90. [18] Y.-C. Wong: “Linear connections and quasi connections on a differentiable manifold”, Tôhoku Math J., Vol. 14, (1962), pp. 49–63.

CEJM 2(5) 2005 840–858

Solitary wave and other solutions for nonlinear heat equations Anatoly G. Nikitin∗, Tetyana A. Barannyk Institute of Mathematics, National Academy of Sciences, 3 Tereshchenkivska Street, Kyiv 4, Ukraine

Received 15 December 2003; accepted 18 June 2004 Abstract: A number of explicit solutions for the heat equation with a polynomial non-linearity and for the Fisher equation is presented. An extended class of non-linear heat equations admitting solitary wave solutions is described. The generalization of the Fisher equation is proposed whose solutions propagate with arbitrary ad hoc fixed velocity c Central European Science Journals. All rights reserved.

Keywords: Exact solutions, Fisher equation, conditional symmetry MSC (2000): 35K55, 35Q51, 35Q80

1

Kolmogorov-Petrovskii-Piskunov equation,

Introduction

Lie groups of transformations which conserve the shape of differential equations present powerfool tools for construction of their exact solutions. The related Lie algebras and their invariants are the main instruments used to find the Ansätze which make it possible to find such solutions in explicit form. In the present paper we search for exact solutions of reaction-diffusion equations using a specific Ansätz which can be related to the classical Lie symmetry and conditional symmetry as well. The nonlinear reaction-diffusion equations play fundamental role in a great number of various models of heat and reaction-diffusion processes, mathematical biology, chemistry, genetics and many, many others. Thus, one of the corner stones of mathematical biology ∗


A.G. Nikitin, T.A. Barannyk / Central European Journal of Mathematics 2(5) 2005 840–858

841

is the Fisher equation [1] ut − uxx = u(1 − u)

(1)

where u = u(x, t) and subscripts denote derivatives w.r.t. the corresponding variable: 2 ut = ∂u , uxx = ∂∂xu2 . ∂t Equation (1) is a particular case of the Kolmogorov-Petrovskii-Piskunov (KPP) equation [2] ut − uxx = f (u) (2) where f (u) is a sufficiently smooth function satisfying the relations f (0) = f (1) = 0, fu (0) = α > 0, fu (u) < α, 0 < u < 1. The reaction-diffusion equation with the cubic polynomial nonlinearity ut − uxx = α(u3 + bu2 + cu)

(3)

where α = ±1, b and c are constants, also has a large application value and includes as particular cases the Fitzhugh-Nagumo equation [3] (α = −1, b = −c − 1, 0 < c < 1) which is used in population genetics, the Newell-Whitehead [4] (for c = α = −1, b = 0) and Huxley [1] (for α = b = −1, c = 0) equations. Notice that the Fitzhugh-Nagumo equation also belongs to the Kolmogorov-Petrovskii-Piskunov type. A nice property of equations (1)-(3) is that they admit plane wave solutions which in many cases can be found in explicit form. Existence of such solutions is caused by the symmetry w.r.t. translations t → t + k, x → x + r with constant parameters k and r. For some special functions f (u) equation admits more extended symmetry groups [5] and, as a result, have exact solutions of more general type than plane waves. We notice that group analysis of (2) for f (u) = 0 was carried out by Sophus Lie more than 130 year ago [6]. The group classification of systems of nonlinear heat equations was presented in papers [8]. The conditional (non-classical) symmetry approach [9], [10], [11] enables to construct new exact solutions of partial differential equations which cannot be found in the framework of Lie theory. In application to the equations of type (2) this approach was successfully used for the case of cubic polynomial nonlinearity (3) only [12], [13]. An effective algorithm for construction of travelling wave solutions together with a number of interesting examples was proposed in the recent paper [14]. However the nonlinear heat equations of the general type (2) were not analyzed in [14]. In our paper we present a specific ”universal” Ansätz which enables to make effective reductions of an extended class of equations (2) which includes (1) and (3). Being applied to (3) this Ansätz makes it possible to obtain all exact solutions found earlier [12], [13]. Moreover, we present solutions of (3) (effectively, an infinite number of them) in explicit forms, i.e., in terms of the Jacobi elliptic functions. In addition, using the unified algebraic method [14] we select such equations of the type (2) which admit solitary wave solutions and construct these solutions explicitly. Finally, we present solutions for the Fisher equation and propose such generalization of it which admit the same exact travelling wave solution as (1.1), but with any ad hoc given velocity of propagation (for solutions of

842


(1.1) this velocity is fixed and equal to √56 ). In spirit of Hirota’s method [15] to achieve these goals we use a special Ansätz which leads to a uniform formulation for all considered equations (which, however, is tri-linear). In addition to equations (1) and (3), this Ansätz makes it possible to reduce an extended class of equations of the type (2). In the following section we discuss the Ansätz which will be used to reduce a class of nonlinear heat equations. In Section 3 we present an infinite set of exact solutions (given explicitly in terms of the Jacobi elliptic functions) for the heat equation with the cubic and cubic polynomial nonlinearity. In Section 4 we describe plane wave solutions for special classes of equations (2). In Section 5 solitary wave solutions for equations (2) are found. Finally, in Sections 6 and 7 we present exact solutions for the Fisher equation and propose a generalization of this equation.

2

The Ans¨ atz and related equations

We start with the reaction-diffusion equation with a power nonlinearity ut − uxx = −λun ,

λ=

2(n + 1) (n − 1)2

(4)

where n is a constant, n 6= 1. For convenience we choose a special presentation for the coupling constant λ. Scaling u one can reduce λ to 1 or to -1 for n > 1 and n < 1 respectively. For any n 6= 1 we set z k 2 x u= , k= (5) z n−1 and transform (4) to the uniform equation 2 z zx ztx − zx zxxx − (k − 1)zxx = zx2 (zt − (2k + 1)zxx ) . (6) In contrast with (4) equation (6) is homogeneous with respect to the dependent variable and includes the cubic non-linearities only while (4) includes u in an arbitrary (fixed) power n. We will show that formulation (6) is very convenient for effective reductions. Ansätz (5) is a particular case of a more general one u = zxk ϕ(z)

(7)

where z is an unknown function of independent variables t, x and ϕ is a function of z. It will be shown in the following that the change (7) presents very effective tools for reductions of equations (2) which cover both classical and conditional symmetry reductions. We notice that (4) is not the only nonlinear equation of type (2) which can be reduced to the tri-linear form via Ansätz (5). A more general equation (2) which admits this procedure is 3−n n+1 ut − uxx = k −(k + 1)un + λ1 u + λ2 u 2 + λ3 u 2 + λ4 u2−n (8)


843

where λ1 , . . . , λ4 are arbitrary constants. Formula (8) defines special but rather extended class of the nonlinear heat equations, which includes all important models enumerated in Introduction and many others. The change (5) transforms (8) to the following form 2 z(zx ztx − zx zxxx − λ3 zzx − λ4 z 2 − (k − 1)zxx )

=

zx2 (zt

(9)

+ λ1 z + λ2 zx − (2k + 1)zxx ).

In contrast with (8) equation (9) is homogeneous w.r.t. the dependent variable and is much more convenient for searching for exact solutions.

3

Infinite sets of solutions

Consider a particular (but important) case of (8) which corresponds to n = 3, λ1 = λ2 = λ3 = 0: ut − uxx = −2u3 . (10) The related equation (9) takes the form z(z˙x − zxxx ) = zx (z˙ − 6zxx ).

(11)

Equation (11) is compatible with the condition Xz = 0 where X=

∂ 3 ∂ − . ∂t x ∂x

(12)

It means [10] that this equation admits conditional symmetry , thus it is reasonable to search for its solutions in the form z = ϕ(y),

y = x2 + 6t

(13)

where y is the invariant variable for symmetry (12). Substituting (13) into (11) we come to the third order differential equation for ϕ ϕϕyyy = 3ϕy ϕyy .

(14)

Dividing the l.h.s. and r.h.s. of (14) by ϕϕyy and integrating we obtain ϕyy = cϕ3 ,

c = ±2

(15)

where c is the integration constant which can be reduced to 2 (for c > 0) or to -2 (for c < 0) by scaling the dependent variable ϕ. We make such scaling to simplify the following formulae. In accordance with (5), (13), any solution ϕ of (15) generates a solution for (10) of the following form 2xϕy u= . (16) ϕ

844


An explicit solution of equation (15) for c = 2 is the Jacobi elliptic function

1 ϕ(y) = ds y, √ 2

, y = x2 + 6t

(17)

so (16) can be represented as 2xcs y, √12 . u = u1 = dn y, √12

(18)

The case c = −2 leads to the same solution as given in (18). The plot of this solution is given by Fig. 1.

Fig. 1 Solution (18) for equation (10), 0 < t ≤ 200.

To construct more elliptic function solutions for (10) we exploit some properties of the elliptic functions formulated in the following assertions. Proposition 3.1. Let ϕ = ϕ(n) be a solution of equation (15) for c = 2 or c = −2. Then (n)

ϕ

(n+1)

ϕy = (n) ϕ

(19)

also satisfies this equation for c = 2. Proof of this and the following propositions is reduced to a direct verification. We notice that equation (15) is equivalent to the following one ϕ(n) y

2

= ϕ(n)

4

+ Cn

(20)


845

where Cn is the integration constant. Then ϕ(n+1) of (19) satisfies the equation (ϕ(n+1) )2 = (ϕ(n+1) )4 + Cn+1 , y y

Cn+1 = −4Cn .

Proposition 3.2. Let ϕ(n) be a solution of equation (20) for Cn > 0. Then this equation is solved also by the following function √ (n)

ϕ˜

=

Cn

ϕ(n)

.

Proposition 3.3. Let ϕ(n) be a solution of equation (20) for Cn = −Bn < 0. Then the function √ Bn (n) ϕˆ = (n) (21) ϕ satisfies equation (15) for c = −2 and the following relation ϕˆ(n) y

2

= − ϕˆ(n)

4

+ Bn2 .

(22)

Using Propositions 3.1 and 3.2 and starting with (18) we obtain infinite sets of solutions for equation (10): un = 2xϕ(n) , n = 0, 1, 2, · · · ,

(23)

and u˜2k+1 =

2k+1 x , k = 0, 1, 2, · · · ϕ˜(2k+1)

(24)

where ϕ˜(2k+1) and ϕ(n) are defined by (21) and the following recurrence relations

ϕ

(n)

(n−1) ϕy 1 (0) = (n−1) , ϕ = ds y, √ . ϕ 2

(25)

For n = 1 we have the solutions (17), while for n = 2, 3, · · · and k = 0, 1, · · · we obtain  cd y, √12 − dc y, √12 1 1 u2 = 2x  − cn y, √ ds y, √ , 2 2 sn y, √12 

2x cs4 y, √12 − dn4 y, √12 √ , u3 = 2 9 2 y, √1 √1 dn y, √12 cs y, √12 2cn − ds y, 4 2 2 ···,

(26)

846

and


1 √ 2xdn y, 2 , u˜1 = cs y, √12 √ 2 9 2 √1 √1 4xdn y, √12 cs y, √12 2cn y, − ds y, 4 2 2 u˜3 = , 4 1 1 4 √ √ cs y, 2 − dn y, 2

(27)

···. Formulae (19), (26) , (27) and the recurrence relations (25) present an infinite number of exact solutions for equation (10). Moreover, taking into account the transparent invariance of (10) with respect to displacements of independent variables t and x we can write more general solutions changing x → x + k1 , t → t + k2 with arbitrary constants k1 and k2 . We stress that all these solutions belong to the class found in [12], [13]. Here we present the explicit analytical expressions for these solutions in terms of the Jacobi elliptic functions. The plots of solution u˜1 is given in Fig. 2.

Fig. 2 Solution u ˜1 (27) for equation (10).

Propositions 3.1 and 3.2 make it possible to construct infinite sets of exact solutions for other equations of the type (8). Consider first equations (8) for n = 3, λ2 = λ3 = λ4 = 0, i.e., ut − uxx = −2 u3 + λ1 u

(28)

where without loss of generality we can set λ1 = ±1. For λ1 = −1 (28) is equivalent to


847

the Newell- Whitehead [4] equation up to scaling variables t and x. The Ansätze u = ξx ϕ(ξ),

ξ = k1 cosh(x + k2 ) exp(3t),

λ1 = 1,

(29)

u = ηx ϕ(η),

η = k1 cos(x + k2 ) exp(3t),

λ1 = −1

(30)

reduces (28) to the form (15). Thus repeating the arguments which follow equation (15) we come to exact solutions for (28). The explicit form of these solutions can be obtained from (18), (23)-(27) via the changes y → ξ, 2x → ξx for λ1 = 1 and y → η, 2x → ηx for λ1 = −1. Finally, we notice that Proposition 3 makes it possible to construct infinite sets of exact solutions for the equations (8) with n = λ4 = −1, λ2 = λ3 = 0, i.e., for the equations ut − uxx = 2u3 (31) and ut − uxx = 2 u3 + εu

(32)

which differ from (10) and (28) by the sign of the l.h.s. terms. Indeed, Ansätze (18) and (29) or (30) reduce the corresponding equations (31) and (32) to the following equation for ϕ ϕ′′ = −2ϕ3 (33) where the double prime denotes the second derivative w.r.t. the corresponding variable (i.e., y, ξ or η). In accordance with Proposition 3, exact solutions for (33) have the form (21) where (n) ϕ are solutions of equations (25) for even n. The related list of exact solutions for equation (31) is given by the following formulae: uˆ0 = xsd y, √12 , 4xsn y, √12 , uˆ2 = 1 1 1 1 1 √ √ √ √ √ (34) cd y, 2 − dc y, 2 − cn y, 2 dn y, 2 sn y, 2 ··· uˆ2k =

2k x ϕ(2k)

where ϕ(2k) are defined by recurrence relations (25). The plots of solutions (33) are given in Figs. 3, 4. Solutions for (32) can be obtained from (34) by changing y → ξ, x → ξx and y → η, x → ηx for ε = 1 and ε = −1 respectively.

4

Solutions for arbitrary n

Let us consider equation (8) with arbitrary n and construct its exact solutions. In this section we restrict ourselves to the case λ3 = λ4 = 0 and use a reduced version of the

848




related potential equation (9) i.e., (n − 1)zt = (n + 3)zxx − (n − 1)(λ1 z + λ2 zx ),

(35)

2 4zx zxxx + (n − 3)zxx − (n − 1)(λ1 zx2 + λ2 zx zxx ) = 0.

(36)

Any solution of the system (36) satisfies (9) with λ3 = λ4 = 0, the inverse is not true. To solve (35), (36) we introduce the new variable y = zzxx . Then, dividing (36) by zx2 x


and using the identity

zxxx zx

849

= yx + y 2 we transform this equation to the Riccatti form

yx +

n + 1 2 (n − 1) y − (λ2 y + λ1 ) = 0. 4 4

(37)

Differentiating y w.r.t. t and using (35) and (37) we obtain the following differential consequence y˙ = Ay 3 + By 2 + Cy + D (38) where

1 (n + 3)(n − 3)(n + 1) A= , 8 n−1

B = λ2

3 2 1 − (n − 1) , 16

(n − 1)2 1 (n − 1)2 λ22 − 2(n + 3)(n − 3)λ1 , D = λ1 λ2 . 16 16 For arbitrary n the system of equations (37), (38) is compatible but has constant solutions only. In three exceptional cases n = ±3 and n = −1 the compatibility conditions for (37), (38) are less restrictive in as much as the related coefficient A in (38) is equal to zero. Let n 6= ±3 and n 6= −1, then y = c1 = const, and equations (37), (38), reduce to the only condition 2 λ1 = −c1 λ2 + (k + 1)c21 , k = . n−1 The corresponding solution for the system (35), (36) is C=

2 2 z = ec1 x+kc1t + c2 e(λ2 c1 −(k+1)c1 )t

(39)

and the related exact solution (5) takes the form ck1 u= k . −c1 x−((2k+1)c21 −λ2 c1 )t 1 + c2 e

(40)

Thus we find exact solutions (40) for the equation ut − uxx = −k(k + 1)un + λ2 ku

n+1 2

+ ((k + 1)c21 − λ2 c1 )ku.

(41)

This equation belongs to the Kolmogorov-Petrovski-Piskunov type provided λ2 = (k + 1)(c1 + 1).

(42)

The corresponding plane wave solution (40) propagates with the velocity v = k + 1 − kc1 . In special cases n = ±3 and n = −1 equations (35), (36) admit more extended classes of exact solutions. In particular, for n = 3 we can recover exact solutions for (3) caused by conditional symmetry and classical Lie symmetry as well. We will not study these special cases here.

850

5


Solitary wave solutions

Consider now the general equation (6) with arbitrary parameters λ1 , λ2 , λ3 and λ4 . It seems to be impossible integrate in closed form the related potential equations (9). Here we search for particular solutions which belong to soliton type and so have good perspectives for various applications. Let us consider solutions for (6) of the form z(t, x) = U(ξ) where ξ = µt + x and µ is an arbitrary (nonzero) constant. Then we come to the following ordinary differential equation for U U[U ′ (µU ′′ − U ′′′ − λ3 U) − λ4 U 2 − (k − 1)(U ′′ )2 ] ′ 2

′

(43)

′′

= (U ) [(µ + λ2 )U + λ1 U − (2k + 1)U ] . where U ′ = dU dξ Let us follow [14] and search for solutions for (43) in the form U = ν0 + ν1 ϕ + ν2 ϕ2 + · · · ,

(44)

where ν0 , ν1 , · · · are constants and ϕ satisfies equation of the following general form p ϕ ′ = ε c0 + c1 ϕ + c2 ϕ 2 + · · · (45) where ε = ±1. In order (44) be compatible with (43) we have to equate separately the terms which include odd and even powers of the square root given by (45). In view of this we come to the following system U ′ (µUU ′′ − λ3 U 2 ) = (U ′ )3 (µ + λ2 ),

(46)

U(U ′ U ′′′ + λ4 U 2 + (k − 1)(U ′′ )2 ) = (U ′ )2 ((2k + 1)U ′′ − λ1 U).

