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a(P). Proof. Let X be an (n - 1)-dimensional complex manifold tangent to M at P to order s (1 < s < + co). We may assume the coordinate w (of formula (2.5.1)) chosen so that X = {(z, w) e U I w = 0}. Now, r(z, 0) vanishes at P to order s + 1. Consequently, D(r) vanishes at P if D involves differentiation of order <s with respect to z, 2 (i.e., no w differentiation). Thus Lemma 2.9 shows that the c coefficient of any F e Ys-1 vanishes at P and hence t(P) > s. Thus t(P) > a(P). 2.12. Lemma. t(P) < a(P). Proof. Suppose that t(P) > m where m is an integer > 1. We may assume that the coordinate w (of formula (2.5.1)) is chosen so that D(r)(P) = 0 where D is any pure differentiation with respect to z or 2 (i.e., no mixture of derivatives with respect to z and 2) of order <m + 1. We will show that w = 0 is tangent to M at P to order >m. The c coefficient of any F e Y,,-, vanishes at P. By Lemma 2.10 we may 2.
CHARACTERIZATION OF POINTS OF TYPE m
177
conclude that (a/az)°(a/az)°r(P) = 0 for a, r any multi-indices satisfying Jul > 1 , Ir I > 1 , Ia l + v < m. (We proceed by inductiononjul + r j using the fact that r,w(P) = r,,(P) = 1. Both statements in Lemma 2.10 are needed.) That is, r(z, 0) vanishes at P to order >m + 1. q.e.d. Lemmas 2.11 and 2.12 complete the proof of Theorem 2.4. 2.13. Corollary. Let M be real analytic and P e M a point of type + Co. Then M contains a complex (n - 1)-dimensional submanifold of a neighborhood of P. Proof. Using the assumption that r is real analytic we may assume the coordinate w chosen so that D(r)(P) = 0 where D is pure differentiation with respect to z or z of any order. Then the reasoning in the proof of Lemma 2.12 shows that {(z, w) I w = 0} is contained in M. 2.14. Counterexamples. The conclusion of Corollary 2.13 need not hold if M is only C. We give two examples : 1. Consider r = 2 Re w + exp (- (I z l' + (Im w)')-') and M = {z, w e C' l r = 0}. Then (0, 0) is a point of type co. However, M is strongly pseudoconvex (type 1) in a deleted neighborhood of (0, 0) and cannot contain a complex submanifold. 2. Consider the formal power series
Re(w-n!z") \
"=2
By a theorem of E. Borel [10, p. 28] there exists a C°° function r in C2 having this series as its formal Taylor series at (0, 0). Let M = {z, w E C' I r(z, w) = 0}. The complex submanifold w = En 2n! z" is tangent to M to order m at (0, 0). However, there is no complex submanifold tangent to M at (0, 0) to infinite order. 3.
The case of a single vector field
As before, M is a real C°° hypersurface in an open subset of C", and P denotes a point of M. denote Let L be a tangential holomorphic vector field to M. We let the C`° module of vector fields spanned by L, L and their commutators of order < p. 3.1. Definition. We say L is of type m at P if there exists F e Y,"(L) such that <ar(P), F(P)> * 0 while for all F E Y,,-,(L) we have
<ar(P), F(P)> = 0 .
We shall use the notation t(L, P) = m. If <ar(P), F(P)> = 0 for all F E and all integers p > 1 we will write t(L, P) = + oo. 3.2.
Proposition.
Suppose there is a 1-dimensional complex submanifold
178
THOMAS BLOOM & IAN GRAHAM
X of a neighborhood of P, tangent to M at P to order s. Then there exists a tangential holomorphic vector field L such that L(P) is tangent to X at P and t(L, P) > s. Proof. Choose coordinates z , z,,_1, w centered at P so that 1.
X={(z,w)Iw=z1=
2.
r = 2 Re (w) + 0 where ¢ vanishes to order > 2 at P.
=zn_2=0},
Consider the tangential holomorphic vector field L,,_1 = r,,
a - rZ, _, a azn_, aw
.
We shall show that L,,_1 is of type >s at p. Now r11 has a zero of order s + 1 at P. Thus the description of the commutators of Ln_1 and L,t_1 contained in Lemmas 2.8 and 2.9 is sufficient to prove the proposition. 3.3. Remarks. 1. If in these coordinates we have D(r)(0, 0) # 0 for some impure differentiation D in z,,_1, of order s + 1, then L,,_1 has type precisely s at P. 2. We do not know if there is a converse to Proposition 3.2. The condition that all nonzero holomorphic vector fields be of finite type is conjectured by Kohn [7] to be necessary and sufficient for the a-Neumann problem to be subelliptic at a boundary point of a pseudoconvex domain. 3.4. The type of a vector field is not determined solely by its value at P. Consider M C C3 defined as the zero set of
P = (0, 0, 0) . Here L, is of type 1, and L2 is of type 3. (L, and L2 are defined by (2.5.4).) r = 2 Re (w) + I z1 J2 - 1z214 ,
Note however that M contains the complex submanifold X = {(w, z1, z2) I w = 0 and z1 = z2}
.
Now L = 2z2L1 + L2 is a tangential holomorphic vector field which restricts to a holomorphic vector field on X. Thus it is of type + oo. Of course L(P) = L2(P) 3.5. It is possible to have a point P e M such that all nonzero holomorphic tangential vector fields are of finite type at P but there are points arbitrarily close to P where these are nonzero holomorphic tangential vector fields not of finite type. We will give one such example with M pseudoconvex.
Let M be given as the zero set of r = 2 Re (w) + I z2,- z2I2 and P= (0, 0, 0). Since r is plurisubharmonic M is pseudoconvex (when considered as
the boundary of r < 0). We will first show that every tangential holomorphic vector field L such that L(P) # 0 is of finite type at P (in fact of type = 0 except for the commutator [[[[[L2i L2], L2], L2], L2], L2]. Thus t(L, P) < 5 (in fact t(L, P)
= 5). Now M contains the complex analytic set X = {w, z1, z2 I w = 0, z1 = z2}
X has a singular point at P, but at all other points it is nonsingular. Thus for any point q e X - P there is a nonzero tangential holomorphic vector field which is not of finite type. 4.
Generic submanifolds of higher codimension
Let M be a real C`° submanifold of dimension 2n - k (k < n) of an open , r, be real-valued C°' functions such that M = subset U of C'. Let r1,
{zEUIr1=
Adr, #0onM.
=r, =0} anddr1A
A ar, # 0 on M. Definition [12]. M is generic if art A This condition is equivalent to dime TI"D(M, q) = n - k for all q E M. , r,,.) This is, of course, the (Hence it is independent of the functions r1, 4.1.
minimum possible dimension for the holomorphic tangent space. 4.2. Definition. A point P G M is of type m (m an integer > 1 or + oo)
if there exists F G Y. such that F(P) 0
P) O+ T°°1(M, P) while
contains no such F. We use the notation t(P) = m. The requirement that F(P) 0 TI"D(M, P) O T°"1(M, P) is equivalent to the , r, are defining functions for M, then <ar2(P), F(P)> # 0 following : if r for some i. 4.3. Remark. This is not the most interesting type condition. Naruki's estimate [11] depends on there being an integer m such that {F(P) I F G Ym} = CT(M, P). The point P is then termed (m + 1)-regular by Naruki. Let X be an (n - k)-dimensional complex submanifold of a neighborhood U of P which is tangent to M at P. 4.4. Definition. X is tangent to M at P to order s (s an integer > 1 or + co) if s = inf It I there exists a real valued C°' function r on U such that
rI, = 0, dr # 0 on M and rI$ vanishes at P to order >t + 1}.
Thus s is the least order of tangency of X with a hypersurface containing M. Note that the roles of X and M cannot be interchanged in this definition, , rk are functions such that M = for dimR X < dimR M. Also whenever r1, {z I rl =
=r, = 0} and dr, A
A dr,k # 0 on M, there is an index i
for which rj I% vanishes at P to order s + 1.
180
THOMAS BLOOM & IAN GRAHAM
We set a(P) = sup {s I there exists an (n - k)-dimensional complex submanifold tangent to M at P to order s}. Thus a(P) is an integer > 1 or + co . 4.5. Theorem. a(P) = t(P). Proof. The proof is analogous to that of Theorem 2.4. Since M is generic, given defining functions r , rk for M we can choose local coordinates , wk at P such that zI, ' , zn-k, w1,
ri=2Re(wi)+¢5
(4.5.1)
,
where ¢i vanishes to order >2 at P. Thus ari
(4.5.2)
aw f
(4.5.3)
ari az;
(P) = ari (P) = d aw,
(P) = ari (P) = o azf
,
,
i, j = 1, ... k
i = 1, ... , k
,
n-k
Consider the vector fields (4.5.4) where E, Eil,
k
a Li=Eazi-- Zj=1Ei' aw,
i= 1 ..
n
, Ex are the cof actors of the elements in the first row of the
(k + 1) x (k + 1) matrix
e
e1
art azi
(4.5.5)
1
ek
...
awk
art
I
I
ark
ark
ark
l azi
awl
awk
Note that Li(r) = 0 for i = 1,
Z;
art awl
...
, n - k, s= 1,
, k since E(ar8/azi) +
E (ar8l awl) is equal to the expansion of the determinant of (4.5.5) when
e = ar8/azi and ej = ar8/aw f. Of course, in that case the matrix has two
identical rows. Now the relations (4.5.2) and (4.5.3) imply that E(P) = 1 while i(P) = 0
The following lemmas are proved in a manner similar to the corresponding lemmas in § 2. Details are omitted for the most part. 4.6. Lemma. P, is generated modulo vector fields vanishing on M as a
C'° module by the commutators of order -x for all three structures. Let IxI be a 8r homogeneous gauge. Our first main result is that the maps x, y -> xy and x, y -* x o y are "asymptotic at infinity" when measured via the gauge, as follows. Theorem 1. There is a positive constant r < 1 so that for any e > 0, (5.1)
Ixy-xoY1 <M,(Ixl+IYI)r
in the set {IxI + IYI > e}. In particular, lim
Ixy-xoyl
1x1+ivi--
IxI + IYI
=0.
The proof of Theorem 1 will be given in § 6. Here we draw some consequences from (5.1). Since x o y - 81,,,(8,,x8,,y) = 81,T(8,,x o 8ry - 8,,xd,y), we deduce that if I x I + I y I > e, then (5.2)
Ix o y - O11r(8rxOry) I < M,r7-1(I xl + MY
In particular, since r < 1, this implies Corollary 1. x o y = lim,__ 8rx o 8ry, it follows directly from (5.2) that if r > e and i x I + I Y I > s, then there is a constant C. so that (5.3)
I or(xy) - (orx)(8rY) I < C.r(l x l + I y I)7
This implies
Corollary 2.
For fixed r, 8r is an "asymptotic automorphism" : lim ixl+ivi--
I8.(xY) - (Srx)(8rY)l
IxI + IYI
0
Our next main result is a comparison of the additive group structure of g and the nilpotent group structure of g. Theorem 2. If e > 0, there is a constant C, so that in the region IxI > s(1 + IYD,
(5.4)
IIPa(xY-x-Y)II xy and x -f x + y are
"asymptotic at infinity" : lim IxI-.o
Ixy-(x+Y)I
=0
IxI
Combining the results of Theorems 1 and 2, we can estimate the action of translation by y on homogeneous functions, as follows. Theorem 3. There are positive numbers M and C with the following property : If f is a C' function on g --- {0} which is 8, homogeneous of degree p, then in the region I x I > M(1 + I Y D, (5.6)
If(xy) - f(x)I < C IIVfII_ IxI"-1 IYI
The proof of Theorem 3, assuming the results of Theorems 1 and 2, goes as follows : Consider the line segments a(t) = (1 - t)x + txy and a,(t) =
(1 - t)x - tx o y, where 0 < t < 1. Using a compactness argument and the fact that 8, is an automorphism of x o y, Koranyi and Vagi [6, Lemma 5.2] show that there exists an N > 0 such that for 0 < t < 1, (5.7)
IxI
- NIYIzz> 1IxI<jco(t)I (IYI + 1) ZZ> Ic(t) - au(t)I G
Ix11
where 0 < r < 1. Combining (5.7) and (5.8), we conclude that there exists a constant M > 1 so that in the region I x I > M(I y I + 1),
M-1 IxI < Ic(t)I < MIxI .
(5.9)
Furthermore, in this region I P2(xY - Y)11
s Iy I in the graded case.
7.
Vector fields with polynomial coefficients
In this section we construct a differential operator realization of the algebraic calculations of § 6. This version is natural to employ when one starts with the group law in exponential coordinates, and will allow us to compare the theorems in § 5 with related results of Auslander, Brezin, and Sacksteder [1]. The first step is to construct some basic filtered nilpotent Lie algebras of differential operators. Let V be a real, finite-dimensional vector space, and let {S,} be a oneparameter group of dilations of V, with spectrum A, as in § 3. Every vector X E V defines a constant-coefficient vector field Dg on V by the formula (7.1)
D,O(v) = d 0(v + tX) dt
c=a
0 E C`°(V), i.e., D is the right regular representation of V considered as an abelian Lie algeba. Denote this space of vector fields by J. Let .9 be the algebra of polynomial functions on V. The dilations S, induce a contragredient action 3* on 9:
FILTRATIONS AND AUTOMORPHISMS
193
o*¢(v) = ¢(or_iv) .
(7.2)
In particular, the spectrum of {o*} on 9 is the semigroup -NA generated by
-A. Let f = 9-9 be the space of all vector fields on V spanned by the operators OD,, with 0 e 9, X e V (the vector fields with polynomial coefficients). Since
D1(9) c 9, it follows that ' is a Lie subalgebra of the Lie algebra of all C`° vector fields on V. The dilations Jr canonically define Lie algebra automorphisms ar of ', and it is easy to calculate that (7.3)
31(¢Dx) = (o*c1)Darx
In particular, a; is diagonalizable on ' and has spectrum
A'={A- Emp,i:2,ure A,m,sre NJ .
Forp eA, set (7.4)
Ijp = {T e Y : J;T = r"T}
Then lj,, is spanned by the vector fields 0Dx, such that 10,v) = rao(v), 3,X and P - a = ji. There is a direct sum decomposition =
Y=Eb,.
(died,).
Since 31 is a Lie algebra automorphism, this decomposition is a gradation of Y. The associated filtration is given by
-Wa =E kp pZa In particular, if we define (7.5)
n = p>0 E p
then n is a finite-dimensional nilpotent Lie algebra of vector fields. Examples. 1. If orV = rv (scalar multiplication), then d = {1}. Hence n = -9, the constant-coefficient vector fields, in this case. Vi), then d = {1, 2}. 2. If V = V, (D V2i and M V I n = Ij, O+ k, with lj, spanned by operators Dx, and f 1Dx2, while +12 is spanned by operators Dxa where Xi e Vi and f, e V. Thus n is two-step nilpotent, of dimension =d1(d2 + 1) + d2 where di = dim Vi. If d2 = 1, n is the (2d1+ 1) dimensional Heisenberg algebra. Suppose now that G is a simply-connected nilpotent Lie group. Then global "canonical coordinates of the first kind" {x,}4=, can be found for G such that the group operations are expressed as
194
(7.6)
ROE GOODMAN
(x-')i = -xi
,
(xY)i = xi -I- Yi -I- fi(xi, ... , xi-1; Y1, ..., Yi-)
where f i is a polynomial function (fi = 0 when i = 1 and 2). The map x H (x1, , xd) is an analytic manifold isomorphism from G to Rd. We use this map to identify G and g with Rd. If X e g, then the straight line {tX}t6R through X is the one-parameter subgroup of G generated by X. Given such a presentation of G, we use the procedure in [1, § 6] to define
inductively a group of dilations or on R1. Namely, we let Xi = (0, , 1, , 0) (1 in ith place), and set 3,Xi = r' Xi. The exponents cri are positive integers chosen successively so that for i = 3, (7.7)
fi(3,x; ory) =
, d,
ra`gi(x; y) + O(r-') .
(cr, and c 2 are arbitrary. In agreement with the normalization of § 3, we take cr, = 1 and cr2 > %.) Theorem 4. The subspaces
g. = span {Xi : cri > n} form a Lie algebra filtration of g. The corresponding graded multiplication is (7.8)
x ° y = lim O,i,(O,X6'y) , r--
and is given in coordinate form by (7.9)
(x 0 Y)i = xi -I- Yi -I- gi(x ; Y)
,
where gi is defined by (7.7). Proof. Let i9 be the Lie algebra of vector fields with polynomial coefficients on Rd, and extend 3r to act as Lie algebra automorphisms S; of Y. Let {1e}Z be the filtration of 9 determined by O. (It is an integral filtration since the cri are integers.) The Lie algebra g can be faithfully represented in i9 via the right regular representation. For Y e g, let y(t) = tY be the one-parameter subgroup of G generated by Y. Define the vector field R(Y) by
R(Y)c(x) =-at d
O(xy(t))
,
¢ e C°(Ra )
To calculate R(Y), write Di = a/axi, and set aya
ci,(x) =
f,(x ; Y) ,
y=o
.
FILTRATIONS AND AUTOMORPHISMS
195
Then by (7.6), (7.10)
R(Xi) = Di + E cijD1 j>i
Differentiating (7.7) we find that Ci,(orx) = eCij(x) + O(rP'-1)
where
f = a; - ai
and
ci;(x) =
(7.11)
aaZ
gg(x; Y)
-o
By formula (7.3) for the action of 8r we thus have
8rR(Xi) - r'iR(Xi)
(7.12)
modulo YQi+l. Hence R(Xi) e
and
8rR(Xi) -- R(orXi)
(7.13)
modulo YQi+l. Note that Jr maps g onto g but is not necessarily a Lie algebra automorphism, while 51 is a Lie algebra automorphism of the larger algebra 2 but may map R(g) out of R(g). To verify the filtration property, observe that since R is a representation, (7.12) implies that
5rR([Xi, X,]) = r'R([Xi, X,]) mod Y9+I, where P = ai + aj. Hence by (7.13), (7.14)
Or([Xi, X j]) = rl [Xi, X j] + O(rP+l)
since R is faithful. (7.14) implies immediately that {g,,} is a Lie algebra filtration. The eigenspaces for Jr furnish complements to g,+, in gn. We map gr (g) linearly onto g using these complements, as in § 4. (7.8) and (7.9) are then consequences of Corollary 1 and (7.7). Remarks. 1. Let {X, Y} be the graded Lie bracket on g determined by the dilations {Or}. Define (7.15)
R(Xi) = Di + F' ci1D, j>i
where ci; is the polynomial defined by (7.11). From the calculations just made we see that R({X, Y}) = [R(X), IZ(Y)] ,
R(orX) = 8rP(X) .
196
ROE GOODMAN
R is the right regular representation of gr (g) on C°°(G), relative to the multiplication x o y. 2. Our Theorem 1 is a quantitative version of the statement in [1, § 6]
that "x o y is a slight change in the group operation of G". References [1] [2]
L. Auslander, J. Brezin & R. Sacksteder, A method in metric Diophantine approximation, J. Differential Geometry 6 (1972) 479-496. G. Birkhoff, Representability of Lie algebras and Lie groups by matrices, Ann. of Math. 38 (1937) 526-532.
[3] [4]
N. Bourbaki, Groupes et algebres de Lie, Hermann, Paris, 1968. J. L. Dyer, A nilpotent Lie algebra with nilpotent automorphism group, Bull. Amer. Math. Soc. 76 (1970) 52-56.
[5] [6]
R. W. Johnson, Homogeneous Lie algebras and expanding automorphisms, preprint. A. Koranyi & S. Vagi, Singular integrals in homogeneous spaces and some problems i n cla ssical an al ysi s, Ann. Scuol a Norm . Su p. Pisa Cl . Sci . 25 (1971) 575-648 . A. W. Knapp & E. M. Stein, Intertwining operators for semisimple Lie groups, Ann.
[7]
of Math. (2) 93 (1971) 489-578.
RUTGERS UNIVERSITY
J. DIFFERENTIAL GEOMETRY 12 (1977) 197-202
LOCAL ISOMETRIC IMBEDDING OF RIEMANNIAN n-MANIFOLDS INTO EUCLIDEAN (n+ 1) -SPACE JAAK VILMS
The problem of isometrically imbedding an n-dimensional Riemannian manifold Mn into Euclidean space En+P has received considerable attention. For example, it is now known that for each n, all infinitely differentiable Mn admit local isometric imbedding into M11+(1/2)n(n+1) and global isometric imbedd-
ing into En+P(n), where p(n) is a certain function whose optimal determination has been the object of recent study. On the other hand, much less progress has been made in discovering necessary and sufficient conditions for a given Mn to be locally or globally isometrically imbeddable into En+P for various fixed values of p G p(n). The known
results are mostly limited to p = 0 and 1. The case p = 0 is of course classical-local isometric imbedding of Mn into En occurs when the curvature is zero, and global imbedding, when the global holonomy group is trivial. For
p = 1, many conditions necessary for global imbedding are known, while sufficient conditions must await further local developments. The basic approach here is also classical. Namely, the fundamental theorem for hypersurfaces [2, p. 47] reduces the question of finding necessary and sufficient conditions for local isometric imbedding of Mn into En+' to the problem of solving the Gauss and Codazzi equations for a suitable second fundamental form tensor, in terms of the curvature tensor of Mn ; therefore, the results obtained will necessarily be in the form of conditions on the curvature tensor. The Gauss and Codazzi equations have been solved by T. Y. Thomas in his fundamental paper [4], and by N. A. Rozenson in her formidable work [3]. Each used different methods and obtained different types of conditions on
the curvature tensor. Due to the quite complicated form of these results, however, the local p = 1 situation is far from being clear and warrants further work.
In the present paper, we use the method of bivectors and a theorem of W. L. Chow [1] to solve the Gauss (and Codazzi) equations in the case of a nonsingular curvature tensor, getting in this case, new necessary and sufficient conditions for local isometric imbedding of Mn into En-11 (cf. Theorem 4 below). We proceed with a precise statement of the problem, in our bivector setting.
Let V be an n-dimensional real vector space with inner product. Let 1IZV deReceived May 10, 1975,
JAAK VILMS
198
note the
(n)-dimensional space of bivectors of V ; it has an inner product in-
duced from V by the definition <x A y, u A v>=<x, u> - <x, v>6 is reduced to the following algebraic question : Given a symmetric linear map R : A2V - A2V, find necessary and sufficient conditions in order that there exist a symmetric linear map L : V V satisfying R = L A L. We now part company with the paths taken by Thomas and Rozenson, and exploit the bivector setting of the problem. Our first theorem uses a result of Chow [1] to establish the existence of a suitable L, modulo the right sign. Nonsingularity of the curvature R is essential to the argument. Theorem 1. Let R be nonsingular and symmetric, and let n > 5. Then
there exists an L such that R = ±L A L if and only if
(1)
R(x, A x) A R(x3 A x4) _ -R(x, A x3) A R(x2 A x4)
for all xi e V
Proof. If R = ±L A L, then (1) follows trivially. So it remains to show that (1) implies R = ±L A L. Let G, denote the subset of A2V consisting of all nonzero decomposable bivectors, i.e., all a in A2V having the form a = x A y, or equivalently, satisfying a A a= 0 in A4V. Since x A y is nonzero if and only if x, y are independent, and since u A v, x A y are proportional if and only if {u, v} = {x, y}, where { . . . } denotes the span of vectors in V, it follows that 2-dimensional subspaces of V correspond biuniquely with those 1-dimensional subspaces of A2V which lie in the subset G2. Hence, if we pass to projective spaces P(V) and P(A2V), denoting the passage by square brackets, then [G2] C P(A2V) is precisely the Grassmann manifold of all projective lines in P(V). We say [a], [9] e [G2] are adjacent if their corresponding projective lines in P(V) intersect. Now Theorem I in [1, p. 38],
LOCAL ISOMETRIC IMBEDDING
199
with r = 1, can be stated in this way: If f : [G2] - [G2] is a bijective mapping which preserves adjacence (both ways), and if dim V > 5, then there exists a nonsingular linear map L : V - V such that f = [L A L] I [G2]. (Remark : The dimension restriction serves to exclude correlations.) Our nonsingular linear map R : A2V - AzV induces a bijection [RI: P(A2V) P(AZV), and we want to apply Chow's result to f = [R] I [G2]. In order to do this, we must verify that [R] maps [G2] onto [G2] and preserves adjacence both ways. To see what this means, we note the analytic meaning of adjacence. Namely, for [a], [P] e [G2], the corresponding projective lines in P(V) are [{x, y}] and
[{u, v}], where a = x A Y, P = u A v. These lines intersect if and only if dim {x, y, u, v} = 3 if and only if a A j3 = 0 in A4V if and only if a, j3 can be represented as a = a A b, R = b A c. Therefore, if we can establish that
(2) aAp=0 if and only ifRaARp=0, for all a,i3eA2V, then it easily follows that R(G2) = G2 and that [R] I [G2] preserves adjacence both ways.
We shall now use our hypothesis (1) and the symmetry of R to establish condition (2). Consider the map h : (V)4 - A4V defined by h(x x2i x x4) = R(x, A x2) A R(x3 A x4) ; clearly h is multilinear, and (1) implies it is alternating. Hence h factors through a linear map A : A4V - A4V, so that A(x, A x2 A x3 A x4) = R(xl A x2) A R(x, A x4). Consequently, A(a A j3) = Ra A Rp for all a, p e AZV. Since A is linear, A(0) = 0, which establishes (2) in one direction. The other part of (2) will follow from the nonsingularity of A, which we establish next, using the symmetry of R.
Namely, let w 1 < r
, as was to be shown. On the other hand, if x, y are dependent, then y = dx, and = = = <x, Ly>. This proves Theorem 2.
It remains to remove the minus sign from R = ±L A L. Let us observe first that the plus and the minus in ±L A L denote two mutually exclusive
LOCAL ISOMETRIC IMBEDDING
201
classes of maps, namely, if n > 3 and L, M are nonsingular, then L A L -M A M. (This follows from the proof on page 44 in [2] by inserting a minus sign ; a contradiction will arise at the end : 1 + c2 = 0, c real). We proceed to state a criterion to distinguish between the two classes of maps +L A L and -L A L on A2V. In order to do this, we must consider coordinate representations for R. If e1, , e. is a basis of V, then ei A ej,
for 1 < i < j < n, is a basis of A2V, and a linear map R : A2V -- A2V has
the coordinates Rki, i < j, k < 1, with respect to this basis. These coordinates can be defined by Rki = (ei A ej)R(ek A el), where ei denotes the dual basis of ei. This formula in fact defines Rki for all values of i, j, k, 1, but it is easy to see that the usual curvature identities hold :
Rki = -Rka = -Rik = Rik
,
Rki = Rkk = 0.
(If one does not want to use the dual basis, then one could, indeed, use these identities to define Rki for arbitrary i, j, k, 1, from its values for i < j, k < 1.) Define ¢(R) and,/r(R) by ¢(R) = RkiRtiQR 1 and 1/r(R) = RkiRURn4, where we sum over all repeated upper and lower indices. The functions ¢(R) and i(R) are scalar invariants of the tensor Rki, i.e., the coordinate expressions remain the same even if a basis change is performed. Hence any coodinates may be used to evaluate ¢(R) and 1p'(R). (Remark : 1/1(R) = 8 trace (R3).) Theorem 3. Let R be nonsingular, R ±L A L, L symmetric, and
n>3.
(i) R = L AL if and only ifO(R)+I*(R)>0 (ii) In case n - 3 (mod 4), R = L A L if and only if det R > 0
.
Proof.
Since L is symmetric, there exists an orthonormal basis ei in V which diagonalizes L : Lei = ,lief, for i = 1, , n. We know that 2i 0 for all i because L is nonsingular (since ±L A L is). Then the basis ei A ej, i < j, diagonalizes R : (4)
R(ei A ej) = ± 2i2j(ei A ej)
,
where ± means either + always or - always. Clearly the expression below is nonzero and is positive or negative according to the sign in (4), that is, the
sign inR= ±LAL:
Z (±2i2j)(±ili2k)(±2jilk) = ± E (A1AjAk)2
i<j0
Moreover, if n = 3 (mod 4), then (ii) can be replaced by det R > 0. References [11
W. L. Chow, On the geometry of algebraic homogeneous spaces, Ann. of Math. 50 (1949) 32-67.
S. Kobayashi & K. Nomizu, Foundations of differential geometry, Vol. II, Interscience, New York, 1969. [ 3 ] N. A. Rozenson, On Riemannian spaces of class one, Izv. Ak. Nauk SSSR Ser. [2]
Math. 4 (1940) 181-192, 5 (1941) 325-351, 7 (1943) 253-284 (in Russian).
[4]
T. Y. Thomas, Riemannian spaces of class one and their characterization, Acta Math. 67 (1936) 169-211. COLORADO STATE UNIVERSITY
J. DIFFERENTIAL GEOMETRY 12 (1977) 203-208
EXISTENCE OF GENERALIZED SYMMETRIC RIEMANNIAN SPACES OF ARBITRARY ORDER OLDRICH KOWALSKI
A Riemannian symmetric space is a Riemmanian manifold (M, g) with the following properties : for each x E M there is a (unique) isometry J., on M such that (a) x is an isolated fixed point of Jx, (J.,)2 (b) = identity. It is also easy to show the following property : for every two points x, y E M we have (c) Jz o Jy = Jz o Jx, where z = Jz(y). The following is a direct generalization of the previous situation. Definition. A Riemannian k-symmetric space (k > 2) is a Riemannian manifold (M, g) on which a family {s,,j E ,, of isometries exists with the following properties : (a) Each x e M is an isolated fixed point of the corresponding sx, (b)
(sz)k = identity for all x e M, and k is the minimum number of this
property, (c) for every x, y e M, s., o sy = sz o sx., where z = s (y).
In fact, Ledger and Obata [3] have proved that for every k > 2 there is a k-symmetric Riemannian space which is not symmetric. The purpose of this paper is to strengthen the previous result in the following sense : for every k > 2 there is a k-symmetric Riemannian space which is not l-symmetric for l = 2, , k - 1. (Such a Riemannian space is said to be generalized symmetric of order k; see [2]). In our further considerations we shall make full use of the original construction by Ledger and Obata. 1. Let M = G/H be a homogeneous Riemannian space. As usual, we suppose G acting effectively on the coset space G/H. Thus the Lie group G can be considered as a group of isometries on M. Let r : G , M denote the canonical prejection. Proposition 1. Let G admit an automorphism a such that (i) H = G° = the fixed point set of a, (ii) all = identity, (iii) the transformation s of M determined by 7r o a = s o 7r is an isometry. Then M is a Riemannian k-symmetric space. Communicated by W. P. A. Klingenberg, May 15, 1975.
OLEIkICH KOWALSKI
204
Proof. For x E M define a transformation sx of M by the formula sx = g. s o g-1, where g E it-'(x). Then sx is independent of the choice of g. In fact,
for each h EH we have Lhoa.L,,_, = aandiroL, = hoir. Hence (hosoh-1)oir = h o (s -;r) o L,,-, = h o (ir o a) o L,,_, =;r o (L,,, o a o L,,-,) = it o a, and consequent-
ly, h o s o h-' = s. Thus for g' = gh we obtain g10S091-1 = gosog-1 It is clear that (sx)k = identity for each x E M. We have to prove that x is an isolated fixed point of s,x. For, it is sufficient to show that the initial point o E M, o = ;r(H), is an isolated fixed point of s. Condition (iii) implies that S*p o it*e = ir*e o U*e on the tangent space Ge. Let X E Mo be such that s*,(X) and hence = X, and let X E G. be a lift of X. Then *e(U*e(X))
U*e(X) = X + Z, where Z E H. Now U*e(Z) = 2, and (U*e)k(X) = X + kZ = X because (U*e)k = identity. Thus 2 = 0 and X is a fixed vector of U*e. We deduce X E H. and X = 0. Because s*, has no nonzero fixed vectors and s is an isometry of M, we conclude that o is an isolated fixed doint of s. Finally, we have to prove the formula sx o sy = sz o sx, z = sx(y). For this purpose we shall identify the elements of G with the corresponding transformations of M. Then we deduce s o g o s-' = U(g). Put sx = gosog-', sy = g' o s o (g')-', where x = g(o) and y = g'(o). Then (gosog-'o g' o s-')(O)
= sx(g'(o)) = s ,,,(y). On the other hand, g o s o g`0 g' o s-' = g o U(g-ig') = g" belongs to G. Consequently, sx o sy = g o s o g-' o g' o s o g"0 S0 (g")-'ogosog-1 = Sss(Y) osx. 2. We shall recall here a class of Riemannian manifolds constructed by
Ledger and Obata (see [3]). Let G be a compact connected nonabelian Lie group, Gk+1 the direct product of G with itself (k + 1)-times, and 4G."+1 the Gk+1. Consider the action of Gk+1 on Gk given by
diagonal of (x1,
, xk+1)(Y1,
, Yk) =
(x1Y1xk+v
, xkYkxk+1)
Then Gk+' acts on Gk transitively and effectively, and 4Gk+1 is the isotropy group at the identity o = (e, , e) of Gk. We get a diffeomorphism between Gk+114Gk+1. Each tangent vector at the identity of Gk Gk and the coset space , , X01 where X1, can be written in a unique way in the form (X1, XkEGe. Now let 0 be an Ad (G)-invariant inner product on Ge, and let 0[k] be the Ad (4Gk+')-invariant inner product on (Gk)o defined by ` ""((XU i=1
. , Xk), (XU ..., Xk))
Z b(Xi - Xj,Xi - X1) . Mi, Xi) + i y(i)) = kcb(X, Y), and V13(Xci> for j for i j.
GENERALIZED SYMMETRIC RIEMANNIAN SPACES
205
The inner product 0[11 can be extended, by the left translations of Gk+', to a Riemannian metric on Gk denoted also by 0[k3. Then Gk+1f4Gkbecomes a homogeneous Riemannian manifold (Gk, 0[k1)
Let a be an automorphism of Gk+1 defined by the rule 6(x1, , xk+1) , xk). Then a satisfies all the conditions of Proposition 1, where
(xk+1i x1,
we write k + 1, Gk+1, 4G k+', Gk instead of k, G, H, M respectively. In particular, condition (iii) can be verified as follows : consider the transformation s of Gk determined by 7r o a = s o 7r. Then for any X E Ge we deduce easily s*o(X ci>) = Xci+u for i = 1, ., k - 1, s*a(Xck>) = -(X' + ... + Xck>) and (P[k1(s*0X`i), s*0Y`p) = O[k](Xci>, Y'i>) for i, j = 1, , k. Thus the Riemannian manifold (Gk, OLkl) is (k + 1)-symmetric. 3. In the remainder of this paper we shall specialize the class of manifolds (Gk, O[k1) in a proper way. Proposition 2. Consider a homogeneous Riemannian manifold (Gk, 0[k]) such that (a) G is simple, (b) Gk+' is the component of unity of the full isometry group I(Gk, 0[k]) Then (Gk, 0[k]) is not l-symmetric for any 1 < k + 1. Proof. Let r be an isometry of (Gk, 0[k]) with the isolated fixed point o = (e,
, e) such that r' = identity. Define an automorphism p of the group
I(Gk, OEk1) by the formula p(g) = r o g o r-'. Then the restriction of is an automorphism p of Gk+' We can easily see that 7r o p = r o 7r. Now Gk+' is a direct product of simple subgroups G*(i), i = 1,
to Gk+l
, k + 1,
all of them being canonically isomorphic to the group G. Then the automorphism p : Gk+1 , Gk+' induces a permutation v of the indices 1, ,k+1 such that p(G*.(i)) = G*(i>, i = 1, , k + 1. Denoting by (pi the restriction of p to we get p(g1, , gk+1) = (cD1(g.0>), ..., (Pk+1(g.ck+ll)) In particular, p(g, ., g) = (cp,(g), .., (Pk+,(g)) Because p(4Gk+') C 4Gk+1 we obtain = (Pk+l under the canonical identification G*"> = . = G* Q+1) = G, and therefore a unique automorphism cp : G -p G such that p(g1, , gk +1) ' ' ., (P(g.(k+u)). Denote by dp (respectively, d(P) the induced automorphism of the Lie algebra gk+' (respectively, g). Then dp(X1, Xk+1) = (dcP(X.(1)), .. , d(P(X.ck+u), X1, ... , Xk+1 E g Now let us recall the following result by Borel and Mostow, [1]. (P1 = cp2
-
Lemma. A semi-simple automorphism A of a nonsolvable Lie algebra g leaves fixed an element X such that ad X is not nilpotent. dcp is a semi-simple automorphism of g because (dcp)' = identity. Let X * 0
be a fixed vector of dcp and suppose l < k + 1. Then the permutation v contains a cycle (i1, , ice) of length m < k + 1. Consider the vector Z = (X1, , Xk+) E gk+1 such that Xi =X for i = i , ,. . . , !and Xi = -X otherwise. Clearly, dp(Z) = Z. Now we can identify gk+' with the tangent space (Gk+')e and dp with the tangent map p*e. We have 7r*e o p*e = r*o o 7r*ef
and thus the projection 7r*e(Z) E (Gk), is a fixed vector with respect to r*0.