(47)

Dividing any term in (46) by µU 2 U ′ we come to the Riccatti equation Y′− for Y =

U′ , U

λ2 2 λ3 Y = µ µ

whose general solutions are Y = Y =

q q

y=

−λ3 λ2 −λ3 λ2

r

√ tanh −λµ2 λ3 ξ + C , √ −1 −λ2 λ3 tanh ξ+C , if λ2 λ3 < 0, µ √

λ2 λ3 ξ+C , if λ2 λ3 > 0, µ µ y=− , if λ3 = 0 λ2 (ξ + C)

λ3 tan λ2

where C is the integration constant.

(48)

(49) (50)


851

Thus all solutions for (43) which can be obtained with using the algebraic method [14] are exhausted by hyperbolic, triangular and rational ones presented by relations (48)(50). Solutions (48), (49) and (50) are compatible with (47) provided

λ3 µ = −λ2 , λ1 = −k , λ4 = (1 − k) λ2

λ3 λ2

2

, λ2 λ3 6= 0

and λ1 = λ4 = 0, λ3 = 0 respectively. Using variables √ 2 λ3 2 τ= t, y = x, σ = −λ2 (n − 1), ν = 2 (n − 1) n−1 λ2 we can rewrite the related equation (8) as follows: n+1 uτ − uyy = 1 + νu1−n −(n + 1)un + ν(n − 3)u + σu 2 .

(51)

The corresponding solutions (5), (48) for equation (51) have the following form 1 u = (−ν) n−1 tanh b y − 1 u = (−ν) n−1 tanh b y − q where ν < 0 and b = (n − 1) −ν , 2 u = (ν) where ν > 0 and b = (n − 1)

1 n−1

pν

u=2

2

+C

+C

√σ t 2

√σ t 2

2 n−1 2 1−n

2 n−1 σ tan b y − √ t + C 2

,

(52) (53)

(54)

, and

n−1

2 1−n σ (n − 1) y − √ τ + C 2

(55)

if ν = 0. 2 For n−2 > 1 formula (52) presents nice solitary wave solutions which propagate with the velocity √σ2 . In the case n = 2 we come to the bell-shaped solitary wave solution which will be discussed in Section 7. 2 If n−2 < −1 then (52) is a singular solution whose physical relevance is doubtful. However, in this case equation (51) admits another solitary wave solutions which are given now by relation (53). We see such solutions exist for the extended class of the nonlinear reaction-diffusion equations defined by formula (51).

852

6


Exact solutions for the Fisher equation

Let us return to Section 4 and consider in more detail the important case n = 2. Setting in (41) λ2 = 0, c1 = −1 and making the change τ = 6t, y =

√

6x

(56)

we come to the Fisher equation (1) for u(τ, y): uτ − uyy = u(1 − u).

(57)

Thus the Fisher equation is a particular case of (41) and so our solutions (40) are valid for (57) provided we make the above mentioned changes of variables and set c1 = −1 in accordance with (42). As a result we recover the well-known Ablowitz-Zeppetella [16] solution 1 (58) u= 2 . y √ − 5τ 6 6 1 + c2 e This solution can be expressed via hyperbolic functions 1 u = u1 = 4

1 − tanh

y 5 √ − τ −c 2 6 12

2

,

c=

2 1 y 5 u = u2 = 1 − coth √ − τ − c 4 2 6 12

1 ln |c2 |, 2

(59)

(60)

for c2 > 0 and c2 < 0 respectively. Taking into account the symmetry of (57) w.r.t. the discrete transformation u → 1−u we obtain two more solutions: u3 = 1 − u1 and u4 = 1 − u2 . Finally, bearing in mind the symmetry of (57) w.r.t. the space reflection y → −y we come to four more exact solutions by changing y → −y in u1, u2 , u3 and u4 . Thus starting with our general formulae (39) and (40) we come to the family of eight exact solutions for the Fisher equation. All of them are plane waves propagating with the velocity ± √56 . To find additional exact solutions we use the Ansätz (7) for n = 2, i.e., u = 3zy2 ϕ(z)

(61)

where ϕ and z are functions to be found. Substituting (61) into (57) we come to the following reduced equations zτ = 5zyy , 4zy zyyy −

2 zyy

(62) =

1 2 z 2 y

and ϕzz = 3ϕ2 .

(63)


853

We see that ϕ has to satisfy the Weierstrass equation (63) which we rewrite in the following equivalent form 1 ϕ˜2z = 4ϕ˜3 − C, ϕ˜ = ϕ (64) 2 where C is the integration constant. Starting with (61) and choosing the following exact solutions of (62) and (64): 2 1 5 (65) z = exp − √ y + τ , ϕ = 6 z+k 6 we come to the Ablowitz-Zeppetella solutions (58) for the Fisher equation. We notice that relations (65) present only a very particular solution of (64) which correspond to zero value of the parameter C. In addition, there exist the infinite number of other solutions corresponding to non-zero C. The related functions (61) are: 1 2 1 5 u = z ℘(z, 0, C), z = exp − √ y + τ + k (66) 2 6 6 where ℘(z, 0, C) is the Weierstrass function satisfying equation (64) for C 6= 0. In order to solutions (61) be bounded it is sufficient to restrict ourselves to the case when − √16 y + 65 τ + k > 0. Such conditions can be satisfied, e.g., for arbitrary positive y and negative τ and k. The related solutions can be interpreted as ones describing the history of the process because the time variable takes arbitrary negative values. The graphics of solutions (61) for some values of the parameter C are given by Figures 5-7.

Fig. 5 Solution (66) with k = 0, C = 102 for the Fisher equation (57).

Thus the Fisher equation admits the infinite set of exact solutions which include the Ablowitz-Zeppetella solutions (58) and also solutions (66) enumerated by two parameters, C 6= 0 and k. All these solutions are plane waves propagating with the same velocity v = √56 .

854




In the following section we consider generalized Fisher equations which admit exact solutions with arbitrary propagation velocities.

7

Generalizations of the Fisher equations

Let us consider equation (51) for n = 2, which takes the following form √ uτ − uyy = (u + ν)(−3u + σ u − ν).

(67)


855

Relation (67) is a formal generalization of the Fisher equation in as much as in the case σ = 0 (67) is equivalent to (1). However, for σ 6= 0 equation (67) admits soliton solutions (52) and (or) solutions (53)-(55) and so has absolutely another nature then (1). Nevertheless for small σ it would be interesting to treat (67) as a perturbed version of (1). Consider equation (67) in more detail. Let ν < 0 then scaling dependent and independent variables we can reduce its value to the following one 3 ν → ν ′ = − , if ν < 0. 2

(68)

Setting then

√ 3 − u, σ = 3ε, t = 3τ, x = − 3y 2 we come to the following relation 12 ! 3 − u˜ . u˜t − u˜xx = u˜ 1 − u˜ + ε 2 u˜ =

(69)

(70)

In the limiting case ε → 0 equation (70) reduces to the Fisher equation in the canonical formulation (1). In accordance with Section 5 equation (67) admits nice bell-shaped traveling wave solution (52) which transforms via changes (68), (69) to the following form u˜ =

3 2 cosh2 21 x −

√ε t 6

+C

.

(71)

Consider now equation (67) for νσ = 0 = 0 and set u = u3˜ . As a result we reduce (67) to the simplest form u˜τ − u˜yy = −˜ u2 . (72) The Ansätz z2 1 u = x2 + 6z 3

r ! 3 zxx 1+ε , 2 z

ε = ±1

leads to the following reduced equations zxxx = 0, Thus we have z =

x2 2

zt = κzxx ,

κ = 5(3 ±

√

6).

+ κt and the related exact solution for (72) is √ √ (3 ± 6)x2 + 10(12 ± 5 6)t u= . √ 2 3 x2 + 10(3 ± 6)t

We notice that this solution can be found also using the classical Lie reduction. Finally, let us consider one more generalization of the Fisher equation given by relation (41) for n = 2. Using notations (56) we rewrite it in the following form 1

u˜τ − u˜yy = u˜(−c1 + (c1 + 1)˜ u 2 − u˜).

(73)

856


For c1 = −1 equation (73) reduces to the Fisher equation (57). In accordance with the results presented in Section 4 equation (73) admits exact solutions (40) which in our notations can be rewritten as u˜1 =

1 4

u˜2 =

1 4

1 + tanh

c√1 y 2 6

c1 (2c1 −3) τ 12

2

−c , 2 1 + coth 2c√16 y + c1 (2c121 −3) τ − c .

+

(74)

Two more solutions can be obtained by changing y → −y in (74). Formulae (74) present the analogies of solutions (59), (60) for equation (73). In contrast with (59), (60) these solutions describe a wave whose propagation velocity is equal to 2c√1 −3 Thus changing parameter c1 in (73) we can obtain solutions (74) with any 6 velocity of propagation given ad hoc. In other words we always can take this velocity in accordance with experimental data. Thus equation (73) reduces to the Fisher equation if the parameter c1 is equal to −1. Moreover, both equations (57) and (73) admit the analogous exact solutions, (59), (60) and (74), which, however, have different propagation velocities.

8

Discussion

There exist well known regular approaches to search for exact solutions of nonlinear partial differential equations - the classical Lie approach [6], the conditional (non-classical) symmetries method [9], [11], [10], [12], generalized conditional symmetries [17], etc. These approaches present effective tools for finding special Ansätze which make it possible to reduce the equation of interest and find its particular solutions. However, sometimes it seems that the Ansätze by themselves are more fundamental than the related symmetries. First, historically, the most famous Ansätze (like the ColeHopt one for the Burgers equation) was found without a scope of a symmetry approach. Secondly, some of Ansätze are effective in rather extended classes of problems characterized by absolutely different symmetries. In addition, in some cases the direct search for the Ansätz is a more straight-forward and effective procedure than search for (conditional) symmetries. We remind that the conditional symmetry approach presupposes search for solutions of nonlinear determining equations which in many cases are not simpler than the equation whose symmetries are investigated [18]. The present paper is based on using special Ansätze (7) which appear to be very effective for the extended class of nonlinear reaction-diffusion equations. In particular, they make it possible to find new exact solutions for the very well studied heat equations with quadratic polynomial non-linearities. Moreover, such Anaätze can be used to reduce wave equations of another type, e.g., hyperbolic equations. We plane to discuss the related results elsewhere. We propose a generalization of the Fisher equation which preserves the type of its exact solutions, but predicts another propagation velocity. This property distinguishes (73) from numerous other generalizations of the Fisher equation refer, e.g., to [19] and


857

references cited therein. Finally, we find soliton solutions for a number of nonlinear equations (8). To make this we use the algebraic method [14] which however was applied not directly to the equation of interest (8) but to the potential equation (9). By this we extend the class of nonintegrable equations which have soliton solutions to the case of appropriate quasi-linear heat equations (2). We stress that all these results were obtained with essential using the Ansätz (7). It seems to be an intriguing problem to find a regular way for searching such ”universal” Ansätze.

References [1] J.D. Murray: Mathematical Biology, Springer, 1991. [2] A.N. Kolmogorov, I.G. Petrovskii and N.S. Piskunov: “A study of the diffusion equation with increase in the quantity of matter, and its application to a biological problem”, Bull. Moscow Univ. Sér. Int. A, Vol. 1(1), (1937). [3] R. Fitzhugh: “Impulses and physiological states in models of nerve membrane”, Biophys. J., Vol. 1(445), 1961; J.S. Nagumo, S. Arimoto and S. Yoshizawa: “An active pulse transmission line simulating nerve axon”, Proc. IRE, Vol. 50(2061), (1962). [4] A.C. Newell and J.A. Whitehead: “Finite bandwidth, finite amplitude convection”, J. Fluid Mech., Vol. 38(279), (1969). [5] V.A. Dorodnitsyn: “On invariant solutions of nonlinear heat equation with source”, Comp. Meth. Phys., Vol. 22(115), (1982). [6] S. Lie: Transformationgruppen, Leipzig, 1883. [7] P. Olver: Application of Lie groups to differential equations, Springer, Berlin, 1986. [8] A.G. Nikitin and R. Wiltshire: “Symmetries of Systems of Nonlinear ReactionDiffusion Equations”, In: A.M. Samoilenko (Ed.): Symmetries in Nonlinear Mathematical Physics, Proc. of the Third Int. Conf. , Kiev, July 12-18, 1999, Inst. of Mathematics of Nat. Acad. Sci. of Ukraine, Kiev, 2000; R. Cherniha and J. King: “Lie symmetries of nonlinear multidimensional reactiondiffusion systems: I”, J. Phys. A, Vol. 33(257), (2000); A.G. Nikitin and R. Wiltshire: “Systems of Reaction Diffusion Equations and their symmetry properties”, J. Math. Phys., Vol. 42(1666), (2001). [9] G.W. Bluman and G.D. Cole: “The general similarity solution of the heat equation”, J. Math. Mech., Vol. 18(1025), (1969). [10] W.I. Fushchych and A.G. Nikitin: Symmetries of Maxwell’s equations, Reidel, Dordrecht, 1987; W.I. Fushchych: “Conditional symmetry of mathematical physics equations”, Ukr. Math. Zh., Vol. 43(1456), 1991. [11] D. Levi and P. Winternitz: “Non-classical symmetry reduction: example of the Boussinesq equation”, J. Phys. A, Vol. 22(2915), (1989). [12] W.I. Fushchich and M.I. Serov: “Conditional invariance and reduction of the nonlinear heat equation”, Dokl. Akad. Nauk Ukr. SSR, Ser. A, Vol. 4(24), (1990).

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[13] P.A. Clarkson and E.L. Mansfield: “Symmetry reductions and exact solutions of a class of nonlinear heat equations”, Physica D, Vol. 70(250), (1993). [14] E. Fan: “Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method”, J. Phys. A, Vol. 35(6853), (2002). [15] R. Hirota and J. Satsuma: “Soliton solutions of a coupled Korteweg-de Vries equation”, Phys. Lett. A, Vol. 85(407), (1981). [16] M.J. Ablowitz and A. Zeppetella: “Explicit solution of Fisher’s equation for a special wave speed”, Bull. Math. Biol., Vol. 41(835), (1979). [17] A.S. Fokas and Q.M. Liu: “Generalized Conditional Symmetries and Exact Solutions of Non Integrable Equations”, Theor. Math. Phys., Vol. 99(371), (1994). [18] R.Z. Zhdanov and V.I. Lahno: “Conditional symmetry of a porous medium equation”, Physica D, Vol. 122(178), (1998). [19] D.J. Needham and A.C. King: “The evolution of travelling waves in the weakly hyperbolic generalized Fisher model”, Proc. Roy. Soc. (London), Vol. 458(1055), (2002). P.S. Bindu, M. Santhivalavan and M. Lakshmanan: “Singularity structure, symmetries and integrability of generalized Fisher-type nonlinear diffusion equation”, J. Phys. A, Vol. 34(L689), (2001).

CEJM 2(5) 2005 859–883

Geometric classes of Goursat flags and the arithmetics of their encoding by small growth vectors∗ Piotr Mormul† Institute of Mathematics, Warsaw University, Banach st. 2, 02-097 Warsaw, Poland

Received 15 December 2003; accepted 10 October 2004 Abstract: Goursat distributions are subbundles, of codimension at least 2, in the tangent bundles to manifolds having the flag of consecutive Lie squares of ranks not depending on a point and growing – very slowly – always by 1. The length of a flag thus equals the corank of the underlying distribution. After the works of, among others, Bryant & Hsu (1993), Jean (1996), and Montgomery & Zhitomirskii (2001), the local behaviours of Goursat flags of any fixed length r ≥ 2 are stratified into geometric classes encoded by words of length r over the alphabet {G, S, T} (Generic, Singular, Tangent) starting with two letters G and having letter(s) T only directly after an S, or directly after another T. It follows from [6] that the Goursat germs sitting in any fixed geometric class have, up to translations by rk D − 2, one and the same small growth vector (at the reference point) that can be computed recursively in terms of the G, S, T code. In the present paper we give explicit solutions to the recursive equations of Jean and show how, thanks to a surprisingly neat underlying arithmetics, one can algorithmically read back the relevant geometric class from a given small growth vector. This gives a secondary, G¨ odel-like super-encoding of the geometric classes of Goursat objects (rather than just a 1-1 correspondence between those classes and small growth vectors). c Central European Science Journals. All rights reserved.