OLDIICH KOWALSKI
206
Moreover, Z E (Gk+1)e is not tangent to the submanifold 4G11 t1 and hence 7r,xe(Z) * 0, a contradiction. This completes the proof. tr (ad X o ad Y) the Proposition 3. For G = SO(3) and 0(X, Y) 2 conditions of Proposition 2 are satisfied. Proof. In the following, the elements of g (respectively, gk) are considered as left invariant vector fields on G (respectively, Gk). First of all, there is a basis {X1, X2, X3} of g such that [X1, X2] = X3, [X2, X3] = X1, [X3, X1] = X2. X(i), a = 1, 2, 3, We have O(Xa, 3 for a, = 1, 2, 3, and the vectors i = 1, , k, form a basis of gk. Now recall formulas (14) of [31: for X, Y E g 1
g(i>Yc9> =
{[X y]c9) - [X Y]">}
2(k + 1) p
1(i)
for i
I,
yci> -- 2 1 [X Y] M
A routine calculation shows the following properties of the curvature tensor R of q)Ek3:
(1)
or a = P = r, R(Xai), X(j))X(k) = 0 whenever a R(X( ), X(j))X( k) and R(Xai), Xaj>)X(k) belong to the subspace generated by X(i), X(j), X(k) .
Let Ho be the component of the unity of the isotropy group of I(Gk, O ) at the origin o, and denote the corresponding Lie algebra by t,. Then ljo has a faithful isotropy representation by endomorphisms of gk = (Gk)o. Clearly, the necessary condition for A E ljo is that A(O ) = A(R) = 0, where A acts as a derivation on the tensor algebra of gk. Let A E ljo and set
(2)
AXai>= E Ea()>XAj>, 't 3
P=1 j=1
i=1,
k,
1,2,3
The relation (Ao[k3)(X(i), X(D) = 0 implies
(3)
k(al(i))a + aiji;) -i*iE aiia - Ei*ja ((j)' = 0
Further, we can calculate easily R(Xa(i),X(i))Xci) = -1X(i) 4 a P A
for a
Consider the relation (AR)(Xai), X(i))Xai) = 0, i.e.,
-4AX") = R(AXai) X('))Xa> + R(Xai) AX())Xa> (4)
+ R(X(i) X(i))AXai)
Let us substitute (2) in (4) and consider a vector X(j), where r * a, R and
GENERALIZED SYMMETRIC RIEMANNIAN SPACES
207
j
i. This vector enters into the left-hand side with the coefficient -4aiJia. According to (1), there is only one term on the right-hand side the evaluation of which can involve X;U>, namely, the term R(Xai>, a«)TX;j>)Xai). Now R(Xai) a((i)aX;J))Xai) =
2)X,i) - X('n]I [4(k + 1)2]
Comparing the coefficients at X; i> we finally get a((;)T = 0. Thus we have proved
aria=0
(5)
j,ap.
fori
Substituting in (3) we get
(6)
a((%))a+ari;=0
force
In particular, for i = j we obtain
(7)
a((i))Q + a((%)); = 0
and hence
(8)
a((i))a = a((2iQ
.. = ack)P
for a $ p
.
Now let us compare the coefficients at X(j), j * i, in the relation (4). X$J) enters into the left-hand side with the coefficient -4a(()P. As for the right-hand side, X(') can be involved only in the evaluations of the terms R(a,():Xa'>, Xci>)Xci) a
R(X(i) a
(9)
After routine calcula-
a ,
tions we obtain (3k
2)a((j) + (k2 + 2k)ai>ia = 0
Writing these relations for (a, Q) _ (1, 2), (2, 3), (3, 1) respectively, we obtain finally
(10)
a(()a = 0
2, 3
Having i = j and a = Q in (3), we deduce from (10) (11)
air>Q = 0 ,
a = 1, 2, 3, i = 1,
If we summarize (5), (10) and then (7),
(8),
,k.
(11), we can see that
On the other hand, the group Gk+l = SO(3)k+l is contained in I(Gk, ([k]) so that d(SO(3)k+i) is contained in Ho. Thus Ho = d(SO(3)k+'), and consequently SO(3)k+l is the component of the unity of I(Gk, as required. Hence we can conclude our paper with Theorem. For each integer k > 2 there exists a compact generalized symmetric Riemannian space (M, g) of order k such that the component of the unity of the full isometry group I(M, g) is semi-simple. llo C
OLDRICH KOWALSKI
208
References [1] [2]
A. Borel & G. D. Mostow, On semi-simple automorphisms of Lie algebras, An. of Math. 61 (1955) 389-405. O. Kowalski, Riemannian manifolds with general symmetries, Math. Z. 136 (1974) 137-150.
[ 3 ] A. J. Ledger & M. Obata, Affine and Riemannian s-manifolds, J. Differential Geometry 2 (1968) 451-459. [4]
J. A. Wolf & A. Gray, Homogeneous spaces defined by Lie group automorphisms.
I, J. Differential Geometry 2 (1968) 77-114. CHARLES UNIVERSITY, PRAGUE
J. DIFFERENTIAL GEOMETRY 12 (1977) 209-227
GAUGE ALGEBRAS, CURVATURE AND SYMPLECTIC STRUCTURE PEDRO L. GARCIA
Introduction
The notion of "gauge algebra" has its origin in the theory of the electromagnetic field. In the most simple case (vacuum space) a electromagnetic field is defined by a 1-form w on the Minkowski space V4 which satisfies the Maxwell equations :
3dw-0, where d is the exterior differential, and 8 is the codifferential with respect the Minkowski metric g. w is called the field potential 1-form. As is known, these equations can be obtained as Lagrange equations of the variational problem defined by the Lagrangian density 2'dx, where dx is the Minkowski volume element, and 2' is the real valued function defined on the 1-jets fibre bundle J'(T*(V4)) by
2'(l1w) = lgjdw, do))
.
In this way we have associated a dynamical theory to the electromagnetic field (Hamilton equations, Poisson algebra, etc.). In particular, an important notion to consider is the Lie algebra of the infinitesimal internal symmetries of the field, that is, the vertical vector fields D on T*(V4) such that their 1-jet extension j'(D) satisfies the condition j'(D)2' = 0, [1]. In our case, this Lie algebra is the abelian real Lie algebra defined by the infinitesimal generators Dr of the uniparametric groups zt of the automorphisms of T*(V4) given by
rt:wy
*co
+t(df).x,
where f runs along the algebra {f} of the real valued differentiable functions on V4. In this way, at the base of the dynamical theory of electromagnetic field we find a special real Lie algebra {f} and a natural representation f E {f} - Dr of this algebra in the vector fields on the space T*(V4). This is the gauge algebra in the electromagnetic field theory. The above formulation gives a very interesting geometric insight which as is Communicated by B. Kostant, May 19, 1975.
210
PEDRO L. GARCIA
proved in [6] corresponds in physics to the fact that an electromagnetic field is the radiation field generated by a moving electric particle. Precisely, an electric
particle is characterized by a variational problem defined on a fibre bundle B = V, X F (which is the direct product of the Minkowski space V, with a real vector space F) which admits the unitary group U(1) as a subgroup of the group of internal symmetries. The corresponding Noether invariant is called the charge-current 3-form of the electric particle. Note that B is associated to the principle bundle P = V, X U(1), whose connections are identified precisely with the 1-forms on V, on which the electromagnetic theory has been built. In this way, one has the following natural equivalences : "electromagnetic fields" Fr "connections"; "Lagrangian of the field" Fr "function of the curvature" ; "Gauge algebra" H "sections of the adjoint fiber bundle of P", etc. All this leads us to define the notion of gauge algebra of an arbitrary principal bundle p : P -> V as the Lie algebra of sections of its ad joint fibre bundle L(P). The object of this paper is now the following. After defining a canonical action of the so defined gauge algebra on the connections of the principal bundle, which locally agrees with the formulas suggested by the physiciens [9], we study the relation between the notions of gauge algebra and curvature. The following are two main results in this sense.
First, the principal bundle p : P -> E induced from p : P -> V on its fibre bundle of connections ir : E -> V by the projection ir has a canonical connection whose curvature 2-form Q defines a special symplectic structure on E such that the gauge algebra is identified with a certain subalgebra of the corresponding Poisson algebra. According to this : every gauge algebra is a subalgebra of a Poisson algebra in a cannonical way. One gets to this result adapting adequately the idea of "pre-quantization" introduced by B. Kostant for ordinary symplectic
manifolds [4]. This result is not only interesting in itself, as it relates to apparently different notions like gauge algebras and Poisson algebras, but opensthe author thinks- the posibility of applying the ideas on "pre-quantization" and "quantization" to the study of unitary representations of gauge algebras. A second main result is an intrinsic characterization of a known result of Utiyama about "admisible lagrangians" in the gauge-invariant classical field theories [8]. Finally, we apply the obtained results to the problem of "combination" of gauge algebras with the so-called "infinitesimal external symmetries" in classical
field theory. Remarks made in this sense can be a good starting point for a differential-geometric approach to this interesting topic for infinite-dimensional Lie algebras of the type of those dealt with in this paper. Concepts and notation in this paper are the ones usually found in any text on modern differential geometry. The reader can refer to the book by J. L. Koszul [5]. All manifolds will be considered paracompact and connected. Differentiability will always mean Cm-differentiability, etc. The author wishes to acknowledge his indebtedness to Professor J. Sancho
ALGEBRAS, CURVATURE AND STRUCTURE
211
for his valuable orientations and effective help and, above all, for his constant and sharp criticism during the preparation of this paper. 1.
The fibre bundle of connections of a principal bundle
Let p : P , V be a principal bundle with structural group G with Lie algebra
9. As it is known [5] that a connection on P can be defined by a splitting a : T -+ Q of the exact sequence of vector bundles on V :
0-*L(P)-*Q-*T-*0 where Q is the vector bundle of G-invariant vector fields on P, L(P) is the subbundle of Q defined by the G-invariant vector fields which are tangent to the fibers of P, and T is the tangent bundle of V. L(P) is a bundle of Lie algebras, where, if D, D' E L(P)y, then [D, D'] is the Lie bracket of D and D'. On the other hand, it is the fibre bundle associated with P by the adjoint representation of G. It is called the ad joint bundle of P. Thus connections of P can be identified with global sections of the affine bundle ir : E , V defined as follows : x E V being given, let E,x be the set of homomorphisms ay : T, , Q, such that px a,, = 1, let E = J ,,. E,,, and let ir be the natural projection of E onto V. Proposition 1.1. ir : E , V has a unique affine bundle structure such that for every connection a on P the mapping a : Hom (T, L(P)) , E defined by h, 6(x) + by is an affine bundle isomorphism on V. Proof. A connection a on P being given, the above said mapping is bijective and makes the following diagram commutative :
Hom (T, L(P)) ° -- E
Then the affine bundle structure of Hom (T, L(P)) defines, by 6, an affine bundle structure on E which, we will see, does not depend on the connection a chosen. Indeed, let a' be another connection. Then a'-1.6: Hom (T, L(P)) , Hom (T, L(P)) is the affine bundle automorphism :
h,xH(6-a')(x)+hy which proves the desired result.
q.e.d. Let F(E) be the vertical bundle of E, i.e., the subbundle of the tangent
bundle of E defined by the vectors tangent to the fibres of E. Corollary 1. There is a canonical vector bundle isomorphism on E between the vertical bundle F(E) of E and the vector bundle lr* Hom (T, L(P)) induced of Hom (T, L(P)) by ir.
PEDRO L. GARCIA
212
Proof. hx E Hom (T, L(P))x being given, let DJ, be the infinitesimal generator of the uniparametric group rt of automorphisms of the fibre E,:
rt(ux) = ax + thx ,
a, e E,x
.
The mapping which assigns to each (ax, hx) E ;c* Hom (T, L(P)) the element (D,)os E F(E) is the desired isomorphism. Corollary 2. E is an affine subbundle of the vector bundle Hom (T, Q).
Proof. A connection a on P being given, it is enough to remark that the isomorphism of Prop. 1.1 is the restriction to the subbundle Hom (T, L(P)) C Hom (T, Q) of the affine bundle automorphism ax H 6(x) + ax of Hom (T, Q). Definition 1.1. The affine bundle E will be called the fibre bundle of connections of the given principal bundle P. 2.
Gauge algebra of a principal bundle and its natural representation on the fibre bundle of connections
Let A be the real algebra of the real valued differentiable functions on V. Definition 2.1. The Lie A-algebra r of global sections of the adjoint bundle L(P) will be called the gauge algebra of the principal bundle P. Examples. (1) If G is abelian, then 9 is also abelian and L(P) can be identified with the trivial bundle V X 9. Thus the gauge algebra is just the abelian Lie algebra of v-valued differentiable functions on V. In particular,
if G = U(1), then 9 = R and r = A, which is the gauge algebra in the electromagnetic field theory. (2)
If P = V X G, then L(P) = V X 9, so r can be identified with the
tensor product A ®9 endowed with the Lie product :
[f ®e, f' ®e7 = (f f') ®[e, e']
,
where f , f' E A and e, e' E 9. One has the so-called "current algebras" introduced by M. Gell-Mann [3].
The sheaf of sections of L(P) gives us a family of gauge algebras (parametrized by the open sets of V) : for every open set U C V, r is the gauge algebra of the principal bundle P. Every element s of the gauge algebra r defines an uniparametric group rt of the vertical automorphisms of the fibre bundle of connections E in the (3)
natural way : rtax = o'x + t[o'x, S] ,
ax e E ,
where [ax, s] E Hom (T, L(P)) is defined by [6x, s]Dx = [6x(Dx), S]
ALGEBRAS, CURVATURE AND STRUCTURE
213
By the canonical isomorphism between F(E) and z* Hom (T, L(P)) (Cor. 1, Prop. 1.1), the infinitesimal generator D, of Tt is the vertical vector field on E : DS : Qx --+ 117X, S]
Theorem 2.1.
.
The mapping s E F H DS is a homomorphism of real Lie
algebras. Proof. Tt is the restriction to E C Hom (T, Q) of the uniparametric group ?t of the vertical automorphisms of Hom (T, Q) :
Ttax = ax + t[a..r, s]
,
ax E Hom (T, Q)
where [ax, s] E Hom (T, L(P))x is defined by [a.z, s]Dx = [ax(Dx), s]
Thus D8 is the restriction to E of the infinitesimal generator DS of it. Accordingly, the theorem would follow automatically if s e P H Ds were a homomorphism of real Lie algebras. We shall see that it is the case. Linearity is immediate. To prove the equality [Ds, Ds,] it will be enough to prove it on functions f of Hom (T, Q) linear on the fibres, because the D3 are vertical. Since for these functions (Dsf)(ax) = f([a,, s]) (it follows that, in particular, the Dsf are also linear on the fibres), the following calculation proves what we want : (D[s,s,,f)(ax) = f([ax, [s, s']]) = f([[ax, s], s']) - f([[ax, s'], s])
_ (Ds,f)([ax, s]) - (Daf)([ax, s']) _ (Ds(D,.f))(ax) - (D,(Dsf))(ax) _ ([D.,, DS-]f)(ax)
q.e.d.
Theorem 2.1 gives us a representation of gauge algebras (by vector fields on a manifold) which we shall call, in what follows, the natural representation of the gauge algebra F of P on the fibre bundle of connections Z: E -> V. Local expression. If U is an open set of V with local coordinates (xi) such that PZ7 U X G and (D j) are the G-invariant vector fields on Pz7 defined by a basis of the Lie algebra 9 in the corresponding isomorphism L(P)Z7 ,;; U x 9, then the functions (xiAij) on Ea, where Qx
a
axi
=
a ax.i
+ Z Ai j(ax)D; ,
ax E Ea
i
define a system of local coordinates on EZ7 C E.
On the other hand, the gauge algebra F can be identified with the Az7module of linear combinations
PEDRO L. GARCIA
214
s=
f j(xi)D j
,
endowed with the Lie product
[s, s] = E fi.fj[Di,Dj] = E fi-fjci-Dk i,j,k
fDk
where (ckj) are the structural constants of 9. In this setting, the vector field ti [a,, s] associated to s can be calculated as follows : DS : [a.x,
S]-'
[a_(
a ), sit
axi
=
axi
+ F_ Aih(as)Dk, Ek h
/ ((L) axx + E C hkAih(as)fk(x) IDJ \1
E
a
from which it follows that
Ds = F
(2.1)
i,j
3.
afj + E chkAihfk h,k
axi
a
aAij
.
A symplectic characterization of gauge algebras by means of curvature
Let p : P - E be the induced bundle of the principal bundle p : P -- V on its fibre bundle of connections it : E -± V by the projection 7r. It is a principal
bundle with structural group G such that the canonical morphism r: P - P is a principal G-bundle morphism, i.e., one has the following commutative diagram :
P
P
where ;c commutes with the action of G.
In this way, if one considers the exact sequence (1.1) corresponding to p : P -- E, then 7r induces a morphism f : Q -- Q of vector bundles, which in turn induces a morphism of exact sequences : 0 ) L(P)
) Q ) T
i i
--I
) 0
0-*L(P) -*Q-*T -*0
ALGEBRAS, CURVATURE AND STRUCTURE
215
We want to remark that L(P) can be identified with the vector bundle 7r*L(P) induced of L(P) by 7r, after f : L(P) -), L(P) coincides with the corresponding canonical morphism. Thus f LIP, is an isomorphism on each fibre. Then the exact sequence 0
has a "canonical splitting" p : Q - L(P) defined by pas(D) = px(fD)
,
vx E E ,
where px is the projector 1 - ax p,*, and px(f D) E L(P), is considered as an element of the fibre L(P)as by the isomorphism f : L(P)oz -), L(P), which we mentioned before. Definition 3.1. We shall call canonical connection of the principal bundle P the connection defined on P by the splitting p. The corresponding connection 1-form will be written 0.
This connection defines a derivation law F in the Lie module I'(L(P)) of sections of L(P). Thus we have an L(P)-valued differential calculus on the manifold E. In what follows we shall use this calculus without explicitly mentioning the derivation law F. Local expression. Let (xiAij) be the system of local coordinates on Eu C E defined in § 2. By the identification of L(P)E. with the induced vector bundle
7r*L(P)u, the basis (D,) of f'(L(P)u) in § 2 defines a basis of F(L(P)E,). A simple local calculation gives for (a°/axi)Dj and (a°/aAik)D; the expressions : v
ap,
(3.1)
ax
D,
¢
=
h,k
ch;AinDk
,
a aAik
D; = 0 .
Now, let Q be the curvature 2-form of the canonical connection. It is an L(P)valued 2-form on the manifold E, whose local expression is, by (3.1),
(3.2) Q
(dA1,
A dxi - 2
k
chx(A,hAik - AihAak)dxt A dxil o D; .
Remark. By what was said in § 1, connections on P are identified with global sections of the fibre bundle 7r: E - * V. Now one observes that the curvature 2-form Q has the following universal property : for every connection a : V - E with curvature 2-form Q° one has Q° = v*Q. In particular, one can obtain from here a simple proof of Weil's theorem on characteristic classes [2]. Proposition 3.1. Q is an L(P)-valued pre-symplectic metric on the manifold E. Proof. From the local expression (3.2) it follows immediately that Q is nonsingular in every point of E. q.e.d. Q is not closed in general. But, if one considers it as an End L(P)-valued 2-form by the rule :
PEDRO L. GARCIA
216
2(D, D')s = [Q(D, D'), s]
,
it becomes closed, for then it coincides with the curvature 2-form of the derivation law 17, which is closed by Bianchi's identity. In what follows, by abuse of language, we shall consider (E, 0) as a symplectic manifold. We will see that this is justified for the ordinary notions of symplectic manifolds can be generalized to (E, Q) in a natural way. By means of the identification of L(P) with the induced vector bundle n*L(P), the gauge algebra T of P is injected onto a A-subalgebra of the Lie algebra T(L(P)). Under these conditions we have the following. Theorem 3.1. If s E T H DS is the natural representation of the gauge algebra T on the fibre bundle of connections n : E - V, then
iDSQ = ds ,
i.e., DS is the hamiltonian vector field of (E. SQ) corresponding to s. T is characterized as the set of sections s E T(L(P)) with a hamiltonian vector field DS which is tangent to the fibres of the morphism it. Proof. By using the local expressions derived in § 2 and § 3, one has
iDSQ = Z iD8(dAij A dxi) o D j=
(DSAi j)dxi o DJ
i,j
(-Lf,- + Z chkAihfkdxi o D j= d Z f j o D j= ds axi
j
h,k
where it is supposed that the local expression for s is s = Z j f j o D j. If s E T, we have just seen that it has a hamiltonian vector field D, tangent to the fibres of i. Conversely, if s E T(L(P)) has a hamiltonian vector field D, tangent to the fibres of ic, then
ds = iDSQ = Z iD,(dAij A dx) o D j = .' (DSAi j)dxi o i,j
Dj
,
ti,j
j f j(xi) o D j, thus proving that s E T . so s has the local expression s Corollary. The kernel of the representation s E T H D, is the ideal To of sections s e T such that ds = 0. To is locally isomorphic with the center of the Lie algebra 9 of the structural group G. In particular, we have two extreme cases : if G is abelian To is globally isomorphic with 9, and if 9 has no center, the representation s e T H D, is faithful. Proof. The first part is an immediate consequence of the theorem. Now, if s = 7, j f j o D j on U C V and (g,) is the basis of 9 defining the (D j) (local expression, § 2), then ds = 0 is equivalent, by (3.1), to the system of equations
af,_ axi
+ Zh,kChkAihf,, = 0
ALGEBRAS, CURVATURE AND STRUCTURE
217
Taking the derivative with respect to A,h, one has E, c; xfx = 0, from which it follows that of jl9xs = 0. Then s e (r,), if and only if s j ,i^, where the 2j are real numbers such that E j ch;A; = 0. The mapping s = E; A1D; E (ro)u H E; J,;gj E 9 establishes the required (local) isomorphism between r, and the center of 9. Now the last part of the corollary is immediate. 4.
Poisson algebra associated to a gauge algebra and prequantization
In § 3 we have seen how the gauge algebra r can be injected canonically into the Lie algebra r(L(P)) in which the differential calculus on the symplectic
r
manifold (E, Sl) is valued. Moreover, r is injected into the A-subalgebra of P(L(P)) defined by the sections s e r(L(P)) which have a hamiltonian vector field. Thus we have the canonical inclusions of Lie A-algebras
rCrCr(L(P)). r C r is always strict, and r c r(L(P)) is strict if dim G > 1. Now on r we should define the notion of "Poisson bracket". We shall see that this can be done in such a way that while preserving all the essential properties of the ordinary Poisson bracket, on r the new product coincides with the old one. In particular, it follows that every gauge algebra can be considered in a canonical way as a subalgebra of a Poisson algebra. The method to follow will be a special adaptation of the idea of "prequantization" introduced by B. Kostant for ordinary symplectic manifolds. In this sense, we shall proceed as follows. The canonical connection of p : P -* E establishes an isomorphism ri : r(L(P)) -9 -* 9(P) between the direct sum r(L(P)) @+ -9 of the modules r(L(P)) of sections of L(P) and -9 of vector fields on E, and the module 9(P) of Ginvariant vector fields on P, by the rule :
ri(s,D)=-s+ D, where b is the horizontal lift of D E -9, and s e r(L(P)) is a G-invariant vector field on P tangent to the fibres of p. Denoting by p* the canonical injection of the L(P)-valued forms on E into the 9-valued forms on P [5] and remembering that we call 0 and Q, respectively, the connection 1-form and the curvature 2-form (as a L(P)-valued 2form on E) of the canonical connection of P, we have the following : Lemma 4.1. If s e r(L(P)) and D E -9, then
L,cs.D)e = p*(iDQ - ds) Proof.
By putting ri = ri(s, D) _ -s +
we shall compute the Lie de-
PEDRO L. GARCIA
218
rivative L,10 = ir)dO + dir)B. Denoting, as it is usual [5], s` for p*s, from ir)B
= B()) = B(-s) = -N we obtain that dir)B = -d& = -p*ds + [B, s"] On the other hand, by the structure equation dO = p*Sl - [0, 0] one has
ir)d. = iDp*Sl + is[0, 0] = p*(iDQ) - [0, s] , so that
L,0 = iy)d0 + diy)B = p*(iDQ - ds)
.
q.e.d.
2'(P) is a real Lie algebra with respect to the Lie bracket of vector fields. This Lie product is expressed with respect to the above parametrization Y) as follows.
Lemma 4.2.
If s,, e I'(L(P)) and D, e 2, i = 1, 2, then
[ri(s1D), (S2D2)] = i2(D1S2 - D2s1 + Q(D1, D) + [St, S211 [D1, D2])
Proof. Let rid = ri(s=Dj) and [y)1, 7)21= ri(s, D) = r). Of course, D = [D1i D2] since D = pri = p[)71, r12] = [P)1, Prl2] = [D1, D2] On the other hando(1)711 )72]) =-S. Then by the structure equation dO =
p*Sl - [0, 0] one has dB(.)1, 722) = (P*OL)(L1, L2) - [s1, `s2]
,
so that N = -O([7)1, 7)2]) = " (D1, D2) - 1)1O(1)2) + 1)20(7)1) - [N1, S2]
.
Thus from Nd = -O(r)Z) and the definition of covariant derivate it follows that N=
»2]) = Q(D1, D2) + D1s2 + [ND N2] - D2s1 - [S2, S1] - [SI5N2]
Now we have the required result by considering that the injection s H N preserves the Lie product. 0, then Corollary. If [r)(sl, D1), 1)(S2, D2)] = 7)([S11 S2] - " (D1, D2), [D1, D21)
Proof. Obvious after Lemmas 4.1 and 4.2. q.e.d. Now we can state the most important result in this paragraph : Theorom 4.1. Let 2'(P, 0) and Y be respectively the real Lie algebras of vector fields ri e 2'(P), such that L,0 = 0, and of hamiltonian vector fields of (E, fl).
ALGEBRAS, CURVATURE AND STRUCTURE
(a)
219
One has the central extension of real Lie algebras :
0,r,,2(P,0)-*
,0
where r, is the kernel of the natural representation s e r -+ D3 of the gauge
algebra r(Cor. of Th. 3.1), r, , 2'(P, 0) is the injection s e ro H -s, and
2(P, 0) , (b)
is defined by the projection p : P , E. The mapping S : r , 2(P, 0) defined by S(s) = 7I(s, D3)
is an isomorphism of real vector spaces. This allows us to endow r with the Lie product { , } induced by the isomorphism 3. The real Lie algebra thus defined (F, { , }) will be called Poisson algebra associated to the gauge algebra r. The Poisson product { , } is given by (4.1)
is, s'} = [s, s']
where D3, D, are the hamiltonian vector fields- corresponding to s, s'.E r. In particular, on r both products [ , ] and { , } coincide. (c) One has the commutative diagram of real Lie algebras:
where ro - r is the inclusion r, c: r, and r
,
is the mapping which assigns to every s e r its corresponding hamiltonian vector field D. Thus the Poisson algebra r is equivalent to the real Lie algebra 2(P, 0) as
a central extension of by P0. In particular, the gauge algebra r is an extension, by P0, of the hamiltonian vector fields tangent to the fibres of ir : E -* V. Proof. .(a) If 7) = r)(s, D) e 2'(P, 0) then, by Lemma 4.1, DO = ds, from which it follows that pry = D e . The mapping 2(P, 0) - * is onto, for, if De , then there exists a sections e r(L(P)) such that DO = ds, from which
we have 7) = r)(s, D) e 2(P, 0) and pry = D. s is determined up to a section so such that dso = 0, i.e., up to an element of P0, thus proving the exactness of the sequence. Mappings are obviously homomorphisms of real Lie algebras.
Last, r, , 2(P, 6) is central by Cor. of Th.' 3.1 and Cor. of Lemma 4.2. (b) It is immediate that j is a homomorphism of real vector spaces. That S(s) = 0 implies s = 0 is obvious also. On the -other hand, if 7) = r)(s, D) e 2(P, 0) then DO = ds, from which we have s e r and DS = D, i.e., S(s) = r).
Thus S is an isomorphism.
PEDRO L. GARCIA
220
By the definition of { , } and Cor. of Lemma 4.2 one has {s, s'} =
=
DS),,J(s, D,.)] s'] - Q(D8, DS-), [DS, DS-])
= [s, s'] - Q(DS, DS.) .
On r both products [ , ] and I, } coincide for, if s, s' E r, then Q(DS, DS,) = 0 because DS, DS, are tangent to the fibres of n : E -* V. Remark. From the preceding theorem one has immediately that if D3, Ds, are the hamiltonian vector fields of s, s' E r, then i[DS, DS,]Q = d{s, s'} ,
that is, [DS, DS,] is the hamiltonian vector field corresponding to {s, s'}. In particular, if s, s' E r then i[DS, DS,]Q = d[s, s']. This gives us a new proof that s E r --+ DS is a representation of real Lie algebras. The Poisson algebra r can be now pre-quantized as in the ordinary case G4].
Let 3:. P , End, (L(P)) defined by 3(s)r = [s, r] + Dar ,
where DS is the hamiltonian vector field of s e r and r E r(L(P)). Theorem 4.2. 8 is a representation of the Poisson algebra r on the real vector space r(L(P)), that is, 3{s, s'} = 3(s) . 3(s') - 8(s')8(s). Moreover, for every r e r(L(P)) one has
ti
8(s)r = S(s)p
(4.2)
Proof. S(s)r
The following calculus gives (4.2) :
= ;2(s, DS)r = (-s + DS)r = [s, r] + Dsr = Is, rl D r = 3(s)r .
It follows immediately from here that 3 is a representation, by observing that
r(L(P)) - r(L(P)) is an isomorphism and that S is a homomorphism of real Lie algebras by Theorem 4.1. q.e.d. In particular, 8 induces a representation of the gauge algebra r on the real vector space r(L(P)), whose local expression is L7 + E ChkAihfk) 8(s)r = [s, r] +i,j,l Ei (\ axi h,k
where s =
agl
-D,
aAiS
t f,(xi) ° D,, and r = Z9 gt(xiAik) o D) on U C E.
,
ALGEBRAS, CURVATURE AND STRUCTURE
5.
221
1-jet extension of the natural,representation and curvature, Utiyama's theorem
Let us suppose that the manifold V is orientable and endowed with an orientation whose volume element is w. A gauge-invariant field on the fibre bundle of connection ;r: E , V can be defined as a variational problem (on the 1-jet fibre bundle J'(E)) with a lagrangian density Yw admitting the natural representation {D8} of the gauge algebra F as a subalgebra of the algebra of infinitesimal internal symmetries [1], i.e., j'(Dg)2' = 0 for every s e F. A natural question is now trying to characterize the lagrangians £ satisfying the above said condition. Settled and solved (locally) the problem by Utiyama [8], we want to see, in this section, its geometrical meaning from the point of view of previously introduced notions. In this sense we shall proceed as follows. The curvature 2-form can be interpreted as a mapping Q : J'(E) -- A' T*(V) ® L(P) by the rule : Q(jc) = (Qo),,
.
This mapping will be called curvature mapping. Proposition 5.1. The curvature mapping Q: J'(E) -- A2 T*(V) ®L(P) is an epimorphism of fibre bundles -on. V, that is, Q is a differentiable projection making the following diagram commutative : J'(E)
AZT*(V) ®L(P)
Proof. It is obvious that Q makes the above diagram commutative. Now taking natural local coordinates (x,, Air, p1 ) and (xi, Rimj), l < m, on J'(E) and AZT*(V) ® L(P), respectively, the mapping Q can be written, by using (3.2), as
xi = x2 ,
RLmj = Pmij - Pcmj
- 2 Z cnk(AahAmk - AmhAlk) 1
Thus Q is differentiable. Now let a point (x°, be given in AZT*(V) L(P), let us consider the local section a : V - E defined by the equations Al Zen aJ,(x,,,, - xm), where a Z are arbitrary constants if m < l and aZ if m > 1. a defines a l-jet j'a at x such that Q(j'xa) _ (x°, R mj). This proves that 12 is an epimorphism. q.e.d. On the other hand, s being a given element of the gauge algebra F, let Xg be the vertical vector field of the vector bundle AZT*(V) ® L(P) such that, for I
every point (w2), and every function f (linear on the fibres) of AZT*(V) ® L(P), one has
222
PEDRO L. GARCIA
(Xsf)(co2).x = -f([s(x), (w2)2,])
where [s(x), (w2)y] is the point in A2T*(V) ® L(P) defined by [s(x), ((o2)z](D, D') = [s(x), (w2),x(D, D')]
Proposition 5.2. The mapping s e I' --+ D3 is a homomorphism of Lie Aalgebras. Proof. Let g e A and let f be a function of A2T*(V) OO L(P) linear on the fibres. Then (w2),x]) (X88f)(w2) _ -f([(gs)(x), (W2)a]) = _ -g(x) f([s(x), (w2),x]) = g((w)z) - (X3f)(w2)"' = ((gX )f)(w2).x
This proves that s e 12 ti X3 is A-linear. Now the equality X[8,8,] = [X8, X8,] can be proved in a way analogous to the proof of Theorem 2.1. q.e.d. In the local coordinate system (x1, Rlm j) in A2T*(V) OO L(P) considered before, the vector field X8 is given by (5.1)
XS = -I,m J,h,k ChkfhRlmk
a
aRamj
where s = E; fjDj. Thus we have two new representations of the gauge algebra r: the 1-jet extension s e r --+ j'(D8) of the natural representation and the representation s e 12 + X8 which we have just defined. The first is a representation of r as a real Lie algebra, and on the other hand the second is a representation of r as a Lie A-algebra. This is the essential difference between both representations. Now Utiyama's theorem can be stated as follows. Theorem 5.1 (Utiyama). A function _W: J'(E) ---*R is gauge-invariant (i.e., it is invariant by the real Lie algebra {j'(D8) I s e I'}) if and only if
Y=YoS2, where 2: A2T*(V) ® L(P) -> R is a function invariant by the Lie A-algebra {X8 I s e I'}, and Q is the curvature mapping. Proof. If s e r has the local expression s = F, j f1(xz)D1, then from (2.1) it follows that
f,D, + E where
a2f
a J Di1 + c1
E,> axaaxm 'Dam,
ALGEBRAS, CURVATURE AND STRUCTURE
D9 = Dig =
k?'
i
a 8A i;
+
h,k,l
k Ch,Aih
k
C
;Ptmrr,
223
a aptmk
a aPSik
Dim; _
a
a
aPimi
apmi;
Thus 2 is gauge-invariant if and only if F is a solution of the system of (local) equations
D j2 = Di J.F = Dtml.F = 0 .
A simple calculus proves that the most general solution of this system is a function 2 = 2(xi, Rimi) where 2 satisfies the conditions (5.2)
1,m,1,k
CkRlmk
a2 aRimf = 0
.