Keywords: Goursat flag, singularity, geometric class, small growth vector, encoding by small growth vectors MSC (2000): 58A17

∗ †

Supported by Polish KBN Grant 2 P03A 010 22. E-mail: [email protected]

860

1

P. Mormul / Central European Journal of Mathematics 2(5) 2005 859–883

Geometric classes of germs of Goursat flags

Goursat flags are certain special nested sequences, say F , of variable length r (2 ≤ r ≤ n − 2) of subbundles in the tangent bundle T M to a smooth (C∞ ) or analytic (C ω ) ndimensional manifold M: D r ⊂ D r−1 ⊂ · · · ⊂ D 1 ⊂ D 0 = T M. Namely, one demands, for l = r, r − 1, . . . , 1 that (a) cork D l = l, and (b) the Lie square of D l be D l−1 . Every member of F save D 1 is called Goursat distribution, r is called the length of F . They naturally generalize the well-known Cartan’s distributions on the jet spaces of functions R → R. The latter (the smallest flag’s member is then of rank 2) satisfy (a) – (b), but display no singularities. While these conditions do admit singularities, as it has been known since 1978 (Giaro-Kumpera-Ruiz). This, very restricted, class of objects was being investigated (intermittently) over the last 110 years, with important contributions by E. von Weber [12] and E. Cartan [2]. They proved independently that every corank-r Goursat distribution D r around a generic point of M locally behaves in a unique way visualised by the chained model – the germ at 0 ∈ Rn (x1 , . . . , xr+2 ; xr+3 , . . . , xn ) of ∂n , . . . , ∂r+3 ; ∂r+2 , ∂1 + x3 ∂2 + x4 ∂3 + · · · + xr+2 ∂r+1

(C)

(these are vector fields generators; effectively used are only first r + 2 coordinates). Chained models can be viewed as the simplest instance of a family of local writings (preliminary normal forms with real parameters of, in general, unknown status) of Goursat distributions, obtained much later by Kumpera and Ruiz in [7] and quoted in Thm. 1.1. We call them KR pseudo-normal forms. Repeating, Kumpera and Ruiz discovered singularities hidden in flags, and pseudo-normal forms were merely a byproduct. Those forms, however, have been an important step in the (still open) problem of the local classification (C∞ or C ω ) of flags. Theorem 1.1 ([7]). For any Goursat flag of length r on a smooth ( C∞ or C ω ) manifold M of dimension n ≥ r + 2, T M = D 0 ⊃ D 1 ⊃ D 2 ⊃ · · · ⊃ D r , around any point p ∈ M there exist local coordinates x1 , x2 , . . . , xr+2 ; xr+3 , . . . , xn centered at p, of the same class as M, such that in these coordinates each D j has around p a Pfaffian description ω 1 = ω 2 = · · · = ω j = 0, j = 1, 2, . . . , r, where ω 1 = dxi1 − x3 dxj1 ,

(i1 , j1 ) = (2, 1)

2

i2

4

j2

(i2 , j2 ) = (3, j1 ) = (3, 1)

3

i3

5

j3

(i3 , j3 ) ∈ {(4, j2 ), (j2 , 4)}

4

i4

ω = dx − x dx , ω = dx − x dx , 6

j4

ω = dx − X dx , ∗ r

ir

ω = dx − X

(i4 , j4 ) ∈ {(5, j3 ), (j3 , 5)} ∗

r+2

jr

dx ,

∗

∗

(ir , jr ) ∈ {(r + 1, jr−1 ), (jr−1 , r + 1)} .

In this writing, for 6 ≤ l ≤ r + 2, X l = cl + xl excepting the cases of inversions (il−2 , jl−2 ) = (jl−3 , l − 1) when simply X l = xl . That is, in these coordinates each


861

D j becomes the germ at 0 ∈ Rn of the indicated corank-j polynomial Pfaffian system. The c6 , c7 , . . . , cr+2 are real constants that are not, in general, uniquely determined by the flag’ germ. In a family of cases, specific simplifications occur in these pseudo-normal forms: when for certain 4 ≤ l ≤ r there is no inversion in the forms ω 3, ω 4 , . . . , ω l , then the constants c6 , c7 , . . . , cl+2 are absent (are zero). Conversely, all pairs of sequences {il } and {jl } ( l = 1, . . . , r) fulfilling the conditions written above, and arbitrary real constants c6 , c7 , . . . , cr+2 (when applicable) are permitted and always give a Goursat flag. We underline that in this theorem all members of the flag of D r simultaneously get polynomial descriptions. The generic model (C) shows up when there is no inversion of differentials (and, in consequence, no constants) in all the Pfaffian equations proposed in Thm. 1.1. Definition 1.2. Small growth vector {nj (p)} of a distribution D at a point p ∈ M is the sequence of dimensions at p of the (local) modules of vector fields Vj , V1 = D, Vj+1 = Vj + [D, Vj ], nj (p) = dim Vj (p) , j = 1, 2, 3, . . . When this sequence attains the value dim M for the first time in its l-th term, nl−1 (p) < nl (p) = dim M, then l is called the nonholonomy degree of D at p. Corollary 1.3. It follows automatically from Theorem 1.1 that Goursat distributions D of arbitrary rank locally are the direct sums of integrable distributions (foliations) D ′ and of Goursat distributions D ′′ of rank 2 invariant with respect to the vector fields taking values in D ′ . In Kumpera-Ruiz coordinates for a germ of D, that integrable summand D ′ gets a clear description dx1 = dx2 = · · · = dxr+2 = 0. As for the small growth vector of D at any point, it is, naturally, the translation (shift) by the value rk D − 2 of the small vector of D ′′ at that point. Constants appearing in Thm. 1.1 often reflect different geometric behaviours of flag’s members. But not all constants, and here is the first nontrivial Example 1.4. For r = 5 and n = r + 2 = 7, in the family of KR pseudo-normal forms dx2 − x3 dx1 , dx3 − x4 dx1 , dx1 − x5 dx4 , dx5 − (c6 + x6 )dx4 , dx6 − (c7 + x7 )dx4 (1) around 0 ∈ R7 , the objects with c6 = 0 are non-equivalent to those with c6 6= 0. Among the former, the value of c7 can be reduced either to 0 or to 1, and these two normalized values are non-equivalent. Among the latter, c6 can be reduced to 1, and (quite unexpectedly; overlooked in [7], rectified in [4] and [3]) c7 to 0. Thus the non-equivalent ‘model’ values of (c6 , c7 ) are just (1, 0), (0, 1) and (0, 0). On the whole, the entire local classification problem for Goursat distributions (still

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open in general) can be worded as follows: to tell what families of constants (meaning the KR pseudo-normal forms with those constants) are equivalent to what other families; when the length r is fixed, these families consist of not more than r − 5 real numbers. Things being so, researchers have tried to approximate the orbits of the classification by analyzing flags’ local geometries – by proposing different stratifications in the vast space of all Goursat germs. An important approach in that direction has been commenced in [6], or rather (if only on an experimental level) in [5]. In the present paper we want to continue and close the work [6], by joining much more tightly the highly original stratification proposed in it with the notion of small growth vector (Def.1.2 above) of a distribution. For, the relation established in [6] (clearly pioneering in the mid 90s) goes only in one direction, establishing unique s. gr. v.’s for the entire Jean’s strata. And not yet answering whether that associating mapping is injective. Our objective is to answer this question in the affirmative (see Main Theorem in Chap. 3), and to do that in the most effective way – by giving an arithmetical algorithm for the retrieval of the original stratum from the small growth vector associated to it. Prior to that, however, Jean’s strata must be carefully described and encoded – by quite natural words over the alphabet {G, S, T}. This contribution (being of independent significance and due to Montgomery & Zhitomirskii) is reported in the present, long chapter. Then the main result of [6] – the recurrences yielding the s. gr. v.’s in an implicit way – is recalled in Chap. 2, followed by a reorganization of the data issuing from it (proposed by us). Our solutions of those recurrences are presented in Chap. 3. The injectivity, hidden in the solutions, is being uncovered in Main Theorem in section 3.2.

1.1 Sandwich Diagram and the definition of geometric classes. A neat geometric clarification is possible to the pseudo-normal forms of Kumpera & Ruiz. It has been due mainly to Jean and Montgomery & Zhitomirskii. Upon closer inspection there emerges, [6], [3], [9], a stratification of germs of flags into canonically defined geometric classes, with strata encoded by words (of length equal to flag’s length) over the alphabet {G, S, T}: Generic, Singular, Tangent, subject to certain restrictions. We want to recall that definition and draw some natural corollaries. The first ingredient is the classical notion, for any distribution D, of the module (or, in the analytic category, sheaf of modules) of Cauchy-characteristic vector fields v with values in D that preserve D, [v, D] ⊂ D. And one of first observations (see, for inst., Lemma 2.1 in [9]) is that for D – Goursat, L(D) is a regular corank two subdistribution of D, rk L(D) = rk D − 2, enjoying one additional (and key) property. Namely, L([D, D]) ⊂ D. Remark 1.5. It becomes clear that L(D) is that integrable direct summand D ′ in D mentioned in Cor. 1.3 (which thus turns out to be unique). For, clearly, [D ′ , D] ⊂ D, hence D ′ ⊂ L(D), while these two subdistributions of D have the same rank rk D − 2.


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Therefore, in any Kumpera-Ruiz coordinates for D – Goursat of corank j on a manifold of dimension n, its Cauchy characteristics L(D) = (∂j+3 , ∂j+4 , . . . , ∂n ). As to the second property L([D, D]) ⊂ D, it is also visible through Thm. 1.1 which supplies local forms simultaneously for D and [D, D], and allows to compute L([D, D]) as well. The second ingredient is putting this all together for a corank-r Goursat distribution D , first done in [9] (p. 464) under the form of the Sandwich Diagram. r

TM

⊃

D1

⊃

D2 ∪

⊃

D3 ∪

⊃ ··· ⊃

D r−1 ∪

⊃

Dr ∪

L(D 1 ) ⊃ L(D 2 ) ⊃ · · · ⊃ L(D r−2 ) ⊃ L(D r−1 ) ⊃ L(D r ). In view of the mentioned properties, all direct inclusions in this diagram are of codimension one. One gets here r − 2 squares (indexed by the upper right vertices) built of inclusions, and in each j-th square (j = 3, 4, . . . , r) the distributions D j and L(D j−2 ) have the same rank. These spaces can be perceived as certain fillings in a sandwich with covers D j−1 and L(D j−1 ) (of not the same dimension). With this interpretation at hand, Montgomery & Zhitomirskii (preceded by Bryant & Hsu with a similar, if given in a less explicit form, idea sketched in [1], p. 455) say that D k is at p in singular position when it coincides at p with L(D k−2 ): D k (p) = L(D k−2 )(p). That is, when the fillings in the k-th sandwich coincide (coalesce) at p. Proposition 1.6. For any Goursat flag F : D 1 ⊃ D 2 ⊃ · · · D r−1 ⊃ D r , the corank-k member D k is at p in singular position iff in any KR pseudo-normal form for F near p there is the inversion of indices (ik , jk ) = (jk−1 , k + 1) in the relevant Pfaffian equation ω k = 0. In such a case the locus Nk of points where D k is in singular position has locally in these KR coordinates the equation xk+2 = 0. Proof. By Rem.1.5, in any chosen KR coordinates, L(D k−2 ) = (∂k+1 , ∂k+2 , . . . , ∂n ). In the absence of inversion in ω k , D k = (∂n , . . . , ∂k+2 , X k+2 ∂k+1 + ∂jk−1 + · · · ) and jk−1 ≤ k, so that D k (0) 6= L(D k−2 )(0). In the presence of inversion, D k = ∂n , . . . , ∂k+2 , ∂k+1 +xk+2 (∂jk−1 +· · · ) and p, recalling, is 0 in that KR chart. In this case, for x near 0, x ∈ Nk ⇔ ( ∂n , . . . , ∂k+2 , ∂k+1 + xk+2 (∂jk−1 + · · · ) ) = (∂n , . . . , ∂k+2 , ∂k+1 ) ⇔ xk+2 = 0. Corollary 1.7. For any flag F of length r, each singularity set Nk , k = 3, . . . , r, is either empty or an embedded codimension-one submanifold of M. Moreover, at every point of Nk , D k is transverse to Nk . Note. Now it is easy to produce examples of flags’ germs belonging only to beforehand prescribed KR singularity classes. The classes can be identified with the subsets of the set {3, 4, . . . , r}, and one just chooses the second alternatives (inversions) in Thm. 1.1

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in the Pfaffian equations ω k = 0 with indices k from the desired subset of {3, 4, . . . , r}. (Constants in the remaining equations, if any, do not matter for any such example.) The construction of the encoding word. Two biggest members of F , D 1 and D 2 , are nowhere in singular positions. Because of that the word starts from the left with two letters G. Further members of F are, or are not, in singular positions at p depending on the KR singularity class of the germ of F at p. (i) For p sitting in M \ (N3 ∪ · · · ∪ Nr ) the encoding word is, by definition, r letters G going in row. Attention. Repeating occurrences of the same letter going in row will henceforth be noted by a subscript expressing the number of occurrences. In the complement of all hypersurfaces (or empty sets) Nk , the local geometry is thus encoded by Gr . (ii) If precisely the members D k1 , D k2 , . . . , 3 ≤ k1 < k2 < · · · ≤ r, are at p in singular position, then the code is being constructed in several steps. It starts with k1 − 1 G’s on the left, and its k1 -th, k2 -th, . . . letters from the left are S. (iii) If p is such that, after step (ii), defined is still not the entire word of r letters, then there is a sequence (possibly, more than one, and possibly of different lengths) of l ≥ 1 blank spaces in it, after certain letter S at the k-th place. Naturally, if this is the last S in the word, then the string of l blank spaces ends the word. How to fill these spaces with letters G and/or T ? Here is the procedure for one blank string. If there are several blank intervals after step (ii) then, naturally, the procedure should be applied several times. Nk ∋ p is smooth by Cor. 1.7. Observe at first that if the (k + 1)-th letter were S (and not a blank space as it actually is) then D k+1 would be transverse to Nk at p. Indeed, in KR coordinates L(D k−1 ) = (∂k+2 , ∂k+3 , . . . , ∂n ) by Rem. 1 and Nk = {xk+2 = 0} by Prop. 1.6. By continuity argument, also in our situation D k+1 can be transverse to Nk at p.‡ If it really is, then the letters No k + 1, k + 2, . . . , k + l in the word are declared G. If, however, D k+1 is not transverse, D k+1(p) ⊂ Tp Nk , then we encounter a deeper, second order (tangent) singularity and we put in the word a letter T at the (k + 1)-th place. (Note here an evident thing that the tangent position of D k+1 at a point is – due to the absence of transversality to Nk – different from the first order singular position. And, needless to say, this tangent position equals D k (p) ∩ Tp Nk , cf. Cor. 1.7. Thus in the sandwich L(D k )(p) ⊂ V ⊂ D k (p) there are two geometrically distinguished positions for a (k + 1)-dimensional space V : the first order singular position L(D k−1 )(p) and the second order tangent position D k (p) ∩ Tp Nk .) Before passing to singularities of higher orders, an analogue of Cor. 1.7 is needed. How ‡

A codimension-1 subspace L(Dk )(p) of Dk+1 is automatically tangent to Nk , because the flows of vector fields in L(Dk ) preserve – by definition – the equality Dk = L(Dk−2 ) holding originally at p. But there is one spare dimension in Dk+1 (p).


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does the locus Nk,k+1, of the singular behaviour ‘ST’ featured by D k and D k+1 together, look like near p ? To answer this, and similar further questions in later stages of the definition, we take again any KR pseudo-normal form (ω 1 , ω 2 , . . . , ω r ) for F near p. (We will resort to KR coordinates on several more occasions, after which our reference point p ∈ M becomes 0 ∈ Rn . But those are, we underline, only technical tools for handling the Jean stratification created by F in the vicinity of p ∈ M. Therefore, the (M, p) and (Rn , 0) contexts will interweave.) In the situation discussed in (iii), by Prop. 1.6, there is an inversion in ω k , while there is no inversion in ω k+1 , . . . , ω k+l: ω k = dxjk−1 − xk+2 dxk+1 , ω k+1 = dxk+2 − (ck+3 + xk+3 )dxk+1 , ω k+2 = dxk+3 − (ck+4 + xk+4 )dxk+1 , ∗ ω

k+l

∗ k+1+l

= dx

∗ − (c

∗ k+2+l

(2)

∗ + xk+2+l )dxk+1 .

So D k+1 = ( ∂n , . . . , ∂k+3 , ∂k+1 + (ck+3 + xk+3 )∂k+2 + xk+2 (∂jk−1 + · · · ) )

(3)

and D k+1(p) ⊂ Tp Nk ⇔ D k+1(0) ⊂ T0 {xk+2 = 0} ⇔ ck+3 = 0 . That is, T follows S in the code iff ck+3 = 0 in any KR coordinates. Substituting this value to (3), now for a general point x near 0, D k+1(x) ⊂ T0 {xk+2 = 0} ⇐⇒ xk+2 = xk+3 = 0 .

(4)

This together with (3) justify, on the level of M, Proposition 1.8. If only nonempty, Nk,k+1 is an embedded codimension-2 submanifold of M. Moreover, at every point of Nk,k+1, D k+1 is transverse in Nk to Nk,k+1. But D k+2 (p) – smaller than D k+1(p) – can either be transverse to Nk,k+1 in Nk [in the KR glasses, think about slightly perturbing at 0 the 1st order singular position L(D k ) = (∂k+3 , . . . , ∂n ) that is clearly transverse to {xk+2 = xk+3 = 0} within {xk+2 = 0}], which replaces M at this stage, because analysed are the points of Nk,k+1 ⊂ Nk , or else tangent to Nk,k+1, thus creating a new third order singularity of F . In the latter case, clearly, D k+2 (p) = D k+1(p) ∩ Tp Nk,k+1. (The transverse alternative notwithstanding the fact that a codimension-1 subspace L(D k+1 ) ⊂ D k+2 is always tangent to Nk,k+1, because the flows of its v. f.’s preserve D k+1 , hence also D k and L(D k−2 ).) At this point we underline, as previously, that L(D k )(p) – transverse to Nk,k+1 at p, and D k+1(p) ∩ Tp Nk,k+1 are two different invariantly (geometrically) defined subspaces, V , in the (k + 2)-th sandwich L(D k+1 )(p) ⊂ V ⊂ D k+1(p). In the transverse case the letters No k+2, k+3, . . . , k+l in the word are declared G. In the tangent case the (k + 2)-th letter is T. In a moment (during the analysis related to the next letter) we will see that this second T appears exactly when ck+3 = ck+4 = 0 in (2).

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Having STT in the code one proceeds further by finding, around p, the locus Nk,k+1,k+2 ⊂ Nk,k+1 of the behaviour ‘STT’ featured by F , and doing this most conveniently in the already chosen KR coordinates. Looking at Thm. 1.1, and at (2) where ck+3 is now absent, there holds D k+2 = ( ∂n , . . . , ∂k+4 , ∂k+1 + (ck+4 + xk+4 )∂k+3 + xk+3 ∂k+2 + xk+2 (∂jk−1 + · · · ) ) . (5) On using the local equations of Nk,k+1 obtained in (4), D k+2 (p) ⊂ Tp Nk,k+1 ⇔ D k+2 (0) ⊂ T0 {xk+2 = xk+3 = 0} ⇔ ck+4 = 0, and this value stands in reality in (5). Now the power of KR pseudo-normal forms plays off as usual: D k+2 (x), read off from (5), is tangent to {xk+2 = xk+3 = 0} iff xk+2 = xk+3 = xk+4 = 0, and these are the local equations of Nk,k+1,k+2 near p. Also, by (5) with now ck+4 = 0, D k+2 (0) sticks out of {xk+2 = xk+3 = xk+4 = 0} inside {xk+2 = xk+3 = 0}. Thus, on the level of M, Proposition 1.9. If only nonempty, Nk,k+1,k+2 is an embedded codimension-3 submanifold of M. Moreover, at every point of Nk,k+1,k+2, the flag member D k+2 is transverse in Nk,k+1 to Nk,k+1,k+2. In this framework, the next step of filling in the word becomes geometrically clear. D can, at 0 in Nk,k+1 = {xk+2 = xk+3 = 0}, be either transverse to Nk,k+1,k+2 = {xk+2 = xk+3 = xk+4 = 0} (when ck+5 6= 0), or tangent to Nk,k+1,k+2 (when ck+5 = 0).§ These alternatives reflect transversality or tangency to the submanifold Nk,k+1,k+2 within bigger submanifold Nk,k+1 (cf. Props 1.8, 1.9). By definition, in the code, in the first alternative, after STT there go l−2 letters G. In the second alternative the letters present at this moment are STTT, and one or more further steps is necessary. k+3

Formally the whole procedure is done inductively, and commences at each step by the observation that the subsequent member, say L(D k+j−2), j ≤ l, of the Cauchycharacteristic subflag is always transverse, in the ‘older’ singularity locus Nk,...,k+j−2, to the assumed in the induction embedded singularity manifold Nk,...,k+j−1 detecting the behaviour ‘STj−1 ’. And hence – that so can be, after a slight perturbation, the flag member D k+j that is assumed in (iii) not in singular position at p. Recalling, this algorithmic pattern starts from • (∂k+3 , ∂k+4 , . . . , ∂n ) = L(D k ) is transverse in Nk = {xk+2 = 0} to Nk,k+1 = {xk+2 = xk+3 = 0} , • (∂k+4 , ∂k+5 , . . . , ∂n ) = L(D k+1 ) is transverse in Nk,k+1 = {xk+2 = xk+3 = 0} to Nk,k+1,k+2 = {xk+2 = xk+3 = xk+4 = 0} , and passes by • (∂k+j+1, . . . , ∂n ) = L(D k+j−2) is transverse in Nk,...,k+j−2 = {xk+2 = · · · = xk+j = 0} to Nk,k+1,...,k+j−1 = {xk+2 = · · · = xk+j+1 = 0} . §

The tangent position of Dk+3 (p) clearly differs from the special transversal position L(Dk+1 )(p), and both these invariant potential positions for Dk+3 (p) find, as well as the pairs of potential positions distinguished at previous steps, a nice interpretation in the systematization of all possible one-step prolongations of Goursat flags proposed in [9].


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Hence D k+j , deviating from L(D k+j−2) at p, is either transverse – then one gives a string of G’s from the (k + j)-th place through the (k + l)-th, or tangent – then STj is in the code, plus further analysis of farther possible tangencies. And that tangent position clearly differing from the first order singular L(D k+j−2) – always two different positions in Montgomery-Zhitomirskii sandwiches and systematization. In the tangent case STj point p sits in the next step singularity locus Nk,...,k+j that is subsequently inspected via the machinery of (2). The # of vanishing constants in the Pfaffian equations grows together with the # of T’s in the code. At the one before last moment one supposes that STl−1 already stands in the code, i. e., that, starting with D k+1 , there occur l − 1 consecutive tangencies in the local behaviour of F at p. Moreover, one supposes known that this behaviour ‘STl−1 ’ of F happens at points of an embedded codimension–l submanifold Nk,...,k+l−1 having around p local equations xk+2 = xk+3 = · · · = xk+l+1 = 0 .

(6)

These equations are meaningful, because for points p that have survived all sieves by alternatives up to this moment there holds in (2) ck+3 = · · · = ck+l+1 = 0 .

(7)

Then D k+l (p) can be – in Nk,...,k+l−2 – either transverse or tangent to Nk,...,k+l−1. Upon writing the vector fields of D k+l in KR glasses and using (6), it becomes clear that ‘transverse’ means ck+l+2 6= 0, and ‘tangent’ means ck+l+2 = 0, independently of the KR coordinates in which all this geometry is watched. Naturally, in the transverse case we write in the code a letter G at the (k + l)-th place, and eventually the whole segment under discussion in (iii) is STl−1 G. In the tangent case – a letter T, with the segment now filled in as STl . Reiterating, in the case of geometry ‘STl ’ at p – on top of (7) there also holds ck+l+2 = 0, i. e., all constants present in (2) vanish. Taking this into account, we draw one last consequence that is not needed now for the continuation of the construction, but will be needed in the next section for a general statement (Prop. 1.13) about the loci of geometric classes of Goursat flags. Namely, the explicit writing of D k+l shows immediately that the locus Nk,...,k+l−1,k+l of the behaviour ‘STl ’ of F , has near p: the local equations (6) of the bigger submanifold Nk,...,k+l−1 in which it sits, plus one new equation xk+l+2 = 0, xk+2 = xk+3 = · · · = xk+l+1 = xk+l+2 = 0 .

(8)

As expected then, when only supposed nonempty, Nk,...,k+l is an embedded codimension(l + 1) submanifold.

1.2 Materializations of geometric classes. Clearly, the codes of geometric classess issuing from this definition are not all 3r words of length r existing over the alphabet G, S, T, but are subject to the following two limitations:

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• a word starts with GG, • never a T goes directly after a G. Observation 1.10. There is F2r−3 words of length these two restrictions. r √fulfilling r 3+ 5 . Therefore, the asymptotics of this number is const 2 Proof. Denote by br the number in question. Naturally, b2 = 1 and b3 = 2 (only the words GGG and GGS). There holds the recurrence relation br+2 = 3br+1 − br , because prolonging to the right any admissible word of length r + 1 by one letter from {G, S, T} yields all admissible words of length r + 2 and br superfluous words |GG {z. .}. GT that are r letters

not allowed. This is precisely the recurrence defining the odd indices’ Fibonacci numbers.

Remark 1.11. Geometric classes of Goursat germs defined in sec. 1.1 are, naturally, finer than KR singularity classes. For any fixed flags’ length r, all F2r−3 geometric classes are non-empty, i. e., the relevant singularities of flags really occur. Indeed, tracing down the long defining procedure one easily writes KR pseudo-normal forms having inversions of differentials only in beforehand prescribed Pfaffian equations ω k = 0, and constants equal to 0 in, also arbitrarily prescribed, numbers l′ , 0 ≤ l′ ≤ l, of the equations that in a KR form directly follow every such ω k = 0. And, if l′ < l, having a non-zero constant in ′ ′ ω k+l +1 (and, if l′ < l − 1, does not matter what constants in ω k+l +2 , . . . , ω k+l ). Example 1.12. In Ex.1.4, the geometric classes that build up the KR class ‘only D 3 in singular position’ are: GGSGG (c6 6= 0 in (1) ), GGSTG (c6 = 0, c7 6= 0), and GGSTT (c6 = c7 = 0). As for the relevant small growth vectors, they are constant within each of these classes (cf. Thm. 2.2 below) but differ among classes. It is • [2, 3, 4, 5, 62 , 7] in GGSGG , • [2, 3, 4, 5, 63 , 7] in GGSTG , • [2, 3, 4, 5, 64 , 7] in GGSTT (subscripts, like in the encoding words, mean the #’s of repetitions of a given integer). Proposition 1.13. Let W be any admissible word of length r, F be a Goursat flag of length r on a manifold M. Let N be the locus of points p ∈ M such that the germ of F at p has the basic geometry (encoded by) W. Then, if only non-empty, N is an embedded submanifold in M of codimension equal to the number of letters S and T in W. Proof. If W = Gr then N = M \ (N3 ∪ · · · ∪ Nr ), see (i) in sec. 1.1. All Nk are embedded codimension-1 submanifolds of M (Cor. 1.7), hence N is open and dense in M. Let us assume now that there are different from G letters in W and that they appear in s ≥ 1 groups ST. . . T, the j-th group (j = 1, . . . , s) encompassing S at the kj -th place in W and lj ≥ 0 letters T following it. Let us also, assuming N non-empty, fix any point p ∈ N. One arbitrarily chosen system of local KR coordinates (Thm. 1.1) x1 , . . . , xn centered at


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p does for showing that, in the vicinity of p, the set N ⊂ M is an embedded submanifold P of codimension c = s + sj=1 lj . Indeed, by (8), for every j ∈ {1, . . . , s} the local equations of the geometry ‘ST. . . T’ (lj letters T) featured near p by flag’s members D kj , D kj +1 , . . . , D kj +lj read xkj +2 = xkj +3 = . . . = xkj +2+lj = 0 . The conjunction over 1 ≤ j ≤ s of all these equations is the local description of N. And it is just the vanishing of c different variables from the chosen coordinate system, with c being equal to the number of the S’s and T’s in W. Corollary 1.14. For any Goursat flag F on M, the basic geometries of F at different points define certain stratification of M by embedded submanifolds. Codimension of the stratum containing any given point p is directly computable from the basic geometry of F at p : it equals the number of letters S and T in the code of the geometric class of the germ of F at p. Important realizations of Goursat distributions of rank-2 and corank-r, and of the geometric classes hidden in them, come from kinematical models of car + attached r − 1 trailers, extensively investigated in the 1990s (to quote but [5], [8]). The configuration r space is Σ = R2 × S 1 and the motion (only kinematical side!) of the system is described by a rank-2 distribution D [that instantly turns out to be Goursat]. Now one takes in Σ an, arbitrarily fixed, point p = (x, y, θ1 , . . . , θr ) (θ1 expresses the angle of the farthest trailer, θr−1 – the angle of the trailer next to the car, θr – of the car itself). The angle θ2 − θ1 between the two most distant trailers is irrelevant for the basic geometry of D at any point, also at p. Of key importance is the sequence (tending to 0) of critical angles a1 = π2 , ai+1 = arctan(sin ai ), i = 1, 2, 3, . . . To begin with, one locates all right angles, disregarding the angle θ2 − θ1 , in the instantaneous position p. Let θkj − θkj −1 = ǫj a1 , kj ≥ 3, j = 1, . . . , s, where the ǫj ’s take values ±1, be all such angles. (If there is none of them, then (D, p) sits in the generic class Gr .) The germ of D at p belongs to the following geometric class C. Proposition 1.15 (Jean, Montgomery & Zhitomirskii). Letters S are precisely at the kj -th places, j = 1, . . . , s. After the letter S at the kj -th place there go exactly lj ≥ 0 letters T, where θkj +1 − θkj = ǫj a2 , θkj +2 − θkj +1 = ǫj a3 , . . . θkj +lj − θkj +lj −1 = ǫj a1+lj , but already θkj +lj +1 − θkj +lj 6= ǫj a2+lj . Note that when the next angle θkj +lj +1 −θkj +lj is right (that is, when kj +lj +1 = kj+1 ) then the sequence of critical values in this proposition is discontinued automatically, and a new sequence of length ≥ 1 starts. Moreover, in each j-th string of consecutive critical angles (1 ≤ j ≤ s), the constant sign ǫj is important. For inst., θkj +lj +1 −θkj +lj = − ǫj a2+lj would, naturally, mean a disconnection in the string.

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2


Jean’s functions β, small growth vectors, and derived functions d

In what follows we will predominantly work on the properties of the small growth vectors (s. gr. v.) of Goursat distributions. And – guided by the last item in Cor. 1.3 – entirely restrict ourselves to rank-2 Goursat distributions. The length r ≥ 2 of the associated flag is kept fixed in this chapter. That is, the distributions under consideration are of corank r and the underlying manifold M has dimension r + 2. Since the appearance of the work [6] we know in general how to compute the small growth vectors for rank-2 Goursat distributions. Prior to it one knew only how to compute s. gr. v.’s for Goursat distributions of small coranks (using plainly Thm. 1.1 and in practice rarely exceeding the corank 6). To be precise, instead of s. gr. v., in [6] used is the function β (clearly depending on the point p ∈ M) that carries precisely the same information. Definition 2.1. For 2 ≤ j ≤ r + 2, β(j) = the number of Lie factors from among X, Y necessary to span at p at least j dimensions. (X, Y being any local basis of sections of D, and β clearly not depending on it.) In fact, Jean computes β only for germs of the car+trailers systems, but this does for all Goursat germs. For, as is explained in detail in Appendix D of [9], car+trailers systems are locally universal for all G. distributions.¶ This universality restricted to a fixed region, say R, in Σ (see sec. 1.2) says that the kinematic model at points of R gives, up to diffeomorphisms, all G. germs in the geometric class represented by R (cf. the interpretation of first order singular positions and consecutive tangencies in terms of car+trailers’ critical angles in Prop. 1.15). Now, via the main theorem in [6] that furnishes the β’s as only functions of regions in Σ for car+trailers, one is able to compute these functions for all Goursat germs. And they depend only on the geometric class represented by the germ in question: Theorem 2.2 ([6]). For each fixed geometric class C of corank-r Goursat germs, the function β = β(C) is one and the same for any rank-2 germ D from C. β(C) is equal to the last term β r in the sequence of functions β 1 , β 2 , . . . , β r constructed for C as follows. For j = 1, 2, . . . , r, the author of [6] is defining a sequence of functions β j : {2, . . . , j + 2} → N (β j , in itself, an integer-valued sequence of length j + 1). He starts from fairly

¶

the Goursat distribution underlying the system ‘car + r−1 trailers’ turns out to be globally equivalent to the r times Cartan prolongation of the tangent bundle to R2 , well known to be locally universal for the rank-2 and corank-r Goursat distributions


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simple β 1 = (1, 2) and β 2 = (1, 2, 3), but continues already recursively in function of C :  j+1  when the (j + 2)-th letter in C is G,  T W O (β ) , β j+2 =

T HREE (β j , β j+1) , when the (j + 2)-th letter in C is S,   ONE (β j , β j+1 ) , when the (j + 2)-th letter in C is T.

Before precising the operations used in this recurrence, note that such a translation of {G, S, T} into {T W O, T HREE, ONE} comes simply from the numeration of cases in the main Jean’s theorem: his 1 concerning the critical angles a2 , a3 , a4 , . . . (consecutive tangent positions in flags); 2 concerning generic values of angles (generic positions of members of flags); 3 concerning the first appearing critical angles a1 , or 1st order singular positions of flag’s members. Compare also more precise Prop. 1.15 above. The simplest is the operation T W O which is one-argument,   3 4 5 ... 2   . T W O (α) =  ↓ ↓ ↓ ↓ . . .     1 α(2) + 1 α(3) + 1 α(4) + 1 . . . One notes that T W O(α) is a sequence by one entry longer than α. The remaining operations are two-argument, with the first argument α being a sequence by one entry shorter than the second argument β, and the outcomes by one entry longer than β:   4 5 6 ... 2 3   T HREE (α, β) =  ↓ ↓ ↓ ... ↓ ↓ ,   1 2 α(2) + β(3) α(3) + β(4) α(4) + β(5) . . . 

2  ONE (α, β) =  ↓  1

3

4

5

↓

↓

↓

2

2β(3) − α(2)

2β(4) − α(3)



6

...  ↓ ... .  2β(5) − α(4) . . .

Example 2.3. Let us illustrate Theorem 2.2 by computing the function β(C) for C = GGSTGS. T HREE

ON E

TWO

T HREE

(1, 2); (1, 2, 3) −→ (1, 2, 3, 5) −→ (1, 2, 3, 4, 7) −→ (1, 2, 3, 4, 5, 8) −→ (1, 2, 3, 5, 7, 9, 15)

The length of the sequence β is r + 1, while the length of the small growth vector sometimes is even exponential in r. How can one retrieve the s. gr. v. from β ? It starts from two universal entries n1 = rk D = 2 and n2 = rk [D, D] = 3 that do not depend on a point. Yet its further terms already do. To begin with, it is observed in [6] (Thm. 3.1,

872


first item) that each β is strictly increasing. This means that all integers from 2 through r + 2 do occur in each small vector of Goursat. In fact, the increment β(3) − β(2) says how many terms in the s. gr. v. are nβ(2) = · · · = nβ(3)−1 = 2 (just one), then β(4) − β(3) says how many terms are nβ(3) = · · · = nβ(4)−1 = 3 (also one); β(5) − β(4) says how many of them are nβ(4) = · · · = nβ(5)−1 = 4 (one or two terms, as β(4) = 3 and β(5) can be either 4 or 5), and so on. The last increment β(r + 2) − β(r + 1) says how many times the one before last value r + 1 occurs in the s. gr. v. : nβ(r+1) = · · · = nβ(r+2)−1 = r + 1. The next entry nβ(r+2) = r + 2 means attaining the full dimension at a given point and the maximal value β(r + 2) of β is the nonholonomy degree (Def.1.2) of D at that point.

Example 2.4. For instance for C in Ex.2.3 one thus knows the #’s of repetitions of the integers 2 through 7 in the relevant s. gr. v. So this vector reads [2, 3, 42 , 52 , 62 , 76 , 8].