The local coordinates (xi, Ai;, Pi;k) and (xi, Rimj) are defined on J'(E)U and (AZT*(V) (& L(P))U respectively, where U is an arbitrary open set of V with local coordinates (xi) on which the fibre bundles under consideration trivialize. This, together with the fact that Q is a fibre bundle epimorphism, implies that the above (local) conditions on F are equivalent to the (global) condition
Y=2oQ, where 2 : AZT*(V) OO L(P) -* R is a function satisfying the (local) conditions
(5.2). But, by (5.1), this last fact is equivalent, in turn, to the fact that 2 be invariant by the Lie A-algebra {XS Is E P}. Thus the theorem is proved. q.e.d. According to this, gauge-invariant fields on the fibre bundle of connections ;r: E - V can be parametrized by functions Y: AZT*(V) OO L(P) - R invariant by the Lie A-algebra {X,s I s E P}. In particular, it is easy to prove that the following functions are of this type : let p be an arbitrary polynomial of the Weil algebra H(G) of G, and let F: AT*(V) - R be an arbitrary function. We define 9: ((o2),. H F(P(w2).x)
where - is the canonical injection of L(P)-valued forms on V into the 9-valued forms on P. If V is endowed with a pseudo-riemannian metric g we can define a function 2 of the above type as follows. We take as p the element of H(G) defined by the Cartan-Killing metric on 9. Then p((b2),x is a 4-form on T.'(V), from which we can obtain its scalar square with respect to the metric g, that is, Y: (w2) H g(P(a2),., P(w2)x)
This Lagrange function has been the almost exclusively used one, up to now, in the physics of free gauge-invariant fields.
Z24
PEDRO L. GARCIA
6.
Gauge algebras and external symmetries
To every classical field defined on a fibre bundle 7r: E -- V by a lagrangian density 9(o one can associate the extension of real Lie algebras : (6.1)
0
) _q"
)
) 27W)
)0
where -9 are the 2r-projectable vector fields on E such that L11(D)2'w = 0, _qv is the ideal of vertical vector fields in -9, and 2r(-q) is the image of _q by the projection ir. w and 2r(-9) are respectively called "infinitesimal internal symmetries" and "infinitesimal external symmetries" of the field under consideration [1]. Now an important question in classical field theory arises : how to determine all possible lagrangians such that their corresponding extension (6.1) (or part
thereof) is given in advance. The problem of Utiyama which we have dealt with in the preceding paragraph, is a typical example of this situation. Nevertheless, in more general situations, it is not likely that such a simple solution can be obtained. In spite of this, it seems that the following general question is a good starting point : Suppose, as it often occurs, that -9v is the natural representation {Ds} of the gauge algebra P. What is the maximal Lie algebra -9 having _qv as an ideal, and what is the corresponding Lie algebra of infinitesimal external symmetries? By definition, _q is the idealizator of {Ds} in the Lie algebra of vector fields on the fibre bundle of connections E. The following result gives a very simple answer to this question. Theorem 6.1. The idealizator _q of the natural representation {Ds} of the gauge algebra P in the Lie algebra of vector fields on the fibre bundle of connections ir : E -- V coincides with the Lie algebra . of hamiltonian 2r-projectable vector fields on the symplectic manifold (E, Q). One has the extension of real Lie algebras : (6.2)
0 - {DS} -) - X (V)
)0
where .1"(V) are all vector fields on V. Proof. First of all, {Ds} is an ideal of _-Yn, for, if DS E {Ds}, Ds, e _-Y. and f is a differentiable function on V, then one has [D3, Ds-]f = D.,D8,f - DS,Dsf = 0 ,
from which, by the remark to Theorem 4.1 and by Theorem 3.1, one gets C -9. On the other hand, it is obvious that [Ds, DS,] = D{,,) e {D3}. So . X(V), which implies {DS} = ker 2 x. Now we go to prove that _-Y _ -9 and our result follows. Let ao : V -- E be a connection on the principal bundle P. A vector field D on V being given, let us consider the section sD e T (L(P)) defined by
225
ALGEBRAS, CURVATURE AND STRUCTURE
SD(Ux) _ (UO)xDx - UxDx
,
UxEE.
We want to prove that sD has a hamiltonian 2r-projectable vector field D such that 2r(D) = D. Indeed, let (xi, Ai j) be the local coordinate system on Er, C E defined in § 2, and let us suppose that ao : V -> E and D have the local expression A ij = xn) and D = Ei gi(x, f ij(x, xn)a/axi with respect to (xi, Ai j). Then sD has the corresponding local expression SD = E j c j(xi, Ai j) o D j, where
Oj(xi, Aij) = Ei (fzj - Aij)gi j
Now a simple calculation proves that the equation iDQ = dSD has as a (unique) solution the vector field D on E whose local expression is a0j + axi
D+
2
> CkxAik ck
a,Ek Chk(AZkAik - AikA1k)91
.
a
aAij
q.e.d.
In order to illustrate the way in which the above result can be employed, let us consider the following. Example. Let p : P -> V be the trivial principal bundle P = R2 X U(1), and let cu = dx1 A dx2 be the euclidean area element of R2. By Utiyama's theorem, a -classical field £w on the fibre bundle of connections r : E -> R2 of P admits
{DS} as internal symmetries if and only if £ = P o Q, where Q : J'(E) -> A2T*(R2) is the curvature mapping and P is an arbitrary function on A2T*(R2).
Now the question is : what is the relation between P and the external symmetries 2r(-9)?
By Theorem 6.1, supposing that the extension corresponding to £w is of type (6.3)
0
7c(-q)-)0
we could start our discussion by considering the case of maximum symmetry : -9 = Win, 7r(-9) = X(V). So we must find all functions £° = P o Q such that, for every D E Win, Lj,(D)'cu = 0, that is, (6.4)
j'(D)Y + Y div 2r(D) = 0
.
By identifying A2T*(R2) with R2 X R by means of the area element cu, (x1, x2, f12)
becomes a (global) coordinate system on A2T*(R2), f12 being the natural coordinate on R, and so we can write P = eL (x1, x2, f12). Now by imposing the invariance condition under where T is the (abelian) Lie algebra of translations of R2, one has P = P(f12) and (6.4) becomes
226
PEDRO L. GARCIA
d! df12
- 2lIdiv 7r(D) = 0 .
Taking D E A e. such that div 7r(D) # 0, one gets 2 = const. f 12. This gives us a trivial lagrangian which does define no variational problem. Thus the maximum
Lie algebra of infinitesimal symmetries must be, in this example, the set of vector fields D on R2 such that div D = 0, and the corresponding lagrangian is P = 2 c Q with 2 = 2(f 12) an arbitrary function. Now we observe that an essential point in the above argument is that the Lie algebra of translations IT is a subalgebra of 7r(-9). With this in mind, the rest of discussion can be carried over without difficulties.
Let us now go back to the general case. Another important question in classical field theory is to determine the splittings of the exact sequence (6.1).
In particular, this allows us to fix the "external Noether invariants" of the field (energy, linear, and angular moments, etc.). In the proof of Theorem 6.1
we see how a connection a.: V , E on the principal bundle P determines a splitting (of real vector spaces) D E 7r(-q) , D of (6.1), D being the hamiltonian vector field corresponding to the section sD defined by the formula (6.5)
sD(a.) = (o'o).D. - a,;D. ,
EE.
In general, this splitting does not preserve Lie brackets. Now an interesting question is to characterize those connections whose corresponding splittings preserve Lie brackets. This would give us in particular, a differential-geometric procedure of "mixing" gauge algebras and external symmetries very close to the physical problem. The following result gives an answer to this question. Theorem 6.2. Let ao : V --> E a connection on the principal bundle P with 2-form of curvature Q, let D, D' be two vector fields on V, and let SD, SD, E T be the sections defined by D, D' according to the above formula (6.5). Then (6.6)
S[D D'] _ {sD, SD,} - Q(D, D') .
Thus a'o defines a splitting (of real Lie algebras) of the exact sequence (6.1) if and only if Q(D, D') E I'o for every pair D, D' of infinitesimal external symmetries. In particular, this is true if 6o is a flat connection. Proof. It will be enough to compute the Poisson bracket {sD, sD'} having in mind the local expression for the 2-form of curvature of a connection. For the last part, it is enough to remember Theorem 4.1(c). q.e.d. According to this, existence of splittings of (6.1) induced by connections should, in general, influence the principal bundle P and, eventually, the splitting itself. Thus, for example, if the splitting is induced by a flat connection, ao, and the base manifold V is simply connected, then P must be isomorphic with the trivial bundle V x G and 6o is isomorphic with the canonical flat connection on V x G [5]. Then the exact sequence (6.1) has, up to equivalences,
ALGEBRAS, CURVATURE AND STRUCTURE
227
a unique splitting, which coincides, in the particular cases dealt with in physics,
with the "trivial combination" of gauge algebras and external symmetries. Simmilarity of this result and O'Raifertaigh's theorems [7] forbidding nontrivial combinations of "internal (finite-dimensional)" and "space-time" symmetries is well apparent. This remark could be a starting point for a differential-geometric approach to this interesting topic for infinite dimensional Lie algebras of the type which this paper deals with. References [1]
P. L. Garcia, The Poincare-Cartdn invariant in the calculus of variations,
Symposia Math. 14 (1974) 219-246. , Connections and 1-jets fibre bundles, Rend. Sem. Mat. Univ. Padova 47 (1972) 227-242. [ 3 ] R. Hermann, Lie algebras and quantum mechanics, Benjamin, New York, 1970. [41 B. Kostant, Quantization and unitary representations. Part I: Pre-quantization, [2]
Lectures in Modern Analysis and Applications. III, Springer, Berlin, 1970, 87-208.
[51 J. L. Koszul, Lectures on fibre bundles and differential geometry, Tata Institute of Fundamental Research, Bombay, 1960. [ 6 ] A. Perez-Rendon, A minimal interaction principle for classical fields, Symposia Math. 14 (1974) 293-321. [7] L. O'Raifeartaigh, Lorentz invariance and internal symmetry, Phys. Rev. 139 (1965) B1052-B1062.
[81 R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. 101 (1956) 1597-1607. [ 9] C. N. Yang & R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96 (1954) 191-195. UNIVERSITY OF SALAMANCA, SPAIN
J. DIFFERENTIAL GEOMETRY 12 (1977) 229-235
CLOSED 2-FORMS AND AN EMBEDDING THEOREM FOR SYMPLECTIC MANIFOLDS DAVID TISCHLER
The existence of universal connections was shown by Narasimhan and Ramanan [5], and Kostant [3] showed that any integral closed 2-form is the curvature form of a connection on some circle bundle. These results can be combined to show the existence of a universal closed 2-form with integral periods. In this paper we will use the symplectic structure of a complex projective space to give an elementary proof of this result ; the precise statement is given in Theorem A. The result of Kostant is in fact a corollary of the existence of a universal closed 2-form, as is indicated below. Another immediate corollary
of Theorem A is the result of Gromov [3] that closed symplectic manifolds can be symplectically immersed in CPn, for large enough n ; see Theorem B. First we indicate why the proof which we are going to give here is a simple and natural generalization of an elementary fact about exact 2-forms. Consider the standard symplectic form 0 = E%=1 dxidyi on R'n. Any exact 2-form on a manifold M can be induced from 0 by a mapping to R211 for some n, since any exact 2-form on M can be written in the form Ei=1 d f i A dgi, where f i, gi are real valued functions on M. CPn has a symplectic structure 0o which is locally given by 0o = E%, dxi A dyi. Furthermore, CPn is the 2n-skeleton of an Eilenberg-MacLane space of type K(Z, 2). It is thus natural to expect that any closed 2-form with integral periods can be induced from 0, by a map to CPn, because there is some map to CPn, for large n, which pulls back QQ to within an exact 2-form of the given closed 2-form. The only complication that is met in CPn to adjusting the map to account for the exact 2-form is that, unlike in R2n, the symplectic charts on CPn have finite radius, so the f i, gi's utilized would have to be bounded. The proof we give of Theorem A depends only on estimating the bounds on f i, gi as n becomes large. A closed k-form on a manifold M will be said to be integral if its de Rham cohomology class is in the image of the canonical coefficient map H'`(M ; Z)
->H'(M;R).
Complex projective space CPn has a Kahlerian structure, and we will denote its Kahler form by Qo. The 2-form On can be chosen to represent a generator in the image of H2(CPn ; Z) -> H2(CPn ; R), and we can assume that i*(Qo+k) = On where i is the standard inclusion of CPn in CPn+k Communicated by R. Bott, May 24, 1975,
230
DAVID TISCHLER
Theorem A. Let M be a closed manifold, and Q an integral closed 2-form on M. Then there exists a map f : M -> CPn, for n sufficiently large, such that D.
Since Qo is the curvature form of a connection on the canonical S' bundle over CPn, a map to CPn which induces a closed 2-form also induces an S' bundle. Hence we obtain Theorem (Kostant [3]). Every integral closed 2-form is the curvature form of a connection on an S' bundle. Definition. Let (M, Q') and (N, S.) denote two manifolds M, N with symplectic forms Q', Q respectively. A map f : M -> N will be called a symplectic map from (M, Q') to (N, D) if f *(,Q) = Q'. Definition. Given a manifold M and a symplectic structure (N, Q), a map f : M -> N such that f *(Q) is a symplectic form on M will be said to be transverse to the symplectic form Q. Any submanifold M of CP11 such that the inclusion i : M-> CPn is transverse to Sao will support a symplectic structure, namely i*(SQo), which is an integral closed 2-form. The converse is also true and resembles Kodaira's embedding theorem, but with Kahlerian weakened to symplectic. Suppose (M, SQ) is a symplectic structure. If S. is an integral closed 2-form, then by Theorem A there is a map f : M -> CPn such that f*(SQo) = D. Since SQ is a nondegenerate 2-forms f is automatically an immersion. This yields the result :
Theorem B (Gromov [2]). If SQ is a symplectic structure on M, and SQ is an integral closed 2-form, then there exists a symplectic immersion of M into CPn for sufficiently large n. Remark. This result can be improved to yield symplectic embeddings in the following way. Assume n is large enough so that the immersions can be approximated arbitrarily closely by embeddings. Choose an embedding g : M CP11 so that g*(SQo) is close to Q. By Moser's theorem on the stability of symplectic forms [4], we conclude that there is a diffeomorphism F of M to itself such that F*(g*(SQo)) = Q. Hence g o F: M -> CPn is the required symplectic embedding. Corollary. Given a symplectic structure (M, SQ), there is, for large enough n, an embedding f ; M -> CPn transverse to Qo, such that f *(SQo) can be made arbitrarily close to SQ in the following sense: given a norm I on closed 2forms and an e > 0, there are a real number k and an embedding f such that Ilk . f*(Qo) - D 11 < s. Proof. Choose a collection of integral closed 2-forms cei, 1 < i < d, which
define a basis for H2(M; R). Any symplectic form 0 can be written as Q = Zf=1 riai + dw for some 1-form w and real numbers ri. Choose rational numbers qi such that SY = Za=1 giai + dw satisfies 11 SQ - SYIJ < s. There is an integer D such that DSO' is an integral 2-form. By Theorem B, DSO' = f *(SQo) for some embedding f : M -> CPn. The corollary follows by setting k = 1 /D.
CLOSED 2-FORMS AND AN EMBEDDING THEOREM
231
Before beginning the proof of Theorem A, we need to establish several notations. Cn will denote n-dimensional complex space, < , > the usual Hermitian inner product on Cn, and I I the corresponding norm. We will consider CPn as the complex lines in Cn+1 passing through the origin, and also as the quotient space of the unit sphere Szn+' in Cn+1 by the action of the complex numbers of norm equal to 1. Given two points p,, p2 in CPn we denote by a(p p) the angle between them viewed as real two-dimensional planes in Cn+1, (cos a = I I / Q Pl Pz
where we are now considering p p, as points in
Cn+1)For
each p in CPn, we make a choice of x in
Sln+l
which represents p. Where it creates no confusion we will speak of x in CPn, and where necessary we will denote the class of x in CPn by Ex]. For each p in CPn the above choice of x allows us to choose a complex hyperplane T, in Cn+1 which passes through x and is orthogonal to x with respect to the Hermitian metric. T, can be identified with the tangent space to CPn at [x]. Let D, be the subset of CPn consisting of those complex lines in Cn+1 which intersect T,. The mapping from D, to T, given by sending a point in D,, to its point of intersection with T, will be denoted by S(x). For a > 0, T,x(s) will denote all points y in T3, such that ly - xj < s, and S-'(x)(T,(s)) will be denoted by V(x, s). Let z = (z0, , zn) be complex coordinates on Cn+'. We can think of Cn as all points z in Cn+1 with z, = 1. Let Bn(r) denote all points (z,, , zn) in Cn such that Ei=, zizi < r2 One can identify Tx with Cn by choosing some unitary transformation of Cn+1 which sends x to (1, 0, , 0) in Cn+1 Composing this map with the mapping (z , zn) -+ (1 + i=1 (z,, zn) yields a diffeomorphism H: T, -+ Bn(1). Consider the closed 2-form Zi, dxi A dyi on Bn(1) where zizi)-1/2
zi = xi +yi. One can show that the Kahler form Qo on D, satisfies Qo = S*(x) o H*(x)(7r-1 (1
dxi A dy), by using the fact that Q = (i/27r)6a log Ei=1 zizi) on the hyperplane z, = 1 viewed as a holomorphic cross-sec-
tion of the canonical line bundle over CPn; see Chern [1] for details of the Kahler structures of CPn. One can think of H(x) o S(x) : D., -+ Bn(1) as a symplectic chart for CPn. There is a natural inclusion i : CPn -+ CPn+1 given by the inclusion i : Cn+1 C11+2 defined by identifying Cn+1 as the first n + 1 coordinates of The choices made above can be made compatible with the inclusion of CPn in CP11 +1 in the following sense. For a point [x] in CPn we can choose Tx, D,, S(x), H(x) as above. We can also let i(x) G Cn+2 represent 1[x], and we have T 3 , = Ti(,, fl Cn+1 and S(i(x)) o I = i o S(x) : D., -+ Ti,,,. One can also choose H(i(x)) so that H(i(x)) o i = i o H(x) : T, - Bn+1(1). With these choices, Cn+2.
11t i=11 dxi A dyi = ((H(i(x)) o S(i(x)))-1)*(Qn,1)
DAVID TISCHLER
232
on B11+1, and also
1 E dxi A dyi = 7ri ((H(x) - S(x))-')*(Qo) it i_1
where it1 is the projection of B"+1(1) onto B"(1) defined by the projection of C11+1 onto the first n coordinates.