2.1 The aim. With the above (mainly terminological) precisions taken into account, Thm. 2.2 says Corollary 2.5. The small growth vector of rank-2 and corank-r Goursat germs well defines a mapping, κ, from the geometric classes of length r to non-decreasing sequences of natural numbers of the type [2, 3, 4, . . . , r + 2]. Thus, staying with the same Ex.2.4, κ(GGSTGS) = [2, 3, 42 , 52 , 62 , 76 , 8]. By inspecting the known data concerning this mapping κ ([3], pp. 145-6, 150 and [10], pp. 94-6), one ascertains that it is surely injective in lengths 2 ≤ r ≤ 7. Then is naturally tempted to conjecture the same in all lengths. And an initial objective of the present work was to prove that much: κ injective in all lengths. In the meantime an imperfect proof of this statement was given in [11]. [We mean Theorem 5. 6 there. The arguments in its proof are not correct in lines 14416−12 , for two absolutely different things are mixed up in that paragraph. Needed is a correct computation of quantities on which Lemma 5. 8 is to be applied.] After that contribution we aimed at describing effectively the range of κ. And, as is illustrated above, these are the increments of β that are essential in retrieving the small growth vector, for they give the #’s of repetitions of all integers present in it. We propose, therefore, to put them in evidence and subsequently work uniquely with them.k k

In the originating contribution [6], as we mentioned already, it was not explained whether all functions


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2.2 Recurrences governing the derived functions d. Definition 2.6. The derived function d : {2, . . . , r + 1} → N of a given strictly increasing function β : {2, . . . , r + 1, r + 2} → N is defined by d(j) = β(j + 1) − β(j) for j = 2, 3, . . . , r + 1. For a fixed geometric class C of corank-r Goursat germs, the vector d(2), d(3), . . . , d(r + 1) obtained from the function β(C) (see Thm. 2.2) is called the derived vector, der(C), of C. We will effectively describe the range of κ in terms of the derived vectors showing up for Goursat distributions. That is to say, in terms of the #’s of repetitions of integers in the small growth vectors of Goursat. Those numbers of repetitions will be effectively computed in all cases – for all geometric classes. Remark 2.7. This entire line of concepts stems from rather elementary Lemmata 1 and 2 in [10] which, in the language of the present work, say that, for r ≥ 3, the derived vector der(C) starts with (1, 1, 1, . . . ) when C ends with G or T, whereas it starts with (1, 1, 2, . . . ) when C ends with S. A natural idea is to extend Thm. 2.2 to the derived functions; the [discrete] derivatives of functions having transparent recurrences should also possess clear recurrences. In fact, the recurrences of the functions d appear slightly simpler than these for the β’s (and this has also been a factor contributing to the solution). In the places of function operators TWO, THREE, ONE governing the β’s, the relevant operators controlling the production of the d’s will be denoted by the very letters G, S, T used to encode the geometric classes (there will be no risk of confusing the two meanings). In fact, Theorem 2.8. The function d associated to any fixed geometric class C of corank-r Goursat germs can be described recursively. It equals the last term d r in the sequence of functions d 1 , d 2 , . . . , d r that are constructed as follows. Defined is, for j = 1, 2, . . . , r, a sequence of functions d j : {2, . . . , j + 1} → N (d j , in itself, an integer-valued sequence of length j). We start by putting d 1 = (1) and d 2 = (1, 1). Then continue recursively in function of C :

d j+2

 j+1  when the (j + 2)-th letter in C is G,  G (d ) , = S (d j , d j+1) , when the (j + 2)-th letter in C is S,   T (d j , d j+1) , when the (j + 2)-th letter in C is T.

β were really different. From a couple of years’ perspective one could say that the sequences of increments of β’s are somehow better suited for demonstrating the same – that they are all different.

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The simplest is the operator G, 

2  G(α) =  ↓  1

3

4

↓

↓

α(2)

α(3)



5

...  ↓ ... .  α(4) . . .

Naturally, G(α) is a sequence by one entry longer than α. The remaining operators S and T are two-argument, with the first argument α being a sequence by one entry shorter than the second argument β, and the outputs being by one entry longer than β,   4 5 6 ... 2 3   S(α, β) =  ↓ ↓ ↓ ... , ↓ ↓   1 1 α(2) + β(3) α(3) + β(4) α(4) + β(5) . . . 

2  T(α, β) =  ↓  1

3

4

5

↓

↓

↓

1

2β(3) − α(2)

2β(4) − α(3)

6



...  ↓ ... .  2β(5) − α(4) . . .

Note a very slight difference between THREE and S, as well as an equally small one between ONE and T. (Yet the operator G is markedly simpler than TWO.) The proof of Thm. 2.8 boils down to a short algebra that is to be done on the data originally supplied by Thm. 2.2 and later reorganized in Def.2.6. 2 Example 2.9. A parallel computation to that in Ex.2.3, now performed in the more efficient language of derived vectors, looks as follows (and offers an illustration of Thm. 2.8. The two computations taken together allow to compare these two parallel Theorems 2.2 and 2.8.) S

T

G

S

(1); (1, 1) −→ (1, 1, 2) −→ (1, 1, 1, 3) −→ (1, 1, 1, 1, 3) −→ (1, 1, 2, 2, 2, 6)

3

Explicit formulas for functions d and Main Theorem

Although the recurrences governing Jean’s functions β do not differ much from the ones governing their derived functions d, the latter, enhanced also by more handy beginning conditions (compare β 1 , β 2 and d 1 , d 2 ), are easier to solve. In the present chapter we will produce explicit solutions of the recurrences stated in Thm. 2.8. We hope they are nice, if involved enough. (They cannot be simpler, given the rich variety of existing geometric classes.) Then, in Main Theorem, by carefully analyzing those solutions, from any one of them we will read back the G, S, T code of the relevant geometric class, thus showing the injectivity of the mapping κ defined in the preceding chapter (in Cor. 2.5). The fact that the vectors der(·) turn out to be all different – is a manifestation of the cipher power of


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geometry. It amazingly ciphers all admissible G, S, T words and yields the derived vectors – the supercodes.

3.1 Hyperclasses of geometric classes and families of integer sequences they define. We will no longer focus on a fixed length (r) of geometric classes. Now we fix the number, say s + 1 ≥ 1 (it will be momentarily clear why so), of letters S in the codes that can otherwise be arbitrarily long. [It should be noted that the characterization, in terms of functions d, of the geometric classes without letters S in their codes is immediate – these functions should be constantly equal to 1. From now on we deal only with geometric classes having at least one letter S.] We firstly introduce necessary letters for the #’s of T’s and G’s in classes’ words going past the letters S, those letters S run – it is important – backwards. Agree that the last S in a code is followed by k1 ≥ 0 letters T, and then by l1 ≥ 0 letters G. Agree also that the one before last S is followed by k2 letters T and l2 letters G, and so on backwards until the first appearing letter S being followed by ks+1 letters T, then by ls+1 letters G. On top of that, let a code start with ls+2 ≥ 2 letters G. Except for this last quantity, all remaining integer parameters may even vanish, as it happens in the ‘Fibonacci’ classes GGSs+1 . (They are so termed because der(GGSs+1) = (F1 , F2 , F3 , . . . , Fs+3 ).) We reiterate that even s can vanish. Now, under the letter E we will understand any s-digit, s ≥ 0, sequence (or hypercode) built out of ciphers 0 and 1, like 010 or 001 for s = 3. The order of ciphers is from the left to right: the cipher 1 in 001 is the third (last) cipher in the hypercode 001. Definition 3.1. The hyperclass having a hypercode E is the union of all geometric classes, of various lengths, having the same number s + 1 of letters S in their codes and having lj = 0 iff the (s + 2 − j)-th cipher in E is 0, for j = s + 1, s, . . . , 2. The ‘physical’ length s of the hypercode E will be called the length of the underlying hyperclass itself. E = ∅ has length zero (s = 0). Likewise for the geometric classes, hyperclasses will often be identified with their [hyper]codes. Note also an important feature that the last (s + 1)-th letter S, in the geometric classes sitting in E, is free in this definition – it leaves no impression in the hypercode E. Superficially, it is much like with the farthest trailer in the kinematic model (see section 1.2) which may move freely and has no impact on the ‘instantaneous’ geometric class. Therefore, and this is also an illustration of the introduced parameters, the hyperclass 010 consists of all geometric classes Gl5 S Tk4 S Tk3 Gl3 S Tk2 S Tk1 Gl1 with l3 > 0, while the hyperclass 001 consists of all geometric classes Gl5 S Tk4 S Tk3 S Tk2 Gl2 S Tk1 Gl1 with l2 > 0.

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We want to first explain what are the derived vectors, in the particular hyperclasses 0s , s ≥ 0, in which the letters G occur in classes’ codes only in the beginning (some ls+2 ≥ 2 of them) and in the end (some l1 ≥ 0 of them). To this end, let us introduce a family of integer sequences • A1 = 1, A2 = 2 + k1 , Aj = Aj−2 + Aj−1(1 + kj−1 ) for 3 ≤ j ≤ s + 2 . This family is parametrized by the non-negative integers k1 , k2 , . . . , ks+1 . These are the integer values that appear in the derived vectors of geometric classes sitting in 0s .∗∗ Yet they appear with their proper multiplicities which are equally important; for instance A1 appears always 2 + k1 + l1 times (cf. now Rem.2.7 above). Later, for hyperclasses more involved than 0s , we will see still more complicated entries in the derived vectors, but the formulas for the #’s of entries’ occurrences (multiplicities) are ‘universal’. Namely Proposition 3.2. For any hyperclass E of length s, in the vectors in d( E) there appear exactly s + 2 different, growing from the left to right, integer values. The first of them (which is always A1 = 1) appears 2 + k1 + l1 times. For 2 ≤ j ≤ s + 1, the j-th new value appears 1 + kj + lj times. The last (s + 2)-th new value appears ls+2 − 1 times. Note the total number of terms in any such derived vector, 2 + k1 + l1 +

s+1 X j=2

1+kj + lj + ls+2 − 1 = s + 1 +

s+1 X j=1

kj +

s+2 X

lj

j=1

= the length of a geometric class in question .

(9)

Attention. In the sequel we will not mention the multiplicities precised in Prop. 3.2 which are, reiterating, universal. Whereas different new values appearing in the derived vectors will always be given in their natural growing order (the derived functions are non-decreasing). Recapitulating the information given up to this point, and sticking already to the note above, Theorem 3.3. In the vectors in der( 0s ), s ≥ 0, there appear the following different values: A1 , A2 , . . . , As+2 . (The parameters k1 , k2 , . . . , ks+1 enter the expressions for the sequence A up to its term As+2 ; the parameters l, in the occurrence only l1 and ls+2 , enter the expressions for multiplicities addressed in Prop. 3.2.) Before passing to further hyperclasses, we need an infinite series of [Gödel-like] sequences of the type A. This series will be parametrized by a natural number N: +N • A+N = 1, A+N = 2 + k1+N (1 ≤ N ≤ s), A+N = A+N for 1 2 j j−2 + Aj−1 (1 + kj−1+N ) 3 ≤ j, N + j ≤ s + 2 . ∗∗

We will denote the family of such vectors by der( 0s ) and, generally, by der(E) the family of derived vectors of geometric classes in a hyperclass E.


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Moreover, agree that in an hyperclass E, always of length s, the last appearing cipher 1 is the n1 -th cipher from the end of the hypercode, the one before last cipher 1 (if it exists in E) is the n2 -th from the end of E, and so on, 1 ≤ n1 < n2 < n3 < · · · ; the first (i. e., closest to the left) cipher 1 is the nq -th cipher in E counted from the right. So, when for inst. E starts with a 1, then nq = s. For the hyperclass 1000001 of length 7 one thus has n1 = 1 and nq = n2 = 7; for 1111111 one has nj = j, j = 1, 2, . . . , 7. With these notations, we are now ready to state a theorem completing Thm. 3.3 to all existing hyperclasses. Theorem 3.4. For any hyperclass E of length s ≥ 1 possessing q, 1 ≤ q ≤ s, ciphers 1 in its hypercode, in the vectors in der(E) there appear s + 2 different values that are listed below, in their growing order, in q + 1 separate rows: • A1 , A2 , . . . , An1 +1 ; 1 1 1 1 • An1 +1 A+n , An1 +1 A+n , An1 +1 A+n , . . . , An1 +1 A+n 2 3 4 n2 −n1 +1 ; +n2 +n2 +n2 1 1 1 • An1 +1 A+n , An1 +1 A+n , . . . , An1 +1 A+n n2 −n1 +1 A2 n2 −n1 +1 A3 n2 −n1 +1 An3 −n2 +1 ; • ··· ··· ··· ··· ··· ··· ··· ··· Qq−1 +nj Qq−1 +nj +nq +nq • An1 +1 j=1 Anj+1 −nj +1 · A2 , An1 +1 j=1 Anj+1 −nj +1 · A3 , . . . , Qq−1 +nj +n An1 +1 j=1 Anj+1 −nj +1 · As−nq q +2 . The values in the last row should be read carefully. Their number is s − nq + 1, so that when, for instance, nq = s in E, then this last group consists of just one value. When nq = s − 1, then it consists of two values (the second and last one being that with +n factor A3 q ), etc. The proof of Theorems 3.3 and 3.4 will be given in author’s subsequent paper.

3.2 Main Theorem (a variation on Gödel’s numeration). We approach, at long last, the procedure of reading back the geometric class code C from a given derived vector v sitting in the range of der (for r equal to the length of v). An analogy with Gödel’s techniques suggests itself. If not the entire propositions in arithmetics are this time being encoded, then at least long and involved words over a three letters’ alphabet {G, S, T} are encoded under the guise of derived vectors. (Thus the encoding software is not artificial – it is geometry itself, cf. Definitions 2.1 and 2.6.) And the reader will see in Thm. 3.5 that a straightforward decoding procedure exists. From the preceding discussion one knows that the Goursat length r is less important than the length, s, of the hyperclass of C that should also be ascertained. And we know s, because, as stated in Prop. 3.2, the number of different values of natural integers in v exceeds s by two. Therefore, we know already the number, s + 1, of letters S in the code(s) C producing v, der(C) = v, and now ask about the values of k1 , l1 , k2 , l2 , . . . , ks+1, ls+1 , ls+2 . This question is being answered in Thm. 3.5 below in parallel with establishing the hyperclass E containing the sought C. This simultaneous (if side) line of conclusions is necessary. If Theorems 3.3 and 3.4, enhanced by Proposition

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3.2, serve as a road map in our arriving at C, then the ascertaining of E is a compass laid on that map. In order to neatly formulate the retrieving algorithm, agree that the different values in v are, in the growing order, M1 (always equal to A1 = 1), M2 (always equal to A2 ), and then M3 , . . . , Ms+2 .†† When s = 0 (only one letter S in the sought C), then the algorithm is particularly simple, but there is no need to separate this case from the algorithm body which runs as follows. Theorem 3.5 (Main Theorem). M2 = 2+k1 and the multiplicity of M1 in v is 2+k1 +l1 , whence the values of k1 , l1 . The multiplicity of Ms+2 in v is ls+2 − 1, whence the value of ls+2 . For 2 ≤ j ≤ s + 1, consider the values Mj and Mj+1 . • If Mj divides Mj+1 then the (j − 1)-th from the right cipher in E is 1 (and assuredly M = 2 + kj . Moreover, the lj > 0). The value of kj is being determined from Mj+1 j multiplicity of Mj in v is 1 + kj + lj , whence the value of lj . •• If Mj does not divide Mj+1 then the (j − 1)-th from the right cipher in E is 0 and so lj = 0. At that, the multiplicity of Mj in v is 1 + kj + lj = 1 + kj , whence the value of kj . Remark 3.6. In the case •• the value of kj can also be determined by dividing Mj+1 by Mj with a remainder. The integer result of the division is then 1 + kj , yielding kj as well. The proof of this theorem is short and directly based on the precise description of all different values that show up in the derived vectors of germs of Goursat distributions (and, naturally, on the knowledge of the multiplicities of those values, that is, the #’s of their occurrences in the derived vectors). If termed ‘main’, because giving finally the sought injectivity of Jean’s mapping κ, it nevertheless rests heavily on the two theorems preceding it. These are Theorems 3.3 and 3.4 that furnish the solutions to the recurrences defined in section 2.2, simplifying the original recurrences of [6]. In short, these solutions are neat enough to allow for Main Theorem above. Example 3.7. One can start to check Thm. 3.5 on the vectors (1, 1, 1, 1, M2 ), M2 = 2, 3, 4, issuing from Ex.1.12. But let us take the derived vector v = (1, 1, 2, 2, 2, 6) computed in Ex.2.9, and try to retrieve its underlying geometric class GGSTGS having appeared earlier in Ex.2.3. Here M1 = 1, M2 = 2, M3 = 6 and s = 1. The easy part of the algorithm gives k1 = l1 = 0 and l3 = 2. Then, taking j = s + 1 = 2, M2 divides M3 , and so one computes in the branch •, knowing before computation that l2 must be 3 positive: k2 = M −2 = 1, and, indeed, the multiplicity of M2 being 3, l2 = 3−1−k2 = 1. M2 Thus G2 ST1 G1 S gets reproduced. ††

Only for the first two of them there are ‘universal’ formulas. The value M3 may already be either A3 or A2 A+1 2 , etc.


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Endly, let us proceed a slightly more involved derived vector (1, 1, 1, 2, 3, 3, 6) excerpted from the list of F11 = 89 different such vectors existing in length 7 (see [10], p. 96). This time M1 = 1, M2 = 2, M3 = 3, M4 = 6, and s = 2. The easy part gives k1 = 0, l1 = 1, l4 = 2. In turn, taking j = 2, M2 does not divide M3 , so that one computes in the branch •• : l2 = 0 automatically, while k2 = the multiplicity of M2 − 1 = 0 (or equivalently – cf. Rem.3.6 – the division of 3 by 2 yields 1 = 1 + k2 with the remainder 1). Then, continuing for j = 3, M3 divides M4 and one jumps to the branch • : k3 = M4 − 2 = 0 and l3 = the multiplicity of M3 − 1 = 1 (as we know from Thm. 3.5, it must M3 have been positive in this branch). The geometric class G2 SG1 SSG1 is produced from its derived vector, as it should be.

4

Appendix

The theory developed in the present paper allows to answer the following Question. Having a [geometric class C and its] small growth vector κ(C), how to get – if it is unique – the small growth vector of the Lie squares of germs in the class C ? asked by M. Zhitomirskii in 2001. The Lie squares of the members of C clearly have one and the same s. gr. v. which is the translation by +1 of the vector κ(C 2 ), where C 2 is the word C with the last letter deleted. So the question boils down to whether this parent vector can be effectively retrieved from the child vector κ(C). It is doable – and visible best – in the language of derived vectors (see Def.2.6). In the present appendix we describe the relevant procedure of obtaining der(C 2 ) from der(C). All steps and conclusions rest, needless to say, on the main theorems of the paper. All particular references to them would, however, make the procedure illegible. Because of that we give references only at few key points. Remark 4.1. This procedure, when carefully read back, allows one to go forward, not backwards, and find the vectors der(C.G), der(C. S), der(C.T) (this last only when the word C.T is admissible) on the sole basis of der(C). That is, to gradually find the derived vectors in one-step recurrences instead of two-step. The operations needed for that are, however, incomparably more involved than the transparent rules G, S, T of section 2.2.

4.1 Notation from Chapter 3. Agree that #(Mj ), for j = 1, 2, . . . , s + 2, means the multiplicity of the value Mj in the vector der(C): Mj goes exactly #(Mj ) times in row in der(C). We begin by focusing on the parameters k1 = M2 − 2 and l1 = #(M1 ) − M2 . With these formulas at hand, it is clear that the class C ends by • G exactly when #(M1 ) > M2 (l1 > 0), • T exactly when #(M1 ) = M2 and M2 > 2 (l1 = 0 and k1 > 0), • S exactly when #(M1 ) = M2 and M2 = 2 (l1 = 0 and k1 = 0). The procedure in case G is trivial, in case T is short, and only in S is longer.

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4.2 Case G. The vector der(C 2 ) is obtained by just leaving out the leftmost entry 1 in the vector der(C).

4.3 Case T. ¯ 1, M ¯ 2, . . . , M ¯ s+2 , where M ¯ 1 = A1 = 1, M ¯2 = The vector der(C 2 ) has different entries M A2 − 1 and, for finding the remaining different entries, it is handy to introduce additio¯ 1 , A¯2 = M ¯ 2 . Then the remaining values M ¯ j are obtained nally new symbols A¯1 = M through the same table of formulas as for der(C) [in Theorem 3.3 or 3.4, depending on the annihilation or not of all the parameters l2 , l3 , . . . , ls+1 ], when one starts from A¯1 , A¯2 in the place of A1 , A2 . More precisely, only the first row in the relevant table undergoes a cardinal change (in Thm. 3.3, however, the first row is the only row – the whole string of different values showing up in der(C)). The last, now seriously modified, entry in that first row is either A¯s+2 or A¯n1 +1 , depending on the # of rows. When it is A¯n1 +1 (i. e., when there is more than one row in the table) then it has an impact on the remaining rows. Namely, they −1 are multiplied by the factor A¯n1 +1 An1 +1 . One can closely trace this process in Ex.4.3 below. ¯ j ’s, they are the same as the multiplicities of the respective As for the multiplicities of M ¯ 1 ) = #(M1 ) − 1 (cf. (9) ). Mj ’s except for the first one, #(M

4.4 Case S. In this case the word C 2 has s letters S, not s + 1 as C. Hence the different values in der(C 2 ) are in number s + 1, not s + 2 as in der(C): ¯1 < M ¯2 < · · · < M ¯s < M ¯ s+1 . M The multiplicities of these are simple enough, ¯ 1 ) = #(M2 ) + 1, #(M

¯ j ) = #(Mj+1 ) for j = 2, 3, . . . , s + 1 #(M

(remember that #(M1 ) = 2 in Case S, hence the sum of multiplicities of Mj exceeds by ¯ j , cf. again (9) ). one the sum of multiplicities of M ¯ j . If the table of different values in der(C) consists It is less simple to give the values M of just one row A1 , A2 , A3 , . . . , As+2 (as is the case in Thm. 3.3; for s = 0 there are just two values), then so is for der(C 2 ) and that only row reads +1 +1 +1 A+1 1 , A2 , A3 , . . . , As+1 (for s = 0 there is but one entry A+1 1 = 1).


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If the table of different values in der(C) consists of q + 1 ≥ 2 rows as in Thm. 3.4, • A1 , A2 , . . . , An1 +1 ; 1 1 1 1 • An1 +1 A+n , An1 +1 A+n , An1 +1 A+n , . . . , An1 +1 A+n 2 3 4 n2 −n1 +1 ; +n2 +n2 +n2 1 1 1 • An1 +1 A+n , An1 +1 A+n , . . . , An1 +1 A+n n2 −n1 +1 An3 −n2 +1 ; n2 −n1 +1 A2 n2 −n1 +1 A3 • ··· ··· ··· ··· ··· ··· ··· ··· Qq−1 +nj Qq−1 +nj +nq +nq • An1 +1 j=1 Anj+1 −nj +1 · A2 , An1 +1 j=1 Anj+1 −nj +1 · A3 , . . . , Qq−1 +nj +n An1 +1 j=1 Anj+1 −nj +1 · As−nq q +2 , then the transformation to the table for der(C 2 ) can be viewed as two-step. Namely, the first row above undergoes a cardinal transformation and becomes +1 +1 +1 A+1 1 , A2 , A3 , . . . , An1 .

(It is by one entry shorter than the first row for der(C).) Then its last entry A+1 n1 exerts its impact on the remaining rows in the table for der(C) only as a factor. In fact, these −1 rows are multiplied by A+1 A : n +1 1 n1 +n1 +1 +n1 +1 +n1 +1 +n1 • An1 A2 , An1 A3 , An1 A4 , . . . , A+1 n1 An2 −n1 +1 ; +n1 +n2 +n1 +n2 +n1 +n2 • A+1 , A+1 , . . . , A+1 n1 An2 −n1 +1 A2 n1 An2 −n1 +1 A3 n1 An2 −n1 +1 An3 −n2 +1 ; • ··· ··· ··· ··· ··· ··· ··· ··· Qq−1 +nj Qq−1 +nj +nq +nq +1 +1 • An1 j=1 Anj+1 −nj +1 · A2 , An1 j=1 Anj+1 −nj +1 · A3 , . . . , Qq−1 +nj +nq A+1 n1 j=1 Anj+1 −nj +1 · As−nq +2 . The procedure of passing from der(C) to der(C 2 ) is now complete. It consists either of Case G, or T, or else S, and the choice of a case uses only the simplest data #(M1 ) and M2 (the # of 1’s and the first entry bigger than 1 in der(C) ). Remark 4.2. In Case S, the two-step transformation of tables assumes the simplest form when the table for der(C) consists of at least two rows and, specifically, the first row contains just two entries A1 and A2 (besides, A2 = 2 in Case S). After the first step that first row becomes just one entry A+1 1 = 1 (which, in the outcome, should formally be merged, or concatenated with the second outputting row). −1 While the factor multiplying the remaining rows in the second step is just A2 , and it erases the first factors A2 in their entries. Summarizing, to get in this situation the table of different values in der(C 2 ), one merges, in the table for der(C), the two first rows, then leaves out the very first entry A1 . All the remaining entries are, in the occurrence, divisible by A2 = 2 + k1 = 2, and one divides them by 2. Example 4.3. We want to illustrate the procedure of finding the vector der(C 2 ) on the sole basis of, longer by one entry, vector der(C), by picking a true derived vector der(C) = 1, 1, 1, 3, 3, 3, 3, 12, 12, 24, 36, 36 (10) and not even attempting to decode the word C,‡‡ only heading directly toward der(C 2 ). We see that #(M1 ) = M2 and M2 > 2, hence it is Case T. ‡‡

The reader may quickly uncover C using Thm. 3.5.

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How does the table of different values in der(C) look like? In the example M2 divides M3 , hence n1 = 1. Likewise M3 divides M4 and so n2 = 2. In fact, with M4 not dividing M5 , the hyperclass of C is 011 and the parameter q from Thm. 3.4 assumes the value 2. Therefore the table consists of three rows, A1 , A2 ; A2 A+1 2 ; +2 +1 +2 A2 A+1 2 A2 , A2 A2 A3 . The table can be written more explicitly as 1, 2 + k1 ; (2 + k1 )(2 + k2 ); (2 + k1 )(2 + k2 )(2 + k3 ), (2 + k1 )(2 + k2 ) 1 + (2 + k3 )(1 + k4 ) .

And still more explicitly as 1, 3; 3 · 4; 3 · 4 · 2,

3 · 4 · 3.

After the transformation, the last entry in the first row gets decreased by one, from 3 to 2. Then it exerts its impact on the remaining rows. Upon multiplying them by 23 , one eventually obtains 1, 2; 2 · 4; 2 · 4 · 2,

2 · 4 · 3.

At last, taking into account also the multiplicities, der(C 2 ) = 1, 1, 2, 2, 2, 2, 8, 8, 16, 24, 24 . Reiterating Remark 4.1, the opposite procedure is also possible (and can be read off from the deprolongation procedure of the present chapter). Performing it over the above vector would give the vector (10), or, in other words, der(C 2.T).

References [1] R.L. Bryant and L. Hsu: “Rigidity of integral curves of rank 2 distributions”, Invent. math., Vol. 114, (1993), pp. 435–461. [2] E. Cartan: “Sur l’équivalence absolue de certains systèmes d’équations différentielles et sur certaines familles de courbes”, Bull. Soc. Math. France, Vol. XLII, (1914), pp. 12–48. [3] M. Cheaito and P. Mormul: “Rank–2 distributions satisfying the Goursat condition: all their local models in dimension 7 and 8”, ESAIM: Control, Optimisation and Calculus of Variations, Vol. 4, (1999), pp. 137–158, (http://www.edpsciences.org/cocv). [4] M. Gaspar: “Sobre la clasificacion de sistemas de Pfaff en bandera”, In: Proceedings of 10th Spanish–Portuguese Conference on Math., University of Murcia, 1985, pp. 67–74 (in Spanish).


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[5] B. Jacquard: Le problème de la voiture ` a 2, 3 et 4 remorques, Preprint, DMI, ENS, Paris, 1993. [6] F. Jean: “The car with N trailers: characterisation of the singular configurations”, ESAIM: Control, Optimisation and Calculus of Variations, Vol. 1, (1996), pp. 241– 266, (http://www.edpsciences/cocv). [7] A. Kumpera and C. Ruiz: “Sur l’équivalence locale des systèmes de Pfaff en drapeau”, In: F. Gherardelli (Ed): Monge–Ampère Equations and Related Topics, Florence, 1980; Ist. Alta Math. F. Severi, Rome, 1982, pp. 201–248. [8] F. Luca and J.-J. Risler: “The maximum of the degree of nonholonomy for the car with N trailers”, In: Proceedings of the 4th IFAC Symposium on Robot Control, Capri, 1994, pp. 165–170. [9] R. Montgomery and M. Zhitomirskii: “Geometric approach to Goursat flags”, Ann. Inst. H. Poincaré – AN, Vol. 18, (2001), pp. 459–493. [10] P. Mormul: “Local classification of rank–2 distributions satisfying the Goursat condition in dimension 9”, In: P. Orro and F. Pelletier (Eds): Singularités et géométrie sous-riemannienne, Chambéry, 1997; Travaux en cours, Vol. 62, Hermann, Paris, 2000, pp. 89–119. [11] W. Pasillas–Lépine and W. Respondek: “On the geometry of Goursat structures”, ESAIM: Control, Optimisation and Calculus of Variations, Vol. 6, (2001), pp. 119– 181, (http://www.edpsciences.org/cocv). [12] E. Von Weber: “Zur Invariantentheorie der Systeme Pfaff’scher Gleichungen”, Berichte Ges. Leipzig, Math–Phys. Classe, Vol. L, (1898), pp. 207–229.

CEJM 2(5) 2004 884–898

Truncated Lie groups and almost Klein models Georges Giraud∗ , Michel Nguiffo Boyom† GTA UMR CNRS 5149, Institut de Mathematiques, Université Montpellier II, Pl. E.Bataillon 34095 Montpellier Cedex 5, France

Received 15 December 2003; accepted 27 April 2004 Abstract: We consider a real analytic dynamical system G × M → M with nonempty fixed point subset M G . Using symmetries of G × M → M , we give some conditions which imply the existence of transitive Lie transformation group with G as isotropy subgroup. c Central European Science Journals. All rights reserved. ° Keywords: Almost Klein model, Klein model, Truncated Lie group MSC (2000): Primary: 54H15, 57S20; Secondary: 22E15, 22E20, 58H10

1

Introduction

Let G be a connected real Lie group acting analytically on a real analytic manifold M . We suppose the fixed point subset M G not to be empty. A natural problem is to know whether there exits a transitive Lie transformation group of M , say H, with G as isotropy subgroup. Under the assumption that M G is nonempty, the above problem is point wise closely related to embedding problem of truncated Lie algebra which has been studied by V. Guillemin- S. Sternberg [4], I. Hayashi [5] and C. Fredfield [3]. Our aim is to give some sufficient conditions under which the pair (G, M ) is induced by a Klein model (H, M ), (a Klein model (H, M ) consists of a transitive Lie transformation group H × M → M , [11]). To every pair (G, M ) which satisfies some additional nice properties may be assigned so called almost Klein models, say E, which are finite dimensional vector spaces of complete vector fields on M . Our approach use the structure of those groups of symmetries of (G, M, E), which act on M with nonempty discrete fixed point set. Let S be such a group of symmetries of (G, M, E) with M S 6= ∅; then ∀x ∈ M S , dS(x) is a subgroup of the linear ∗ †

Email: [email protected] Email: [email protected]

G. Giraud, M.N. Boyom / Central European Journal of Mathematics 2(5) 2004 884–898

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group GL(Tx M ). The set of dS(x)-fixed points in Tx (M ) is denoted by H 0 (dS(x), Tx M ). The following statement is labeled central theorem. Theorem 1.1. Let E be a (G, M )-almost Klein model. Suppose that there exists a semisimple Lie group S of symmetries of (G, M, E) with the following properties: (i) M S is nonempty and discrete, (ii) ∀x ∈ M S , H 0 (dS(x), Tx M ) = 0 and the isotopic dimensions of the dS(x)-module Tx (M ) are greater than 3. Then, there is a Klein model (H, M ) with G as isotropy subgroup.

2

Truncated Lie groups

Let V be a finite dimensional real vector space and let g be a Lie subalgebra of the Lie algebra gl (V ) of linear endomorphisms of V. To every integer ` is assigned the `th prolongation of g , say g` , [4], [13], [6], which is defined as it follows. The dual vector space of V is denote by V ∗ and g` is given by g` = 0 ∀` < −1 ; g−1 = V ; go = g ; g`+1 = g` ⊗ V ∗ ∩ V ⊗ S `+2 V ∗ ∀` > 0.

(1)

Every s ∈ g`+1 will be regarded as a linear map from V to g` . Thus ∀v ∈ V the evaluation of s at v is denoted by hs, vi . The Koszul-Spencer cochain complex of g is the direct sum of the vector spaces C p,q (g) = gp ⊗ Λq V ∗ , [13], [7]. The coboundary operator d : gp ⊗ Λq V ∗ → gp−1 ⊗ Λq+1 V ∗

is defined as it follows. Let θ ∈ C p,q (g) and let v0 ∧· · ·∧vq ∈ ∧q+1 V . Then dθ (v0 , · · · , vq ) is given by dθ (v0 , · · · , vq ) =

q X i=0

(−1)i+1 hθ (v0 , · · · vî , · · · vq ) , vi i .

(2)

At the level Cp,q (g) the cohomology space of (2) is denoted by Hp,q (g) . In particular, H−1,2 (g) is the quotient space V ⊗ ∧2 V ∗ /d (g⊗V ∗ ) . Let ∧ (V ∗ ) be the exterior algebra of the vector space V ∗ . We set Alt (V ) = V ⊗∧ (V ∗ ) and we equip the vector space Alt (V ) with the graduation Altp (V ) = V ⊗ Λp+1 V ∗ . Let (p, q) be a pair of non negative integers and let Sp+q+1 be the symmetry group of {0, . . . , p + q} . Then, Sq,p is the subset of σ ∈ Sp+q+1 such that σ (0) < σ (1) < · · · < σ (q) and σ (q + 1) < σ (q + 2) < · · · < σ (q + p) . The vector space Alt(V ) is a graded algebra whose multiplication is defined as it follows, (see [10]) ; ∀f ∈ Alt p (V ) , ∀g ∈ Altq (V ), then fog is the element of Altp+q (V ) such that ∀ u0 ∧ · · · ∧ up+q ∈ ∧p+q+1 V, we have f og (u0 , .., up+q ) =

X

σ∈Sq,p

¡ ¡ ¢ ¢ sig (σ) f g uσ(0) , .., uσ(q) , uσ(q+1) , .., uσ(q+p) .

(3)

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Now, let us set [f, g] = f og − (−1)pq gof.

(4)

With the bracket defined by (4), Alt (V ) is a graded Lie algebra. Actually, we have a natural action of g on the complex C (g) ; that action is compatible with the coboundary operator. Let H g (g) be the subspace of g-invariant cohomology classes in H (g), [8], [9], [13]. Every homogeneous subspace Hp,q (g) is a submodule of the g-module H (g) , in particular g H−1,2 (g) is g-stable. A direct calculation shows that if f ∈ f¯ ∈ H−1,p (g) and g ∈ g¯ ∈ g H−1,q (g) , then the cohomology class [f, g] is well defined and doesn’t depend on the g choice of f ∈ f¯ and g ∈ g¯. Thereby, we can define a (Lie) bracket on H−1,2 (g) by setting £

¤ f¯, g¯ = [f, f ].