The function f will be constructed in stages; the jth stage will be denoted f,, where 0 < j < p for some p to be chosen later. Choose f,: M - CP"z for n sufficiently large, so that fo (S2) and SZ are cohomologous. This can be done since CP"z can be taken to be the 2n-skeleton of an Eilenberg-MacLane space of type K(Z, 2). Hence SZ = f o*(Q) + dw for some 1-form w on M. We need a couple of lemmas before we can construct the f j's. Lemma 1. Given R > a > 0, there exists a 6 > 0 such that Proof of Theorem A.
V(x, s, 6) = {y e CP11 I a(y, x') G 8 for some x' E V(x, s)} C S-1(x)(T,(R))
Furthermore, 6 can be chosen independently of n. Proof of Lemma 1. The lemma follows easily from the facts that Tz(s) C
Tx(R) and that, for 0 < 0 < -7r, {y e D,x I a(x, y) < 0} = S-1{z E T,x I cos 0 < zI-'J .
From now on we fix a choice of a, R, 6 satisfying Lemma 1. We also choose
a p> 0 such that 1- p> cos28. Lemma 2.
Given a 1-form w on a closed manifold M, a finite open cover
{W1} of M, an R > 0, and a p such that 1 > p > 0, there exist real valued functions hk, t, 1 G k G p such that (1) Ek=1 A, A dtk = dw, (2) {W'}, (3) (4)
each pair (hk, tk) has support contained in some element of the cover
r[ k=1 (1 + K2(hk2 + tk2)) < 1 / (1 - p), where K2 = 1 + R2, hk2 + tk2 + R2/(1 + R2) < 1. Proof of Lemma 2. There exists some choice of functions hk, tk, 1 < k G p, such that Elk'=1 dhk A dtk = dw. This can be seen by choosing a partition of unity {cpk} subordinate to some finite coordinate cover {Ui} of M. Then dw = d(E cpk(O), and d(cpk(o) _ E,71=, dhk A dtk for each k and some choice of hi, tk with support in Ui, where m = dimension of M. Hence (1) can be satisfied. Now choose a partition of unity {?i}, 0 G i G c, subordinate to {Wi}. Then k=1
dhk A dtk = E E E d(Vihk) A d(Y' jtk) k=1 j=1 i-1
and (2) can also be satisfied by taking the Tih,c as the h,c's and the 'jtk as the
CLOSED 2-FORMS AND AN EMBEDDING THEOREM
233
tk's. By replacing hk and tk by N copies of hk/N and tk/N respectively, and using the fact that lim,__ (1 + n-2)n = 1, we see that we can choose the hk's and tk's to satisfy condition (3). By a similar argument, the hk's and tk's can be chosen small enough so that condition (4) is satisfied as well, and the proof of the lemma is complete. M has an open cover given by {fo-'(V(x, e))}, [x] E CPn. Fix a finite subcover {Wi} of this cover. Fix a choice of {hk, tk}, 1 < k < p, satisfying Lemma 2 applied to our fixed choices of e, R, S, p, {Wi}, and such that P
1 -E Wk Adtk)=dw
where do) - Q - fo (Q )
For each k, 1 < k < p, we choose a Wk in the cover {Wi}, such that the support of hk and tk are contained in Wk. Recall that Wk = fo'(V(xk, e)) for some xk E Cn+'
For each j, 1 < j < p, let us assume the two induction hypotheses (i) There is a map f,-,: M- CPn+>-' such that f*-1(Q0+j 1)
= fa
:
1 E1 (dhk) A (dtk) 7C k=1
(ii) fi(W5) C V(x;, R), for all i < j -
1.
If we show that (i) is true for 19, we will be done since fP*(Q0+P) = fa (Qo) +
I
E (dhk) A (dtk) = fo (Q0) + do) = Q
7C i=1
.
We already have (i) and (ii) satisfied for j = 1 ; (i) is true vacuously and (ii) follows from the fact that V(xf, e) C V(xf, R). Hence it suffices to show that given f;_, satisfying (i) and (ii) there is an fj satisfying (i) and (ii). Define ff as follows : (a) On M - W p set f ; = i o f _ , where i : CPn+>-' --> CPn+ f is the inclusion. (b) On W;, we define first a map gf : Wj --> B"+'(1) given by 7r,gf =
H(x' .) oS(x,) o ff_, with values in Bn+>-'(1), and by 7r2gj = hj + ti with values in B'(1), where 7r 7C2 are the projections of B"+'(1) onto Bn+>-'(1) and B'(1) respectively, induced by the projections of Cn+> onto its first n + j - 1 coordinates and last coordinate respectively. We can now define fj = S-'(i(xj)) o H-'(i(xj)) o gj, (we are taking the choices of H(x), H(i(x)), to be compatible in the sense described just before the beginning of the proof of Theorem A). By property (4) of Lemma 2 we have that I (7r2gj) 12 < (1 - R2/(1 + R2))
in B'(1). By induction hypothesis (ii) applied to ff_, and by the fact that
H(xj)(T,x.(R)) C Bn+>-'R(1 + R2)-"2 we have that 17r,(gj)J2 < RZ1(1 + R2) in Bn+J-'(1). Hence we can conclude that gp : Wf --> Bn+'(1) is well defined,
DAVID TISCHLER
234
and consequently that f j is well defined on W j. By Lemma 2, part (2), we can conclude that f j is well defined on all of M. On W j
f;
!1
_
g*
n+j
1 n+j
7r
i=1
i=1 1 n+j-1
i=1
n+
=
n+j
(7r1g;)*(1 E dxi A dyi) + (7r2gj)*(- E dxi A dyi
_ (H(xj) o S(xj) o f j_1)*(( 7r
=
- i.1E dxi A dyi \7r 1
i
(+j) = g*((H(i(x) ° S(I(x)))-1\* /
f,_1(S*(xj) ° H*(xj)(1 7r
1
i=1
fj*-1(Qo+s-1) + 1 (dhs A dt;)
dxi A dyi) + -(dhj A dtj) 7r
dxi n dyi 11 +
1
(dh, 3 A dt,)
7r
.
7r
This equality follows from the compatibility conditions on H(x j) and H(i(x j)) discussed just before the beginning of the proof of Theorem A. Hence we have shown that induction hypothesis (i) is satisfied for f j. Therefore we mill be done if we can show that f j(Wk) C V(xk, R) for all k > j. For any x E Wk and 0 < i < j, set Ai = S(xi+1)(fi(x)) and Bi = S(xi+1)(fi+1(x)). We consider the Ai, Bi as all contained in Cn+J, (note that Ai is a scalar multiple of Bi_1). We now add another induction hypothesis for each j, 1 < j < p, (iii) =0 for all i' < i < j - 1. If hypothesis (iii) is true for j - 1, it is seen to hold for j, since B; - Aj is
perpendicular to C11+; in Cn+;+1, using the construction of f j as above, and by the compatibility conditions given before the proof of Theorem A. (Hypothesis (iii) is vacuously satisfied for f o.) Given Ai, Bi as above and our fixed p, we will show that cos' of j-1 > 1 - p, where cri = T([A,], [Bi]). We have (I f Cos, 01i = ( J J 2 =
AI. Bid 1/
1
by induction hypothesis (iii), and this expression is equal to (cost Ti _
) I A 1 12 / I Bi 12.
Since IBi12 = IAiJ2 + IBi -Ail' and JAil > 1, we have that lAi12/1BiI2 > 1/(1 + JBi - Ai12). However IBi - Ai12 < K2(hk2 + tk2) with K2= 1 +R2, by the construction of fi+1, the definition of the map H(xi+), and the fact that Bi and Ai are in Tx1}1(R). Hence we have cost ai > cost 01i and so cost Q' j -1
(1 + K2(hk2 + tk2))-1,
j-1 F1f (1 + K2(h k2 + tk2)-1) > 1k=1
which is greater than 1 - p by part (3) of Lemma 2. Since we chose p such
CLOSED 2-FORMS AND AN EMBEDDING THEOREM
235
that 1 - p > cos' 8, we have cr;, < 8. Since Ao is contained in V(xk, s), we get that B1_1 is contained in V(xk, e, 8) which is contained in V(xk, R) by Lemma 1. Hence fi(x) is contained in V(xk, R) for all x in Wk. This shows that fj satisfies induction hypothesis (ii), and the proof of Theorem A is complete. References
[1] [2]
S. S. Chern, Complex manifolds without potential theory, Van Nostrand, Princeton, New Jersey, 1967.
M. L. Gromov, A topological technique for the construction of solutions of diflerential equations and inequalities, Actes Congres Intern. Math. (Nice,
1970), Gauthier-Villars, Paris, No. 2, 1971, 221-225. B. Kostant, Quantization and unitary representations, Lectures in Modern Analysis and Appl. III, Lecture Notes in Math. Vol. 170, Springer, Berlin, 1970, 87-207. [ 4 ] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286-294. [ 5 ] M. S. Narasimhan & S. Ramanan, Existence of universal connections, Amer. J. Math. 83 (1961) 563-572.
[3]
QUEENS COLLEGE, CITY UNIVERSITY OF NEW YORK
J. DIFFERENTIAL GEOMETRY 12 (1977) 237-246
THE HOMOLOGY OF SUBMANIFOLDS OF COMPACT KAHLER MANIFOLDS GERALD LEONARD GORDON 1.
Introduction
In this article we study certain topological properties of submanifolds of compact Kahler manifolds. Specifically, let i = X C Y be the inclusion of a compact manifold X of complex dimension n into a compact Kahler manifold of complex dimension n + q. Let I : HP+2q(Y) , HP(X) be the map given by transverse intersection, where the coefficients are in K, a fixed field of characteristic zero. Then we ask when do we have the decomposition HP(X) = Ker i, O+ I(HP+2q(Y)) such that if p = n, each direct summand is nondegenerate with
respect to the intersection pairing. In cohgmology this states that HP(Y) i*HP(Y) O+ RHP+2q-'(Y - X), where R is the Leray-Norguet residue operator.
If n = 1, then a corollary of this result is that if X, and X2 have this decomposition in Y, i, and i2 are the inclusions, and Ij is the intersections, then the following diagram
H,+2q(Y) -± 121
HI(X2)
1(h)* (I-Z
H1(Y)
commutes when restricted to coimage I, fl coimage 72, i.e., to the set of 0 for j = 1 and 2. r e H1+2q(Y) such that I j(r) In this article we shall show for n = 1 and 2 that this decomposition exists for p = n, as well as for submanifolds of complete intersections of CPN. However, for p > 3 and any n > 2, q > 1 we shall give counterexamples. This problem arose from questions about the local invariant cycle problem ; cf. Griffiths [5, p. 249]. Namely, this decomposition for n = 1 = p is precisely what one needs to prove the problem when one has 2 surfaces intersecting in a double curve [5, p. 292]. In Gordon [4], it is shown that this decomposition for n = 1 = p is essentially what proves the local invariant problem for Kahler surfaces. Furthermore, these counterexamples to the decomposition allows us to construct projective varieties which cannot be embedded in a oneReceived February 24, 1975, and, in revised form, July 3, 1975. This work was
partially supported by NSF contract GP 38964A #1.
238
GERALD LEONARD GORDON
dimensional analytic deformation whose generic fibre is a nonsingular compact Kahler manifold ; cf. [4]. In § 5 we pose the analogous question about schemes, which should, if true, have applications to studying the monodromy for schemes, over arbitrary algebraically closed fields. The author would like to thank the referee for pointing out a mistake in the original proof of Corollary 3.2. 2.
Definition of 4P(X, Y)
2.1. In this section Y will always denote a nonsingular connected, compact Kahler manifold and X a nonsingular connected, compact submanifold of complex dimension n, where the complex codimension of X in Y is q. i : X C Y will denote the inclusion map. The Poincare dual class of 0 # [X] E H2n(Y) will be denoted by Six E H2q(Y),
where the coefficients are in K, a fixed field of characteristic zero. Then we have a mapping AQx : HP(Y) - HP+2q(y)
2.1.1. Definition. Let AP(X, Y) denote the proposition that Ker {ASQx : HP(Y) - HP+22(Y)} = Ker {i* : HP(Y) - HP(X)}.
Let I : Hp+zq(Y) - HP(X) denote the map given by transverse intersection ; it is the vector space dual of the Thom-Gysin map of the normal bundle of X in Y. 2.2.
(2.2.1)
(2.2.2)
Proposition.
AP(X, Y) = HP(X) = Ker i* + I(HP+2q(Y))
AP(X, Y), A,.-P(X Y)
Proof of (2.2.1).
jHp(X) = Ker i* O+ I(HP+2q(Y)) , H2,i_P(X) = Ker i* EE I(H2n-P+2q(Y))
In cohomology, we have the following communative
diagram :
Hzn_P(X) -> Hzn-P(o Dx l j
Dy1
HP(X) -> HP+2q(y)
PI HP(Y)
where Dw denotes Poincare duality in W, and I* is dual (as vector space) of I. Thus applying Hom to the above diagram, where we identify HP(X) (HP(X))* = Homx (HP(X), K) via integration, we get
HOMOLOGY OF SUBMANIFOLDS
239
Hen-P(X) K- Hen-P(Y)
Dxll
I? Dy
HP(X) * Hp+2q(Y)
I Y a HP(Y) where fl Qx is cup product. Then AP(X, Y) implies that im ix = im fl Qx. Thus, if a E HP(X) with i*a # 0, then i*a =n Q1(S). But by the communative diagram, i*I(p) = i*a, hence I(s) - a E ker i*, i.e., a = r + I(p)
for some r E Ker i*. Proof of (2.2.2). Suppose we have AP(X, Y) and A27L_P(X, Y). If rP E HP(X) with rP = I (rP+2q), then by the above diagram i*(Dy(rp+2q)) = Dx (rp) E
Hen-P(X), and A2 _P(X, Y) implies that AQx(Dy(rp}2q)) # 0. But by the first communative diagram in the proof of (2.2.1), we have AQx(Dy(rp}2q)) # 0 i*rp # 0. Hence HP(X) = Ker i* Q I(HP}21(Y)). By duality we have the direct summand decomposition for H27t_P(X). The converse of (2.2.2) is clear. 2.3. Proposition. If X is a positive hypersurface of Y, then AP(X, Y) is
true for all p. This is an immediate consequence of the hard Lefschetz theorem. 2.3.1. The difficulty is that when one wants to work with problems as the local invariant cycle problem, one wants to apply AP(X, Y) when X is a hypersurface which comes from a monoidal transform, and hence is very negative. 3.
Study of AP(X, Y)
3.1. Proposition. Let PP, denote he primitive cohomology of HP(W ; C) for any compact K2hler manifold W. For i : X C Y, if i*PY C PX for all q < p < n, then AP(X, Y) is true. Proof. We first prove AP(X, Y) for complex coefficients. Consider the following diagram : Hen-P(X
*x
F
C) -
; C) - -
12 Homx
H2n-P(X ; C) D, 12
- HP(X ; C)
Homy J-I2
H2n_P(Y ; C)
I2 Dr HP+2q(Y; C)
GERALD LEONARD GORDON
240
where Hom is vector space duality via integration, and * W is the usual real star operator on forms on a manifold W, which induces an isomorphism on means harmonic forms ; *W is complex conjugation followed by *W. The the diagram commutes where Hom o WW = *w for W compact follows from the definition of *W and the fact that
aA9= Ja. p
JW
Fw(P)
Furthermore Hom$ is natural in the sense that if 0 # a) e H"-P(X) fl Im i*,
then i*(Homx)-1(a)) # 0. To see this, let w = i*W' and a = (Hom1)-1((y). Then
f
Z«(a)
' = f i*(w') = f m= (Hom$)-1(w)(a) = 1. a
Hence i*(a) # 0, as it has nonzero periods. The converse is also true in the sense that if a e H2i,_P(X) with i*a # 0, then the projection of Hom$ (a) onto the subspace Im i* is nonzero. Thus by the commutative diagram to show AP(X, Y) it suffices to show that
if i*w # 0, then ,QX(w) # 0. Since i* respects (r, s) type, it suffices to consider forms of pure type. Suppose i*(D8,P-8 # 0, for ws,P-s E H8,P-8(Y; C). We must show that i*Dxi*(08,P-s # 0. By the above remark, this follows if we show that
Hom$ o DXi*(,s,P-8 e Image i*, i.e.,
*Xi*W1,P-8 e Image i*
.
LrWs-r,P-s-r By the Hodge decomposition theorem, we can write w8,P-8= i*a)s-r,P-s-r E where the (Us-r,P-s-r E Pp 2r. Then, since p - 2r < p < n,
Px 2r by hypothesis. Hence, by a standard identity for compact Kahler manifold (cf. Well [8, p. 23]), r 1)1/2(P-2r) (P-2r+1)
r P
(_ 1)1/2 (P-2r)(P-2r+1)
r!
(n-p-!-r)!
(n - p + r)
LX
(- 1)P(N -
1)3LY(Us-r,P-s-r/
where we have used the identity LXi* = i*LY and the fact that LX and i* are real operators, and where C is the Weil operator. But complex conjugation HP-s-r,s-r(Y ; C) and AY, the sends H8-r,P-s-r(Y ; C) isomorphically onto adjoint of LY, is a real operator, hence @P-r,s-P-r G PP.72r.
Furthermore p < n, so that 0 < n - p < n + s - p = dime Y - p : hence LY Pl
(- 1)1/2(P-2r)(P-2r+1)
r
r!
(n-p+1)!
0 /l
HOMOLOGY OF SUBMANIFOLDS
241
by the hard Lefschetz theorem and the uniqueness of the Hodge decomposition ;
cf. Well [8, p. 75]. But if K is any subfield of C, then H* (X ; C) = H* (H ; K) (x x C, while the operators i* and AQx are integral operators, hence are defined on any subfield of C. Thus Ker i* I HP(Y; C) = Ker AQx I HP(Y; C) implies that Ker (i* I HP(Y ; K)) = Ker A Qx I HP(Y ; K). 3.2. Corollary. For all X and Y, AP(X, Y) is true for p < 2. Proof. For p = 0, PY = Px = 0, while for p = 1, P'x = H'(X ; C).