(5)

Now, we are in position to recall the notion of truncated Lie algebra structure (TLA). Definition 2.1. [4], [5] Let V be a real vector space and let g be a subalgebra of the linear Lie algebra gl (V ) . A (V, g)-truncated Lie algebra structure is a Koszul-Spencer g cohomology class c¯ ∈ H−1,2 (g) such that [¯ c, c¯] = 0. Now, let M be a connected real analytic manifold. If ϕ is an analytic diffeomorphism of M , then, the differential at x ∈ M of ϕ is denoted by dϕ (x). Definition 2.2. A M -truncated Lie algebra is an analytic map assigning to every x ∈ M a subalgebra g (x) of the Lie algebra gl (Tx M ) and a (Tx M, g (x))-truncated Lie-algebra g(x) structure c¯ (x) ∈ H−1,2 (g (x)). Definition 2.3. A M -truncated Lie algebra x → (g (x) , c¯ (x)) is locally transitive if ∀x, y ∈ M, there exists an analytic local diffeomorphism ϕ of M such that (i)

ϕ (x) = y ; dϕ(x)g (x) (dϕ(x))−1 = g (y) ,

(6)

(ii) dϕ(x)¯ c (x) = c¯ (y) dϕ(x). The property (ii) of Definition 2.3 has the following meaning. Let c ∈ c¯ (x) , ∀u, v ∈ Ty M, let z ∈ C −1,2 (g (y)) defined as it follows

then z ∈ c¯ (y).

¡ ¢ z (u, v) = dϕ(x)c (x) (dϕ(x))−1 (u) ∧ (dϕ(x))−1 (v) ;

Before pursuing, let us give some classical examples of the notions which are just defined above. Example 2.4. Let g be a subalgebra of a Lie algebra h and let V¯ = h/g. We choose a subspace V ⊂ h such that h = g⊕V. The canonical projection of h onto V¯ ., say u → u¯, induces an isomorphism from V onto V¯ . Given u, v ∈ V, A ∈ g, let us write


887

[u, v] = [u, v]V + [u, v]g , [A, u] = [A, u]V + [A, u]g . The vector space V¯ is actually a g-module under the adjoint action of g on h. Let us put c (¯ u, v¯) = [u, v]V , ³ ´ ¡ ¢ sa (¯ u) = ad [A, u]g ∈ End V¯ , ³ ´ ¡ ¢ ω (¯ u, u¯) = ad [u, v]g ∈ End V¯ , ¡ ¢ A=ad (g) ⊂ End V¯ .

Thus we have the Koszul-Spencer complex

⊕ Ap ⊗ ∧q V¯ ∗

p,q

The Jacobi identity of h induces the following relations

[c, c] (¯ u, v¯, w) ¯ − dw (¯ u, v¯, w) ¯ =0

(7)

(a.c) (¯ u, v¯) = dsa (¯ u, v¯) .

¡ ¢ Thus c defines a (−1, 2)-class in H A (A) , say c¯. So, (7) shows that c¯ is a V¯ , A truncated Lie algebra structure. Example 2.5. Let G be a connected closed subgroup of a connected Lie group H. We denote by M the analytic manifold H/G. Let h be the Lie algebra of left invariant vector fields on H. To every x ∈ M is assigned the isotropy algebra g (x) .. The image of g (x) under the isotropy representation is denoted by A (x) . Let us identify h/g (x) with T x M. Thereby, applying Example 1 to the pair (h, g (x)) , we canonically get a (Tx M, A (x))-truncated A(x) Lie algebra structure c¯ (x) ∈ H−1,2 (A (x)). Those considerations yield to the M-truncated Lie algebra x → (A (x) , c¯ (x)) . The example above is a transitive M-truncated Lie algebra. Following [11], the pair (G, H) is called a Klein model. Let M be a connected real analytic manifold. Let G be a connected real Lie group acting analytically on M with nonempty fixed points set. Definition 2.6. The action G × M → M is called 1-regular at x if the canonical differential operator g → j 1 g is injective at x. We identify g ∈ G with the diffeomorphism x → g (x) .

888


Of course, a 1-regular action G × M → M means that g → jx1 g is injective ∀x ∈ M . Let M and G be as above. Let g be the Lie algebra of G. Definition 2.7. Let G × M → M be a 1-regular analytic ³ −áction with nonempty fixed G points set M ; a (G, M )-truncated Lie group is a pair ρ, c consisting of (i) a map ρ assigning to every x ∈ M a linear representation ρx of G in Tx M such that ∀x ∈ M G , ρx (g) = dg (x), (ii) ∀x ∈ M, c¯ (x) is a (Tx M, dρx (g))-truncated Lie algebra structure. Example 2.8. Let (G, H) be a Klein model. Let M = H/G; then, ∀x ∈ M, the isotropy representation at x may be regarded as a linear representation of G in Tx M ; combining that with Example 2.5 yields to a truncated Lie group. Let (ρ, c¯) be a (G, M )-truncated Lie group. A local isomorphism of (ρ, c¯) is a local isomorphism of the associated M -truncated Lie algebra (see Definition 2.3). Truncated Lie groups which come from Klein models are transitive. Question rises to know under what conditions a Klein model is uniquely determinated by a given truncated Lie group.To answer the last question one might need extra hypotheses. For instance, if we drop the connectedness property of G, then a pair (G, M ) may come from two non isomorphic Lie groups H, H 0 . Example 2.9. Let H = C2 . The group structure of H is defined as it follows. Let us write z = x +

√ −1y ∈ C. Then

(z1 , z2 ) . (z10 , z20 )

=

µ

z1 +

z10

√ ¶ π −1 0 exp y2 , z 2 + z 2 . 2

(8)

The formula (8) defines a real analytic Lie group structure in C2 . Now, let Γ be the closed subgroup of H consisting of √ √ ¢ ¡ √ ¢2 ¡ γ = p1 + −1q1 , p2 + 4 −1q2 ∈ Z+ −1Z .

Z being the ring of integers. Then, we consider Γ acting in H by left translation. So, given γ as above and h = (z1 , z2 ) , we have γh = γ + h.

(9)

Therefore, the analytic manifold M = Γ\H is diffeomorphic to the flat torus T 4 . Of course, the non commutative group H acts transitively on M. The commutative Lie group R4 = C ⊕ C also acts transitively on M. Both H and C ⊕ C contain Γ as closed subgroup and Γ\C ⊕ C =Γ\H. Let G×M → M be a 1-regular analytic action with nonempty fixed points set M G . Let (ρ, c¯) be a (G, M )-truncated Lie group which is locally transitive. Let n = dim M +dim G ∼ and let g ⊂ X(M ) be the space of vector fields ξ˜ which are induced by ξ ∈ g, g is the Lie algebra of G.


889

Definition 2.10. Let (ρ, c¯) be a (G, M )-truncated Lie group as above. An almost Klein model associated to (ρ, c¯) is a n-dimensional vector space E consisting of analytic complete vector fields on M such that ∼ ∼ (i) g ⊂ E, [g, E] ⊂ E, the map ξ → jx1 ξ, is injective, ξ ∈ E. (ii) (normal form) ∀x ∈ M,l et {α1 · · · αm β1 · · · βn−m } be a base of E such that βj (x) = 0 1 6 j 6 n − m and £ ¤ then, c = ckij ∈ c¯ (x) .

[αi , αj ] (x) = Σckij αk (x) ,

Definition 2.11. If the truncated Lie group is deduced from an effective Klein model (G, H) , then the vector space e h of fundamental vector fields given by the Lie algebra h of H is an almost Klein model. An almost Klein model E is called integrable if it is a sub-algebra of the Lie algebra X(M ). A local isomorphism of an almost Klein (ρ, c¯) is a local diffeomorphism ϕ of (ρ, c¯) such that dϕ (E) = E.

3

Some useful results

The proof of our central result will involve some technical lemmas. This section is devoted to those auxiliary results. Let g be a Lie algebra and let V be a g-module. We shall denote the ChevalleyEilenberg complex of V -valued cochains by Cg∗ (V ) [2]. The `th cohomology space of Cg∗ (V ) is denoted by H ` (g,V ) . Given a vector space V and a subalgebra A of the Lie algebra gl (V ) , we shall focus our attention on groups of A-homomorphisms of V . Lemma 3.1. Let A ⊂ gl (V ) be as above. If S is a semi-simple Lie algebra of Ahomomorphisms of V such that H 0 (S, V ) = 0, then H 0 (S, C−1,2 (A)) is isomorphic to H 0 (S, H−1,2 (A)). Proof. Let us consider the following exact sequence 0 → Ker (d) → C0,1 (A) → d (C0,1 (A)) → 0.

(10)

Actually, (10) is an exact sequence of S-modules. To prove lemma 3.1, it suffices to prove that H 0 (S, d (C0,1 (A))) = 0. Let us regard A as an S-module under the trivial action. Let ϕ ∈ H 0 (S, C0,1 (A)) . Then, ϕ is a S-homomorphism from V to A. Thus, ∀u ∈ V ,∀s ∈ S, we have [s, ϕ (u)] = ϕ (sv) = 0. The hypothesis H 0 (S, V ) = 0 yields to S.V = V. Thus we must have ϕ (u) = 0 ∀u ∈ V, viz ϕ = 0. To end the proof, we consider the following exact sequence of Smodules

890


0 → d (C0,1 (A)) → C−1,2 (A) → H−1,2 (A) → 0.

(11)

According to the vanishing of H 0 (S, C0,1 (A)) , the cohomology sequence deduced from (11) yields H 0 (S, d (C0,1 (A))) = 0. Thus, the cohomology sequence deduced from (11) is reduced to 0 → H 0 (C−1,2 (A)) → H 0 (S, H−1,2 (A)) → 0 That ends the proof of Lemma 3.1. Lemma 3.2. Let c¯ be a (V, A)-truncated Lie algebra structure. Let S be a semi-simple Lie algebra of A-homomorphism of V such that H 0 (S, V ) = 0. If the isotopic dimensions (A) of the S-module V are greater than 3, then the space Z0,2 of (0, 2)-Koszul-Spencer cocycles of C∗∗ (A) is 0. Proof. The vector space Z0,2 (A) is the kernel of d : C0,2 (A) → C−1,3 (A) . Let θ ∈ Z0,2 (A) ; ∀u, v, w ∈ V, we have I < θ (u, v) , w >= 0. Without loss of generality, we shall suppose the ground field to be the field C of complex numbers. Let τ be an irreducible submodule of the S-module V. The τ -type isotopic component of V is denoted by Vτ . Let us consider θ ∈ Z0,2 (A) − 0. Let us fix a Cartan subalgebra in S and the associated weights-system Then, there are two different weights λ, µ (of the S-module V ) and uλ , uµ ∈ V such that θ (uλ , uµ ) 6= 0. Let ν be an another weight which is different from both λ and µ. Then we have I < hθuλ , uµ , uν i = 0. The subspace corresponding to the weight λ is denoted by V λ . Then, hθ (u, v) , uλ i ∈ V λ , ∀u, v ∈ V, ∀uλ ∈ V λ . So, we must concludee that hθ (uλ , uµ ) , uν i = 0 whenever λ, µ, ν are pairwise different. Since all of the V λ are 1-dimensional, we have hθ (u, v) , wi = 0 ∀u, v, w. The last result is in contradiction with θ ∈ Z0,2 (A) − 0. Lemma 3.2 is proved. Remark 3.3. The following exact sequence is a straight consequence of Lemma 3.2. 0 → H 0 (S, C0,2 (A)) → H 0 (S, C−1,3 (A)) → H 0 (S, H−1,3 (A)) → 0.

(12)


891

To every pair (c, ω) ∈ C−1,2 (A) × C0,2 (A) we shall assign the element i (c) ω ∈ C0,3 (A) which is defined by I i (c) ω (u, v, w) = ω (c (u, v) , w) . (13) Lemma 3.4. Let (c, ω) ∈ H 0 (A,C−1,2 (A)) × C0,2 (A) satisfying the relation c2 = dω. Then i (c) ω ∈ Z0,3 (A). Proof. The relation c2 = dω implies the following identity c2 (c (u, v) , w, t) = − hω (c (u, v) , w) , ti + hω (c (u, v) , t) wi

(14)

− hω (w, t) , c (u, v)i .

At the other side, since c ∈ H 0 (A,C−1,2 (A)) , we get ω (w, t) .c (u, v) = c (hω (w, t) , ui , v) + c (u, hω (w, t) , vi)

(15)

Now, let us calculate the Koszul-Spencer coboundary of i (c) ω. d (i (c) ω) (u, v, w, t) = − hi (c) ω (v, w, t) , ui + hi (c) ω (u, w, t) , vi − hi (c) ω (u, v, t) , wi + hi (c) ω (u, v, w) , ti . Before pursuing, let us set ω ˆ (u, v, w) = dω (u, v, w) ; ω ˆ c (u, v, w) = i (c) ω (u, v, w) . Then, we have d (i (c) ω) (u, v, w, t) =

I

hˆ ω (u, v, w) , ti −

I

hˆ ωc (u, v, w) , ti

(16)

According to the formula (4), the right member of (16) is nothing but (coc 2 − c2 oc) (u, v, w, t). Thereby, we have ¤ £ di (c) ω = c, c2

Since the degree of c is 1, we have c2 = 2 [c, c] So, we get £

¤ c, c2 = 2 [c, [c, c]]

By the virtue of graded Jacobi identity, we must conclude that £

¤ c, c2 = 0.

892


That ends the proof-of-Lemma 3.4. Definition 3.5. A realization of a (V, A)-truncated Lie algebra c¯ is a (dim V + dim A)dimensional Lie algebra h such that A is a 1-regular subalgebra of h and the canonical ¢ ¡ h/A, ad (A) -truncated Lie algebra of (A, h) is isomorphic to the (V, A)- truncated Lie algebra c¯. For other considerations regarding realizations of truncated Lie algebras, the reader may consult [3], [4], [5]. The realizations we are concerned with are closely related to our central theorem. We need sufficient conditions to integrate almost Klein models. Our central result relates the integrability problem to groups of isomorphisms of almost Klein models. The algebraic counterpart of the problem of “going from truncated Lie groups to Klein models” is the following realization theorem. Theorem 3.6. Let c¯ be a (V, A)-truncated Lie algebra, (T LA), admitting a semi-simple Lie group of automorphism, say S ⊂ GL (V ). Suppose that S satisfies the following conditions. (i) S commutes with A ; H 0 (S, V ) = 0. (ii) The isotopic dimensions of the S-module V are greater than 3. Then (V, A,¯ c) admits a realization h. Moreover S can be left to a group of automorphism of h. Demonstration 3.7. Let S be the Lie algebra of S. Elements of S are “derivations” of (V, A,¯ c) ; in particular c¯ ∈ H 0 (S, H−1,2 (A)) . By the virtue of Lemma 3.1, there is a unique c ∈ H 0 (S, C−1,2 (A)) such that c ∈ c¯. A direct consequence of Lemma 3.2 is that the Koszul-Spencer coboundary operator induces an isomorphism from C0,2 (A) to C−1,3 (A). Thus, there exists a unique ω ∈ C0,2 (A) such that c2 = dω. In other words, we have the following identity I

c (c (u, v) , w) −

I

hω (u, v) , wi = 0.

The canonical map p : C−1,2 (A) → H−1,2 (A) is a A-homomorphism of S-modules. Thus, X.c ∈ H 0 (S, C−1,2 (A)) ∀X ∈ A. Since p (X.c) = X.¯ c = 0, we must conclude that 0 c ∈ H (A,C−1,2 (A)). At the present step, we can apply Lemma 3.4 to the pair (c, ω); viz di(c)ω = 0. An other direct consequence of Lemma 3.2. is that A` = 0 ∀` > 1. Thus, we have the following exact sequence. d

0 → C0,3 (A) → C−1,4 (A) → H−1,4 (A) → 0.

(17)

By taking into account that i(c)ω ∈ C0,3 (A) , we must deduce from (17) that i(c)ω = 0. We are now in position to construct a realization h of (V, A,¯ c) . Let us consider the vector space V ⊕ A. Elements of V ⊕ A are presented as it follows: X = (u, a) ∈ V × A. Let us set


[(u, a) , (v, b)] = (c (u, v) + a.v − b.u, ω (u, v) + [a, b]A ) .

893

(18)

A direct calculation shows that the Jacobi identity of [(u, a) , (v, b)] is equivalent to the following system    c ∈ H 0 (A, C−1,2 ) (A) ;    (19) i (c) ω = 0 ;      c2 − dω = 0. By the virtue of prelimary lemmas, we know that (19) holds. Thus, h = (V ⊕ A, [, ]) is a Lie algebra. That proves the first conclusion of Theorem 3.6. Now, elements of S are automorphisms of the Koszul-Spencer complex C∗∗ (A). So, the relations c2 = 2 [c, c] = dω imply that d (X.ω) = 0 ∀X ∈ S. By the virtue of Lemma 3.2, X.ω = 0 ∀X ∈ S. So, we get the following identities:    X.c = 0    (20) X.ω = 0      [X, A] Gl(V ) = 0 The system (20) above shows that X is a derivation of h. That ends the demonstration of Theorem 3.6. Example 3.8. Let A be a Lie subalgebra of gl (p, R) . Let q be a non negative integer. We set V = Rpq . We regard V as a direct sum of p copies of Rq , say V = Rq1 ⊕ Rq2 .... ⊕ Rqp . We consider the canonical base of V, say ¿ e11 , . . . , e1q ; e21 , . . . , e2q ; . . . ; ep1 , . . . , epq À . We denote by εij the linear map from Rqj to Rqi which is defined by εij (ejk ) = eik

1 6 k 6 q.

(21)

Let [aij ] ∈ gl (p, R), we identify [aij ] with the linear endomorphism A ∈ gl (V ) given by P A= aij εij . i,j

Now, let S be a semi-simple irreducible Lie subgroup of GL (Rq ) . Let ϕ ∈ S, we consider the direct sum of p copies of ϕ,viz ϕp = (ϕ, . . . , ϕ) . Thus ϕp is a linear auL q tomorphism of V, preserving the decomposition V = Ri . It is easily seen that ϕp is i

894


a A-homomorphism of V. If q > 3, then every (V, A)-truncated Lie algebra admits a realization. Let us consider an other sample (from geometry view point). Example 3.9. Let R1 (M ) be the bundle of first order linear frames over a m-dimensional smooth manifold M. Let G be a Lie subgroup of GL (m, R) . We consider a G-reduction p E → M of R1 (M ) . Let E 1 be the Singer-Sternberg first prolongation of E, [4], [13]. Then, E 1 is a principal bundle over E whose fiber over z ∈ E, say Ez1 is defined by © ª Ez1 = Hz ⊂ Tz E / dPz : Hz → Tp(z) M is an isomorphism .