Similarly H3'0(X ; C) (1 Px = H3'0(X ; C) and the same for H°'2(X ; C). Hence to prove the corollary, it suffices to consider Ply,'. 3.2.1. Lemma. w"' E P'Y' either i*co"' E P'X' or aLx(1) for a E C where 1 is a generator of H°(X ; C). Note. This says that the restriction of a primitive 2-form does not split up into two nontrivial components in the Hodge decomposition of the subspace. Proof. If it is not so, then we have aLx(1) + mx where 0 * wx E PX' and 0 # a e C. Then aLx(1) = i*(aL,(1)), hence 0 aLy(1)) E P'X'. But by the uniqueness of the Hodge decomposition, aLy(1))
0 P". Thus we have a nonprimitive form on Y whose restriction to X is primitive, which is impossible. q.e.d. for Lemma 3.2.1. Suppose we have w E H'>'(Y; C) with w = w'"' + pLY(1) for 9 E C, 1 a generator of H°(Y; C), and m'°' E Ply,'. Then if i*o) # 0, we must show AQ1(w)
0 ; by the diagram in (3.1) and the remarks after the diagram, it suffices to show *xi*(w) E Im i*. If i*o)',' E PX', then by Proposition 3.1 we are done. Hence by Lemma 3.2.1
it suffices to assume that i*w',' = aLx(1). But then i*w = aLx(1) + i*pLy(1) = (a + p)Lx(1), and a-I 3 * 0 by hypothesis that i*c 0. Hence *xi*w _ i*(-(a + Q)LY(1)). q.e.d. for Corollary 3.2. 3.3. Corollary. If Y is a complete intersection in CPN, then AP(X, Y) is true for p < dime Y for any submani f old X in Y. This follows because PY = 0 for p < dime Y. 3.4. Proposition. Let n > 2, and let p and q be fixed such that 3 < p < 2n - 1 and q > 1. Then there exist projective algebraic manifolds X and Y such that AP(X, Y) is false. Proof. The first case to consider is p = 3, n = 2 and q = 1. Let T C CP3 be the nonsingular elliptic curve of degree 3, and let H : Y CP3 be the
monoidal transform with center T. Let X = II-'(T) and i : X C Y be the inclusion, where Y is projective algebraic.
Then by Seminaire Geometrie Algebrique 5, vii i* : H3(X) -- H3(Y) Hl(T) O+ H3(CP3) = K (+ K and H1(Y) = 0. Then by Poincare duality, H'(Y)
= 0. But then i*: H3(Y) _- H3(X) and AQx : H3(Y) -- (H5(Y) = 0) is the zero map, hence A3(X, Y) is false. Next consider p = 3, any n, and q > 1. All we need to do is to take X X CP11-2 and Y X CP11-2 X CP4-1. Then by the Kunneth formula, i* is still an
242
GERALD LEONARD GORDON
isomorphism for p = 3, but dim,, (Ker A Q1) = 2 for p = 3. For p = 4 and n > 3 and any q, we need only consider X X
and Y X CPn-3 X T X CPq-1. Let 0
co e H3(Y), 0
0, while AQ1((o, 0, r, 0) = 0. In general, if p = 2k + 3, for any n and q, take X X X CPq-' and consider ((o, rk, 0) for 0 w e H3(Y) and 0 to get a counterexample to AP(X X CPn-2, Y X X
CPn-3
X7
r e H'(T). Then
i*(c), 0, r, 0)
CPn-2
CPn-2 and Y X
CPn-2
rk e H2k(CPn-3) CPq-1)
Finally, if p = 2k + 4, for any n and q, take X X CPn-3 X T and Y X CPn-3 X T X CPq-1 and (c), rk, r, 0) will give the counterexample. 3.4.1. The counterexamples for p > n arise from the fact that one has an G C. But nco22,-P and *xi*(OP = am22 -P for 0 coP e HP(Y) with i*coP = LX e.g., one could have there is nothing to guarantee that 0)2nX-P e Image i*, Axi*. H2"-P(Y; C) = 0. The basic reason for this is that i*A, For p < n, the problem arises because we no longer have Proposition 3.1.,
i.e., the restriction of primitive forms need not be primitive for p > 2, e.g.,
in our example for p = 3 = n and q = 1 we have X X CP' c Y X CP'. Then H1(Y X CPI; C) = 0, so that H3(Y X CP' ; C) ~ CE) C is all contained in the primitive cohomology. But b1(X x CP') = 2, b3(X x CPI) = 4 and the map
LxxcP1 : H'(X X CP'; C) , H3(X X CP'; C) H3(X ; C) E (H'(X; C) © H2(CP' ; C)) has Lxxcpl(a) = (Lxa, 0, a, 0), Lxxcpi(p) = (0, Lxp, 0, I3) for a, (3 generators of H'(X ; C). Thus, if we take c)2,1 e H2"'(Y X CP' ; C), then i*co2,1 = aLxco'"0 + 72,1 ,1 where 0 a G C, oP3 and m''0 G PX. Then Wxi*0o2°' _(aLxc)'°0 - )72,1), which is not in the image of i* because of the change of sign before )72,1. In homology this states that we have a finite 3-cycle Ti and a nonfinite 3-cycle 72 in a subspace which are homologous when injected into the ambient '72,1 G
space.
4.
Some consequences of An(X, Y)
4.1. Corollary. Suppose A,n(X, Y) is true. In particular, if n = 1 or 2 or if Y is a complete intersection, then Hn(X) = Ker i* QQ I(Hn12q(Y)) ,
Hn(X) = i*Hf(Y) 0
RHn+2q-1(Y - X)
Furthermore, the restriction of the intersection pairing to each of the summands is nondegenerate (equivalently, the restriction of cup product on Hn(X) is nondegenerate on each of the summands). Proof. The decomposition for homology follows from Proposition 2.2 and
243
HOMOLOGY OF SUBMANIFOLDS
Corollary 3.2. The Thom-Gysin sequence in homology for X C Y can be written Hn+2q(Y)
y
Hn(X) y
2q
(Y)
R
Hn(X)
Hn+2q-1(Y)
- X)
Hn+2q-1(Y)
y
where we take vector space duality via integration to get the vertical isomorphisms, c denotes compact support, F denotes closed support and R is the Leray-Norguet-Poincare residue. The duality via integration between homology with compact support and
forms with closed support was proven for q = 1 by Leray [6]. For q > 1, this was done by Norguet. For an exposition of the dualities between homology with compact support and cohomology with closed support, the reader is referred to Fotiadi, et al. [1, part III].
It can be shown, cf., e.g., Poly [3], that every cohomology class a of HF+zq-1(Y - X) can be represented by a closed C°° form of the type 6 A Kx
+
where 6 and
are CW forms with singularities on X. Furthermore
R(a) = [61 X] where 61 X is closed. Hence Image I ^_ Ker r --Image R, so that the decomposition in cohomology follows. The cup product pairing is nondegenerate on each summand because in the proof of Proposition 3.1, we showed if w E i*Hn(Y), then Homx o D1(o)) E i*Hn(Y), but fX w A Homx o D%(w) > 0. Also, if D* = Homx o D%, then
(D*)2 = (- 1)n Id, where Id is the identity on Hn(X), hence this gives the nondegeneracy on
RH11+2q-1(Y
- X).
4.1.1. For n = 1 = q, the nondegenerate decomposition also can be proven by the Poincare complete reducibility theorem : the map I*: HI(X) -' H3(Y) is derived from the map of Albanese varieties with the nondegenerate cup product structure, hence the Poincare complete reducibility theorem states that the image has a direct summand which respects the nondegenerate structure.
4.2. Corollary. Let X;, j = 1, , k, be nonsingular submanifolds of complex dimension 1 in Y, a compact Kdhler manifold of complex dimension 1 + q. Let i : U X> C Y and ij : Xj C Y be the inclusions. If r1+2, E H1+2q(Y)
is such that 0 * r1+2, n x, = r1,; E H1(X;) for j = 1,
, k, then (ij,)*r1,j,
_ (i12)*r1,>2 for 1 G j1 C j2< k. Proof. It suffices to assume k = 2 by looking at the Xj two by two.
Let X = U;-1 Xj, which is a subvariety of Y and i : X C Y the inclusion. In Gordon [1, Chapter 4] it is shown that one has the diagram of exact rows
244
GERALD LEONARD GORDON
H,(X),d
H1+2q(Y)
H1+2q-1(Y - X)
I
H2+1(y X)
H'(Y)
I
i*
H1(Y)
y H1(X)
a* (
I H2+1(Y, X)
where the first row is isomorphic to the second row by either Poincare-Lefschetz duality or by the duality theorem proven in Gordon [3], where a definition of
H1(X), is also given. Basically, H1(X), are those cycles in X over which one can construct "tubes" in Y - X. Thus they are the cycles which lie in the non-
singular part of X or intersect transversally the singular locus of X. The second is isomorphic to the third row by vector space duality.
means the
diagram commutes. H1(X) C O+, H1(Xj) (1 Ho(X12) by the Maier-Vietoris sequence for X1 U X21 where X12 = X1 fl X2. By Gordon [1, Corollary 4.13] H1(X), -- O+j H1(X;), Q+ rX12. Also H1(Xj), C H1(X,) and rX12 is generated by tubes over classes in X127 i.e., if 0 I(r1+2q) has a representative which is
homologous to zero in X, then this representative can be chosen so that it is a tube over a lower dimensional cycle in X12. Furthermore, under the isomorphism H1(X) H1(X)4, H1(X) r) H,(X12) - rX12 If I(r1+2q) 0, then I(r1+2q) e O+ H1(Xi), or I(r1+2q) E rX12. For if not, this would give nontrivial relations among the H1(Xj), and rX12 in Hi+2q-1(Y - X).
But Gordon [2, Corollary 4.19] has shown that if one looks at the Leray spectral sequence of the inclusion map j : Y - X C Y, then E2'' 7> (rHr+s-2q+1(X)1C Hr+s(Y - X))
and in particular, El 2q-1
(O H1(X,i), C
HI(Xj))
while Ez+2q-1-s,s
rX12
for s > 2q - 1
But since we are working over a field, there can be no nontrivial relations between El,2q-' and
EL+2q-1-s,s for s > 2q
- 1. Hence
(Image I) fl (D H1(Xj) -- (coimage i*) fl (B H1(Xj) s
.1
by the exactness of the sequences and duality of vector spaces. Moreover, the isomorphism is given by Hom o Dj, where Dj is Poincare duality on Xj. Let D4- = Hom o Di. Then by Corollary 4.1, if
HOMOLOGY OF SUBMANIFOLDS
245
n x, = r1,, # o
r1+2q
there is a rl+zq E H1+zq(Y)
with
rl+zq n x, = D*r1,j Hence
D*r1,1 - D2r1,z 0 Ker r, i.e.,
r(D*r1,1 - D*r1,z) = 2rD*r1,1 # 0.
Thus (D1 ° D*)r1,1- (D2 ° Da )r1z 4 coker a*, i. e.,
(D*)zr1,1- (Dz*)zr1 z E Image a*
.
But (D*)z = -Id, where Id is the identity map on H1(Xi). 5.
A question on schemes
5.1. Suppose that Y is an integral algebraic k-scheme, where k is an arbitrary fixed algebraically closed field of any characteristic. We assume that Y is a smooth subscheme of projective space PN(k), and dimension of Y is n + q. Suppose furthur that X1 and Xz are smooth subschemes of Y of dimension q, and i,: X, C Y is.the inclusions. Consider the following diagram HP(Y)
.
f2 I
HP(X) I G,
H.(X ) 2
G2
) Hll+zq(Y)
where the Gi are the Gysin maps, where we are facing the l-adic cohomology, for l prime to the characteristic of k. 5.1.1. Question. When does the diagram commute with respect to coim i* n coim i2 , i.e., if i*r, i2 *r # 0, does G1i*r = Gziz*r for n = 1?
Over the complex numbers, this is the dual statement in cohomology to Corollary 4.2. The reason one believes it might be true for n = 1, is that one needs essentially only the strong Lefschetz theorem to prove Corollary 4.2., but the analogue of the strong Lefschetz theorem is true in etale-cohomology. However, the Kahler identities do not have an immediate analogue. If the answer to question 5.1.1 is true for n = 1, one could probably prove the local invarient cycle problem for deformations of smooth schemes of dimension 2, using the analogues of the geometric constructions in [4].
GERALD LEONARD GORDON
246
Some of the results in this paper have been generalized ; see the author's paper, On the primitive cohomology of submanifolds, to appear in Illinois J. Math.
Added in Proof.
References [1]
D. Fotiadi, M. Froissart, J. Lascoux & F. Pham, Application of an isotropy
[2]
G. L. Gordon, The residue calculus in several complex variables, Trans. Amer. Math. Soc. 213 (1975) 127-176.
theorem, Topology 4 (1965) 159-191.
[ 3 ] -, A Poincare duality type theorem for polyhedra, Ann. Inst.
Fourier
(Grenoble) 22 (1972) 47-58.
[ 4 ] -, A geometric study of the monodromy of complex analytic surfaces, Invent. Math. 40 (1977) 11-36.
[5]
P. A. Griffiths, Periods of integrals on algebraic manifolds, Bull. Amer. Math. Soc.
76 (1970) 228-296.
complexe [6] J. Leray, Le calcul dilerentiel et integral sur une variete analytique 81-180.
(problem de Cauchy. 111), Bull. Soc. Math. France 87 (1959)
[7] J. B. Poly, Sur un theoreme de J. Leray en theorie des residues, C. R. Acad. Sci. [8]
Paris Ser. A, 274 (1972) 171-174. A. Weil, Varietes k&hlerienne, Hermann, Paris, 1958.
UNIVERSITY OF ILLINOIS, CHICAGO
J. DIFFERENTIAL GEOMETRY 12 (1977) 247-252
ISOMETRY TO SPHERES OF RIEMANNIAN MANIFOLDS ADMITTING A CONFORMAL TRANSFORMATION GROUP KRISHNA AMUR & S. S. PUJAR
1.
Introduction
Let M be an orientable smooth Riemannian manifold of dimension n with Riemannian metric gij. Let K,ti Jk, KiJ and K denote the Riemann curvature tensor, the Ricci tensor and the scalar curvature of M respectively. Let X be an infinitesimal conformal transformation of M so that (1.1)
(Lxg)i1 = 2pgi1 ,
where p is a function on M, and Lx denotes the Lie derivative with respect to X. Recently Yano and Hiramatu [3], [4] have obtained conditions for M to be isometric to a sphere without assuming any condition on the scalar curvature function. The purpose of the present paper is to extend the study of the above authors. Among the four lemmas which we shall prove, two (Lemmas 1.1 and 1.2) relate to some of the main results of [3] and [4]. Also Theorems 1.1 and 1.2 in this paper generalize some of the results of [3] and [4].
The tensor fields G, Z [2] and W [1] required in our study are given by
Gif =Kif -
(1.2)
(1.3)
(1.4)
Zhifk = Khifk -
K -gif n
K
n(n - 1) (ghkgi f - ghJ gik)
,
WhiJk = aZhifk + b,ghkGiJ - b2gh.fGik + b3gifGhk - b4gi kGhi + b5ghiGJk - b6gfkGhi ,
where a, b -
, be are constants, and W was first introduced by Hsiung. As usual 17 denotes covariant differentiation on M. We denote lip by pi and g1JVJp by pi. Dp denotes the vector field on M associated with the differential
1-form dp. The Laplace-Beltrami operator on M is given by d = gi'1i11. Received June 17, 1975. The first author was supported partially by Department of Atomic Energy Project No. BRNS/Maths/11/74, and the second author by C.S.I.R. JRF No. 7 / 101(105) /74-GAU. I.
KRISHNA AMUR & S. S. PUJAR
248
For the sake of easy reference we list some known formulas (for details see [1] and [2]):
LxK = -2(n - 1)dp - 2Kp
(1.5)
[X, Dp]K = LXLDPK - LDPLxK ,
(1.6)
Lx(WhijkWhijk) _ -4pWhijkW7",x - 2cGij1ipj
(1.7)
where c > 0 is given by c - 4a2 = 2a
n- 2
[(-1)i-,bil2
bi +
J
1
(1.8)
6
+ (n - 1)
b2 - 2(b,bs + b2b4 - b5bc) i=1
We prove the following lemmas and theorems. Lemma 1.1. Let M be a compact orientable smooth Riemannian manifold of dimension n > 2 admitting an infinitesimal conformal transformation X satisfying (1.1). Then (1.9)
pKLXKdV = (n - 1) SM
LDPLxKdV SM
1 SM (LXK)2dV . 2
L emma 1.2 (Yano and Hiramatu [4]). same properties as in Lemma 1.1 we have
For a manifold M having the
SM KpipidV (1.10)
=
4n(n
1
- 1) f m
[4(n - 1)[X, Dp] K + 2(n-1)(n + 2)LDPLXK + 4nK2p2 - n(LXK)2]dV .
Lemma 1.3. 1.1 we have
For a manifold M having the same properties as in Lemma
f", [KjpipJ c
+
- 4n(nl-
fx 1
2
1)
(2Kp +
p,W,,j,WhijkdV
fM
[KPIPI
LXK)2J dV
- 21 -nc f m LxLx(W hi jkW
2n(n1 - 1)
hijk)dV
f 2nK2p2 + (n + 2)KpLxK + (LXK)2}]dV ,
ISOMETRY OF RIEMANNIAN MANIFOLDS
249
where c is given by (1.8) and is assumed to be positive. Lemma 1.4. For a manifold M having the same properties as in Lemma 1.1 we have Jar
{Kp1p1
(1.12)
c Jar
4n(n1-
1)
LXK)2I dV
(2Kp +
p2WhijkWhr1kdV -
xLxLx(WnMjkWhzik)dV
2nc J
+ Zn f [X, Dp]KdV , where c is given by (1.8) and is assumed to be positive. Theorem 1.1. If a compact orientable smooth Riemannian manifold M of dimension n > 2 admits an infinitesimal nonhomothetic conformal transformation X satisfying (1.1) such that LxLx(WMjkWhaik)dV (1.13)
ar fm
- nc far [Kpipi
- 2n(n1-
1)
{2nK2p2 + (n + 2)KpL1K
+
(L$K)2}I dV
0, then M is isometric to a sphere. Theorem 1.2. For a manifold M having the same properties as in Lemma 1.1 with c > 0 we have (1.14)
far [LxLx(WhjjkWhijk) - c[X, Dp]K]dV > 0
(c > 0)
,
where the equality holds if and only if M is isometric to a sphere. Remark. Theorems 1.1 and 1.2 are equivalent and generalize [3, Proposition 12] and [4, Proposition 3] respectively. We need the following known lemmas and theorem.