(22)

Let g be the Lie algebra of G and Hz ∈ Ez1 . The tangent space Tz E may be identified with Hz ⊕ g. The structural algebra of the bundle E 1 → E may be identified with the first prolongation g1 of g. Let c¯1 be the Chern-Bernard invariant of E 1 → E, [1], [13]. Then ¡ ¢ ¡ ¢ c¯1 ∈ H−1,2 g1 = (Rm ⊕ g) ⊗ ∧2 (Rm ⊕ g)∗ /d g1 ⊗ (Rm ⊕ g)∗ . Let c1 ∈ c¯1 , c1 ∈ (Rm ⊕ g) ⊗ ∧2 (Rm ⊕ g)∗ . Given (u, a) , (v, b) ∈ Rm × g, we may write c1 as it follows c1 ((u, a) , (v, b)) = (c (u, v) + av − bu, ω (u, v) + [a, b]) .

(23)

Really, the crux terms of c1 ((u, a) , (v, b)) are c (u, v) and ω (u, v) . If g1 = 0, then c¯1 = c1 and the relevant geometry informations are carried by the pair (c, ω); that is the case whenever G is compact. So, if E 1 is transitive, then we get the (Rn , g)-truncated Lie algebra structure c¯. Our realization theorem say that the Lie pseudogroup of local automorphisms of E 1 contains a Lie group transformations of M acting locally simply transitively on E 1 . In such a situation, the conjecture studied by C. Fredfield holds, [3]. Remark 3.10. According to Guillemin-Stenberg [4] and Hayashi [5], a realization of a truncated Lie algebra (Rm , g, ¯c) is a transitive filtered Lie algebra L (viz a transitive Lie algebra of formal vector fields at the origin of Rn ) whose first order truncated Lie algebra is (Rm , g, c¯) . Guillemin and Sternberg solve the problem of finding such a L under some acyclicity hypothesis, [4]. Then, rises the question to know whether do finite-dimensional realizations exist. This last problem is the main concern of Fredfield work, [3]. In the present work, the existence of semi-simple group of automorphisms of (R m , g, c¯) with some extra properties supplies the acyclicity hypothesis of Guillemin-Sternberg in [4] and yields solutions of the conjecture studied by Fredfield.

4

Demonstration of the central theorem

Let us recall the notation of the central theorem : (ρ, c¯) is a (G, M )-truncated Lie group and E is a 1-regular (ρ, c¯)-almost Klein model ; S is a semi simple Lie group of E-


895

preserving diffeomorphisms of (ρ, c¯) . We plan showing that E is a Lie algebra of analytic vector fields on M. Step 1. To begin, we observe that M S ⊂ M G . Indeed, let ϕ ∈ S and x ∈ M S .Then ∀g ∈ G we have g.ϕ (x) = ϕ (gx) = g.x. Since M S is discrete and G is connected, we get g.x = x ∀x ∈ M S , ∀g ∈ G. Since X → j 1 X is injective, we may identify E with Tx M ⊕ g (x) . The linear group d S(x) = {dϕ (x) , ϕ ∈ S} is a group of automorphism of g(x) the (Tx M, g (x))-truncated Lie algebra structure c¯ (x) ∈ H−1,2 (g (x)) . Moreover d S(x) commutes with g (x) . Thereby, we apply the realization theorem to (Tx M, g (x) , c¯ (x)) . Thus E is equipped with a Lie algebra structure whose bracket is denoted by [ , ]x such that the linear map jx1 is an isomorphism from (E, [ , ]x ) to the realization of (Tx M, g (x) , c¯ (x)) given by (18). Step 2. Let x0 be an element of M S . According to our hypothesis, E is locally transitive. Thereby, all of the (Tx M, g (x) , c¯ (x)) look alike. So, ∀x ∈ M, there is a local isomorphism ϕ of E such that ϕ (x0 ) = x. Of course, the group ϕ Sϕ¯1 is a E-preserving group, with the same properties as S. Then, the realization theorem may be applied to (Tx M, g (x) , c¯ (x)) . We denote again by (E, [ , ]x ) the Lie algebra structure which is isomorphic to the realisation of (Tx M, g (x) , c¯ (x)) given by Theorem 2.5. Step 3. According to the two first steps above, we have a map x → (E, [ , ]x ) from M to the set L (E) of Lie algebra structures on E. Up to Lie algebra isomorphism the map x → (E, [ , ]x ) is constant. Let us denote the bracket of smooth vector fields by [ , ]p . We intend proving the auxiliary result which follows. Lemma 4.1. Let X, Y ∈ E. Then ∀x ∈ M S , we have the equality [X, Y ]p = [X, Y ]x . ∼

Proof. Let us remind that the Lie algebra g of vector fields induced by hthe iaction p ∼ G × M → M is included in E. Since M S ⊂ M G , it is easy to see that g, ξ = (x) h∼ i g, ξ (x) , ∀x ∈ M s . x

∼

At the other side, we have g (x) = 0 ∀x ∈ M S . Then, we shall consider a normal form at x of E. So, consider a base (α1 · · · αn , β1 · · · βn ) of E such that βj (x) = 0. Let E x be the vector space spanned by α1 , . . . , αm . Every u ∈ Tx M corresponds a unique X (u) ∈ E x such that X (u) (x) = u. The conclusion of Lemma 3.1 holds iff [X (u) , X (v)]p (x) = [X (u) , X (v)]x (x) , ∀u, v ∈ Tx M . Since E is analytic, the equality above is equivalent to the following one jx∞ [X (u) , X (v)]p = jx∞ [X (u) , X (v)]x . Now, let us set ξ (u) = jx∞ X (u) = lim

N P

N →∞ k=0

(24)

ξ k (u) .

Let (x1 , . . . , xm ) be an analytic (local) coordinate functions such that xi (x) = 0. Let us identify jx∞ X (u) with the Taylor serie at x of X (u) . Then ξ k (u) is a homogeneous

896


polynome of degree k in variables x1 , . . . , xm . We may write ξ k (u) = £ ¤ bracket ξ k (u), ξ ` (v) is given by the following formula £

k

`

¤

ξ (u) , ξ (v) =

Xµ i,j

ξjk

∂ξ k (u) ∂ξ ` (v) − ξj` (v) (u) ∂xj ∂xj

¶

m P

i=1

∂ ∂xi

ξik (u) ∂x∂ i . The

(25)

The classical definition of the bracket of formal vector fields is [jx∞ X (u) , jx∞ X (v)]p = jx∞ [X (u) , X (v)]p . ∼

∼

We notice that jx∞ E = Tx M +jx∞ g. So we can write ξ (u) = u+ζ (u) , with ζ (u) ∈ jx∞ g. ∼ ∼ ∼ Let g (x) be the linear component at x of elements of E ; thus, g (x) is nothing but jx1 g. For every integer k > 1, the homogeneous (1 + k)th component of ξ (u) is included in ∼ ∼ the k th prolongation of jx1 g. According to Lemme 2.2, the first prolongation of jx1 g is 0. So, with respect to coordinates functions x1 · · · xn , all of the ξ 0s (u) are affine vector fields, viz we have ξ (u) = u + ζ (u) , where ζ (u) = ξ 1 (u) is a linear vector fields. Thus [ξ (u) , ξ (v)]p = [u + ζ (u) , v + ζ (v)]p . The left member of the last equality is nothing but jx∞ [X (u) , X (v)]p (x) while the right member belongs to [E, E]x (x) . Lemma 3.1 is proved. Step 4. Using the analyticity argument together with lemma 3.1, we conclude that [E, E]p (y) = [E, E]y (y) for all y near the point x. The (local) transitivity of (ρ, c¯) allows us to conclude that E is a subalgebra of the Lie algebra X (M ) of analytic vector fields on M. That ends the demonstration of the algebraic counterpart of central Theorem. A relevant remark is the fact that under the hypotheses of Theorem 3.6 the (G, M )truncated Lie group (ρ, c¯) is induced by a Klein model. Theorem 4.2 below is nothing but our central theorem. Theorem 4.2. Let (ρ, c¯) be an analytic (G.M )-truncated Lie group where G is connected and M is simply connected. Every (ρ, c¯)-almost Klein model E admitting a semi-simple Lie group of automorphisms, say S, satisfying the assumptions of the central Theorem 1.1 is induced by a transitive Lie transformations group of M . Demonstration 4.3. By the virtue of Theorem 3.6, E is a Lie algebra of complete vector fields. According to the R. Palais formalism, [12], there exists a simply connected Lie group H with an analytic effective action H × M → M whose Lie algebra of fundamental vector fields is E. Of course, since G is connected, the action H × M → M is transitive and effective. An almost Klein model E is called Riemannian if elements of E are infinitesimal isometries of a Riemannian structure (M, g). The following statement is a straight corollary of Theorem 4.2. Theorem 4.4. Under the hypothesis of Theorem 4.2, let us suppose E to be Riemannian. If M is compact, then up to finite covering, the underlying Riemannian manifold (M, g)


897

is isometric to the flat torus. Demonstration 4.5. Let x ∈ M S ; using the differential operator jx1 we shall identify E with the vector space Tx M ⊕ g(x), with g(x) = jx1 ge. We equip E with the bracket [ , ]x , that is to say [u + A, v + B] = c(x)(u, v) + Av − Bu + ω(u, v) + [A, B].

(26)

Since E is Riemannian, the cocycle c must be exact in the Koszul-Spencer complex C∗∗ (g(x)), ( c is cohomologue to the torsion of any Riemannian connection). By the virtue of Lemma 3.1, one has c=0. We know that c and ω ( in (26)) are related by c2 = dω. Thereby, ω is a Koszul-Spencer cocycle which is g(x)-invariant. Using Lemma 3.2, we must conclude that ω = 0. So, (E, [ , ]x ) is a subalgebra of the standard affine algebra f be the connected simply connected Lie group whose Lie algebra gl(Tx M ) × Tx M . Let H e on (M, g). We can is E. Theorem 3.2 tells that E comes from a transitive action of H e acts effectively on (M, g). Thus we can identify H e with a G-principal suppose that H 1 bundle of orthogonal linear frames of (M, g). Thus jx G = dG(x) contains the unit component of the holonomy group of the Levi-Civita connection of (M, g). From (26) we π e → deduce that the principal connection on the bundle H M which corresponds to the −1 Levi-Civita connection of (M, g) is flat along the fiber π (x). The transitivity argument implies that (M, g) is flat. Thus, (M, g) is a finite quotient of the flat torus, [14], [15].

References [1] D. Bernard: “Sur la géométrie différentielle des G-structures”, Ann. Institut Fourier, Vol. 10, (1960), pp. 151–270. [2] C. Chevalley and Eilenberg: “The cohomology theory of Lie groups and Lie algebras”, Trans. Amer. Math. Soc., VOl. 63, (1948), pp. 85–124. [3] C. Fredfield: “A conjecture concerning transitive subalgebra of Lie algebras”, Bull of the Amer. Math. Soc., Vol. 76, (1970), pp. 331–333. [4] V.W. Guillemin, S. Sternberg: “An algebraic model for transitive differential geometry”, Bull of the Amer. Math. Soc., Vol. 70, (1964), pp. 16–47. [5] I. Hayashi: “Embedding and existence theorem of infinite Lie algebras”, J. of Math. Soc. of Japan, Vol. 22, (1970), pp. 1–14. [6] S. Kobayashi, K. Nagano: “Filtred Lie algebras and geometric structures III”, J. of Math. and Mech., Vol. 14, (1965), pp. 679–706. [7] J.L. Koszul: “Multiplicateurs et classes caractéristiques”, Trans. Amer. Math. Soc., Vol. 89, (1958), pp. 256–266. [8] M. Nguiffo Boyom: “Déformations des structures d’algèbre de Lie tronquée”, CRAS Paris, Vol. 273, (1973), pp. 859–862. [9] M. Nguiffo Boyom: “Weakley maximal submodules of some S(V)-modules, Geometric applications”, Indaga Math., Vol. 1, (1990), pp. 179–200.

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[10] A. Nijenhuis: “Deformations of Lie algebra structures”, J.Math. and Mech, Vol. 17, (1967), pp. 89–106. [11] A.L. Onishchik (ed.) Lie groups and Lie algebras I. Foundations of Lie theory. Lie transformation groups. in Encyclopaedia of Mathematical Sciences. Vol. 20, SpringerVerlag., Berlin, 1993. [12] R.S. Palais: “Global formulation of the Lie transformation groups”, Mem Amer. Math. Soc., Vol. 22, pp. 178–265. [13] I.M. Singer and S. Sternberg: “The infinite groups of Lie and Cartan”, Jour. Analyse Math. Jerusalem, Vol. 15, (1965), pp. 1–114. [14] J.A. Wolf: Spaces of constant curvature, 3rd ed., Mass.: Publish Perish, Inc. XV, Boston, 1974. [15] J.A. Wolf: “The geometry and structure of isotropic irreducible homogeneous spaces”, Acta Math., Vol. 120, (1968), pp. 59–148.

Preface

This volume constitute the Proceedings of the ”5th Conference Geometry and Topology of Manifolds” which was held in Krynica-Zdrój (Poland) from April 27 to May 3, 2003. There were over fifty participants and almost forty lectures and communications were delivered during this event. The conference is one of a series initiated with a meeting in Konopnica in 1998. It was organized by • Institute of Mathematics of the Technical University of Lodz (Jan Kubarski, chairman), • Faculty of Applied Mathematics at AGH University of Science and Technology, Cracow, (Tomasz Rybicki), • Institute of Mathematics of the Jagiellonian University, Cracow (Robert Wolak). The conference was held under the auspices of Prof. Jan Krysi´ nski, Rector of the Technical University of Lodz. The members of the Scientific Committee were D.Alekseevsky (GB), I.Belko, (Belarus), R.Brown (UK), S.Brzychczy (Poland), J.Cari˜ nena (Spain), J.Grabowski (Poland), A.Kock (Denmark), P.Lecomte (Belgium), A.S. Mishchenko (Russia), P.Nagy (Hungary), A.Nikitin (Ukraine), J.Pradines (France), J.Szenthe (Hungary), W.Tulczyjew (Italy), Y.Yu (China), W.Zhang (China) together with the members of the organizing committee. Further information is available on our WWW pages: http://im0.p.lodz.pl/konferencje/krynica2003 The main purpose of the conference is to present an overview of principal directions of researches conducted in differential geometry, topology and analysis on manifolds and their applications, mainly (though not restricted) to Lie algebroids and related topics. Special attention was also paid to symplectic and Poisson geometry, Riemannian and CR manifolds, index theorems and elliptic operators, Lie groups, Lie groupoids and Lie algebroids, foliations, characteristic classes. The works presented at the meeting constitute a representative sample of the subjects discussed during the conference. The papers in the present volume reflect only partially the scientific activities of the conference. For the record, we give a list of the invited lectures presented during the conference.

D. Alekseevsky, Geometry of quaternionic and para-quaternionic CR manifolds, I. Belko, The fundamental form on a Lie groupoid of diffeomorphisms, S. Haller, Harmonic cohomology of symplectic manifolds, A. Kock, Second neighbourhood of the diagonal, and the foundation for conformal geometry, P. Lecomte, Bordemann’s proof of the existence of projectively equivariant quantizations, W. Mikulski (coauthor J. Kurek ), Symplectic structures of the tangent bundles of symplectic and cosymplectic manifolds, A.S. Mishchenko (coauthor J. Kubarski), Transitive Lie algebroids: spectral sequences and signature, P. Nagy, The classification of compact smooth Bol loops, L. Plachta, Essential tori in link complements in standard positions: geometric and combinatorial aspects, N. Poncin (coauthor J.Grabowski), Lie algebras of differential operators, J. Pradines, Lie Groupoids as generalized atlasses for spaces of leaves, (opening lecture), J. Szenthe, On spherically symmetric space-times, W. Tulczyjew, Analytical Mechanics with Discontinuities and Disipation. The prevailing opinion of the participants was that the conference was a great success. The beautiful surroundings of Krynica contributed to a very creative atmosphere and simultaneously offered various possibilities for recreation. The organizers express their sincere thanks to all the participants for coming to KrynicaZdrój and for their inspiring lectures, discussions and comments. We would especially like to thank the plenary or invited lectures. One of the aims of the conference was to invite mathematicians from middle and eastern Europe. There were thirteen mathematicians from those countries, all of whom received full or partial support from the organizers. We would like to thank all professors who attended the conference with students. It was very important for students to see and hear different approaches to geometry and topology. We would also like to thank the young mathematicians for their lectures and discussions. Finally, we would like to thank the young collaborators Bogdan Balcerzak and Witek Walas for the electronic preparation of the abstracts and printing them, for preparing our WEB-pages and very heartily, for their assistance during the conference. The organizers of the conference are grateful to the following sponsors: – Rector of the Technical University of Lodz, – Dean of the Faculty of Technical Physics, Computer Science and Applied Mathematics at the Technical University of Lodz, – Dean of the Faculty of Mathematics, Physics and Computer Science at the Jagiellonian University, – Dean of the Faculty of Applied Mathematics at the AGH University of Science and Technology for their substancial financial support. The organizers would like to express their deep gratitude to the Editors and Redactors of the Central European Journal of Mathematics for making this publication possible and

their kind assistance throughout the whole editorial process, especially Editor-in-Chief Prof. Grigoriy Margulis, Managing Editor Monika Sperling (as well as theirs predecessors Prof. Andrzej Bialynicki-Birula and Prof. Stanislaw Betley, respectively), Editor Prof. Wlodzimierz Jelonek, Publishing Director Jacek Ciesielski and Bartlomiej Gierszyn. We would also like to express special thanks to the referees of the papers in this volume. We hope that this volume will stimulate new and further interest in the rapidly developing interactions between different areas of geometry and topology of manifolds. Jan Kubarski, Tomasz Rybicki, Robert Wolak

Central European Journal of Mathematics

Central European Journal of Mathematics

Central European Journal of Mathematics