Lemma A (Yano and Sawaki [5]).
If a compact orientable smooth
Riemannian manifold M of dimension n admits an infinitesimal conformal transformation X satisfying (1.1), then for any smooth function f on M we have
far pfdV = -_1n
Lemma B (Yano and Hiramatu [4]). properties as in Lemma A we have
SM
LxfdV
For a manifold M having the same
KRISHNA AMUR & S. S. PUJAR
250
(1.15)
-n
pp'PiKdV = n f p2zKdV = fm L$LDPKdV , 2
fm
-f (dp)LxKdV = fM LDPLxKdV .
(1.16)
For a manifold M having the same
Lemma C (Yano and Hiramatu [4]). properties as in Lemma A we have
_f
(1.17)
(dp)2dV = f piVi(dp)dV . if
Theorem A (Yano and Hiramatu [3]). If a compact orientable Riemannian manifold M of dimension n > 2 admits an infinitesimal nonhomothetic conformal transformation X satisfying (1.1), then (1.18)
fm KijpipJdV < 4n(n1-
1)
fm (2Kp + LxK)2dV ,
equality holding if and only if M is isometric to a sphere. 2.
Proofs of lemmas and theorems
Proof of Lemma 1.1. Multiplying (1.5) by L1K, integrating over M and using (1.16) we obtain (1.9). Proof of Lemma 1.2. Using (1.5) and (1.6) we have
[X, Dp]K = LILDPK + 2(n - 1)piVi(dp) + 2pp1PiK + 2Kpipz Integrating over M and using (1.15) and (1.17) we get
fM
KpipidV =
2 far [X, Dp]KdV -
n-2
fM
LxLDPKdV
+ (n - 1) fm (dp)2dV , which in view of (1.5) and (1.6) takes the form
f KpipidV = M
f 1n fm [X, Dp]KdV - n-2 2n
LDPLxKdV
M
(2.1)
+ 4(n 1 Now by Lemma 1.1 we have
1)
fm (2Kp + LxK)2dV .
ISOMETRY OF RIEMANNIAN MANIFOLDS
(2.2)
251
f x (2Kp + LXK)2dV = JM [4K2p2 + 4(n - 1)LDPLXK - (L1K)2]dV .
Substituting (2.2) in (2.1) we obtain (1.10). Proof of Lemma 1.3. From (1.7) it follows that (2.3)
KijVipi = - 2 pWkjihWkjih -
2c
La(WxjihWkjih) +
Kn dp
On the other hand, using ViKji = 2ViK we have (2.4)
V (Kijppi) = 2(V K)ppi + Kijpipj + pKijVipi
Also
(2.5)
Vi(Kppi) = (ViK)ppi + Kpipi + Kpdp
.
Eliminating KijVipj and (ViK)ppi from (2.3), (2.4) and (2.5), integrating over M and using (1.5) and Lemma A we obtain
f KijpipidV = 2 fH p2WkjihWxjihdV c
+i
1
2 f. KpipidV
2nc fm LxLx(WkjihWkjih)drjJ
n-2 4n(n - 1) Subtracting
4n(n
1
- 1)
fm (2Kp
f
M
Kp(2Kp + L1K)dV.
LxK)ZdV from both sides of (2.6) we obtain
(1.11).
Proof of Lemma 1.4. Eliminating f KpipidV from (1.10) and (1.11) and m using (1.9) we obtain (1.12). Proof of Theorem 1.1. Assumption (1.13) of the theorem and Lemma 1.3 lead to the inequality
JM {KPJ
4n(n1-
1)
(2Kp +
L1K)2]dV
> 0,
which by Theorem A implies that M is isometric to a sphere. Proof of Theorem 1.2. From (1.12) we have CJM
p2Whi jkW hijkdV + fM L
(2.7)
=
1
2nc
M
4n(n1
- 1) (2Kp + L1K)2 - Ki jpipj]dV
{LxLx(W hi;kW hijx) - c[X, Dp]K}dV]
KRISHNA AMUR & S. S. PUJAR
252
Theorem 1.2 follows from (2.7), Theorem A and the assumption that c > 0. References [1] [2]
C. C. Hsiung & L. W. Stern, Conformality and isometry of Riemannian manifolds to spheres, Trans. Amer. Math. Soc. 163 (1972) 65-73. K. Yano, Integral formulas in Riemannian geometry, Marcel Dekkar, New York, 1970.
[3]
K. Yano & H. Hiramatu, Riemannian manifolds admitting an infinitesimal conformal transformation. J. Differential Geometry 10 (1975) 23-38.
[ 4 ] -, Isometry of Riemannian manifolds to spheres, to appear in J. Differential Geometry.
[5]
K. Yano & S. Sawaki, On Riemannian manifolds admitting a conformal transformation group. J. Differential Geometry 2 (1968) 161-184. KARNATAK UNIVERSITY, DHARWAR, INDIA
J. DIFFERENTIAL GEOMETRY 12 (1977) 253-300
LES VARIETES DE POISSON ET LEURS ALGEBRES DE LIE ASSOCIEES ANDRE LICHNEROWICZ
Introduction
On sait l'interet present porte aux varietes symplectiques. L'origine de cet interet est double, d'une part elaboration d'une dynamique geometrique adaptee aux problemes globaux de la mecanique analytique classique, qu'il s'agisse de systemes a liaisons independantes on dependant du temps, en vue d'applications a la mecanique quantique (Kostant, Maslov, Leray), d'autre part etude de Tune des plus interessantes parmi les algebres de Lie infinies classiques (Arnold, Gelfand; voir aussi [3], [4], [11]). A partir de 1'etude des transformations canoniques, j'ai ete amen recemment a introduire la.notion gepmetri=
que nouvelle de variete canonique (voir [12], [9]) et a etudier certaines des algebres de Lie associees. Variete symplectique et variete canonique sont dos cas particuliers d'une structure geometrique plus generale, celle de variete de Poisson qui a ete introduite episodiquement dans [9]. Il s'agit, grosso modo, de la structure geometrique la plus generale qui permet de definir, sur l'espace des fonctions a valeurs reelles definies sur une variete, un crochet de Poisson generalise.
Ce papier est consacre a 1'etude generale des varietes de Poisson qui posent des problemes de geometrie differentielle naturels qui sont loin d'etre triviaux. Apres avoir defini la structure (W, G) de variete de Poisson a partir d'un 2-tenseur contravariant antisymetrique G de rang constant, verifiant [G, G] = 0 an sens du crochet de Schouten, ainsi que la G-cohomologie correspondante sur les tenseurs contravariants antisymetriques, on etudie les differentes algebres de Lie attachees a une variete de Poisson et on determine leurs derivations. Plus generalement on etudie la cohomologie 1-differentiable de l'algebre de Lie dynamique N d'une variete de Poisson. Conjointement avec un theoreme important concernant les 1-cochaines de N a cobord d-differentiable (section III), cette etude permet celle des deformations de.1'algebre de Lie N. Ces differents resultats englobent nos resultats anterieurs [4], [9] concernnnt varietes symplectiques et varietes canoniques, a quelques particularites pras.
En vue d'applications a la theorie quantique des champs, Dirac [7] a'developpe, dans un contexte local et non invariant, une theorie que nous repxenons : it s'agit de la dynamique associee a une sous-variete largement arbitraire Communicated September 22, 1975.
254
ANDRE LICHNEROWICZ
d'une variete symplectique (section VI) et elle conduit a la mise en evidence du crochet de Dirac que nous interpretons geometriquement. Cette approche conduit a la mise en evidence naturelle d'une structure de Poisson et a 1'etude de deformations lineaires rigoureuses faisant passer du crochet de Poisson au crochet de Dirac, au moms sur un ouvert de R2 . Je remercie vivement M. Flato et D. Sternheimer pour d'utiles discussions au cours de l'elaboration de cet article.
VARIETES DE POISSON
1.
1.
Notion de variete de Poisson
(a) Soit W une variete differentiable connexe, paracompacte, de dimension m et classe C. Tous les elements consideres ici sont supposes C. Nous po, m) une carte sons N = C`°(W ; R) et designons par {xA} (A, B, - - = 1, locale de W de domaine U. Pour abreger nous appelons i-tenseur un tenseur contravariant antisymetrique d'ordre i. Sur de tels tenseurs, Schouten [15] et Nijenhuis [14] ont introduit un crochet (le crochet de Schouten-Nijenhuis) qui, a tout couple A, B d'un i-tenseur et d'un j-tenseur fait correspondre un (i + j - 1)-tenseur note [A, B] qui peut etre defini de la maniere suivante : pour toute (i + j - 1)-forme f ermee , on a
i([A, B])p = (-1)ti+'i(A)di(B)(3 + (-1)ii(B)di(A) ,
(1.1)
of i(.) est le produit interieur. Pour i = 1, [A, B] = 2(A)B, ou 2( . ) est l'operateur de derivation de Lie. On verifie immediatement sur (1.1) qui l'on a
[A, B] = (-1)Q[B, A] .
(1.2)
De plus si C est un k-tenseur, on a "l'identite de Jacobi" (1.3)
(-1)z'[[B, C], A] + (-1)ik[[C, A], B] + (- 1)1"[[A, B], C] = 0
Un calcul elementaire fournit pour composantes de [A, B] sur le domaine d'une carte locale arbitraire : [At
B]x2...xt+7
=
1
"a: A,i
J.J,AEI2...I'aRBJI ...JJ
(1.4)
+
i!(j_1)!
BRJa...JjaRAZi...ry
ou a est l'indicateur antisymetrique de Kronecker et of aR = a/axR.
LES VARIETES DE POISSON
255
(b) Donnons-nous sur W un 2-tenseur G partout de rang 2n (Gn+' = 0, Gn est partout # 0). Nous posons dans !a suite h = m - 2n ; nous notons
p, q, r, s des indices prenant 2n valeurs, a, b, c, des indices prenant h valeurs. Sur l'espace N = C°°(W ; R), introduisons l'application bilineaire alternee (on crochet de Poisson generalise) N x N -- N define par {u, v}G = i(G)(du A dv)
(1.5)
,
(u, v e N)
.
Si u, v, w E N, etudions la fonction t = S{{U, v}G, WIG
oii S design la sommation apres permutation circulaire. On a explicitement sur U {{u, v}G, w}G = GDCaD(GABaAUaBv)aCW
Un calcul direct donne tlu = (GRAaRGBC + GRBaRGCA + GxcaRGAB)aAUaBVaCW
.
Or it resulte de (1.4) que les composante de [G, G] sur U sent donnees par (1.6)
2[G,
G]ABC = GRAaRGBC + GRBaRGCA + GRCaRGAB
On a ainsi t l u = 2 G, G] ABCaAU6BV6CW
.
Ainsi.pour que (1.5) verifie l'identite de Jacobi, it faut et it suffit que [G, G] = 0. Nous sommes ainsi conduits a la definition suivante Definition. On appelle variete de Poisson une variete W, de dimension m, munie d'un 2-tenseur G de rang constant 2n (avec h = m - 2n) verifiant (1.7)
[G,G]=0.
Sur une variete de Poisson (W, G), le crochet (1.5) defini par G determine sur N une structure d'algebre de Lie. Cette algebre est dite l'algebre de Lie dynamique de la variete de Poisson. De telles variete ont ete introduites et etudiees sommairement dans [9, § 3]. Les varietes symplectiques (h = 0) et les varietes canoniques (h = 1) sont des cas particuliers des varietes de Poisson. Dans la suite, nous supprimerons dans (1.5) l'indice G lorsqu'aucune confusion n'est a craindre. (c) Examinons le cas des varietes symplectiques. Une structure symplectique est define en general sur une variete W de dimension 2n par une 2-forme F de rang 2n, f ermee (dF = 0). Nous notons p : TW - T*W l'isomorphisme de fibres vectoriels defini par p(X) = - i(X)F, ouu X e TW. Cet isomorphisme
256
ANDRE LICHNEROWICZ
s'etend naturellement aux fibres tensoriels. Soit G le 2-tenseur u-'(F) de rang 2n ; le crochet de Poisson de (W, F) est defini par (1.5) et G verifie (1.7). Inversement une structure symplectique peut etre define sur W de dimension 2n par un 2-tenseur G de rang 2n verifiant (1.7) (definition comme variete de Poisson) ; G definit directement u-' et par suite u. De plus si A est un i-tenseur, on a (voir [11, § 3])
u([G, A]) = du(A) .
(1.8)
On sait qu'il existe sur une variete symplectique des atlas de cartes canoniques , n ; ce = a + n) ; dans une telle carte, F a pour {XP} = {x°, x8} (a = 1, seules composantes non nulles
Fa,, =-F8a=1, d'oi 1'on deduit un resultat analogue pour G. On a le lemme suivant, utile sous cette forme. Lemme. Soit V un domaine de RZ'I muni d'un 2-tenseur G de rang 2n verifiant [G, G] = 0. Si x E V, it existe une carte {xP} = {x x8} de domaine U C V, of x e U, tel que G admette pour seules composantes non nulles :
Gab = -G8a = 1
(1.9)
.
2.
Feuilletage et coordonnees canoniques pour une variete de Poisson
(a)
Soit (W, G) une variete de Poisson telle que h # 0. Si U est un domaine
contractile de W it existe sur U, d'apres la condition portant sur le rang de G, h 1-formes info,) (a = 1,
, h) lineairement independantes, telles que
(2.1)
GDA(0( ) = 0
On deduit de (1.7) a partir de (1.6) : GDBaDGCA, w(Aa) + _ GDCaDGAB , W(Aa) = 0 ,
soit, d'apres (2.1), GBDG A(aDw(Aa) - aAWD))
=0
Vest-a-dire, (2.2)
GBDGCE(dW(a))DE
=0
Adoptons sur U des coreperes privilegies 1(,),A )I de la forme {w`1), w(P)I. On a
dans ces coreperes GBa = 0 et det (GP4) # 0. La relation invariante (2.2) s' ecrit
LES VARIETES DE POISSON GPTGgs(dco 1. En substituant le developpement precedent dans (6.3), on voit qu'il suffit d'etudier les elements u de N(U) des trois types suivants :
(I)
u,=Ncp(x7)
(I # 1)
avec
N = (Yl)kl ... (yh)kn(xl)1i(xI)tl ... (xn)1n(x)tn oil
k1+... +kh+11+1,+... +1,
11>1.
LES VARIETES DE POISSON
(II)
UII1 =
(xl)d+22*(xA)
265
,
Urn = (YI)k1 ... (V'h)khX(xI)
(III)
oil
k1+...+kh=d+1,
k1>1.
Pour obtenir une fonction v1 telle que {(x1)2, v1} = u1i c'est-a-dire telle que 2x'81 v1 = Ncp(x') it suffit de prendre V1 =
1
V") k1 1
2(l1 + 1)
(y h ) k h(x l )
L11 ) 1+ (
1
... /xn)L,/x 9t)Lx(p(xI)
Avec ce choix, on a d' apres (6.1) appliquee a T , : T ,(u) = TU{(x1)2, v1} = {TU((x1)2), VI} + {(x1)2, TdU(V)j + l a((xI)2, v)
D'apres le lemme 2, TU((xl)2) E d(U) et le premier terme du dernier membre est nul. On en deduit qu'en xa
(T ,(u))(x) = 0 Considerons maintenant la fonction uII; pour que {(xi)2, v11} = u11 it faut et it suffit que 81vII = _ (xl)d+lr^(xA) Si Yr (XA) est une primitive en x1 de 'lj,(XA), on peut prendre
VII = - 1 (xI)d+lr(xA)
Avec ce choix, on a d'apres (6.1) appliqueee a TU TU(uI1) = T ,{(x')2, vII} = IT ,((xI)2), VII} + {(xi)2, T
,(v11)} + Cd((xI)2, vII)
On voit encore qu'en xa (l'U(u11))(xo) = 0 .
Cherchons enfin une fonction',v111 telle que {y',,'x1, v,,,),= y1{x1, v111} = u111, soit a1vIII = (V,1)ki-1(,2)k2 ... V"h)kh%(xI)
Nous prenons (xI) VIII = (yl)ki-l(Vy2)k2 ... (yh)khxIg Va
oil le facteur de
est de deegre d + 1. On
,
d'apres (6.1) applique a Tdu
ANDRE L1CHNEROW1CZ
266
d a d d Tu(un1) = Tu{Y x , vn1} = {Tu(Y x ), vn1} + {Y x , Tu(vnz)} 1
1
1
1
1
1
Ca(Y1x1,
vn1
et l'on en deduit qu'en xo (TU(u111))(xo) = 0 ,
ce qui demontre le lemme. (d) On en deduit la proposition suivante. Proposition, Si Tu est un endomorphisme de N(U) tel que 8Tu soit une 2-cochaine d-differentiable (d > 1) de N(U), on a
Tu=Pu, oh Pu est un operateur differentiel d'ordre d sur N(U). Soit x un point arbitraire de U, {xA} une carte canonique de domaine U. A Tu correspond par cette carte l'operateur differentiel Pu d'ordre d tel que
l'endomorphisme TU = Tu - Pu annule les polynomes de degre d en les coordonnees canoniques (lemme 1).
Soit u un element de N(U). Il existe sur U un polynome a, de degre d en les coordonnees canoniques, tel que la(u)(x) = 3a(u)(x)
Il resulte du lemme 3 que l'on a (TU(u))(x) = (TU(u))(x) = 0 .
Ainsi TU(u) est nul en x, donc sur U et, pour tout u e N(U), on a Tu(u) _ Pu(u), ou Pu est un operateur differentiel d'ordre d. 7.
1-cochaine locale de N a cobord d-differentiable
Une i-cochaine locale de N est dite d-differentiable si elle induit, pour tout domaine U de W, une i-cochaine d-differentiable de N(U). Considerons une cochaine locale T de N telle que 8T soit une 2-cochaine d-differentiable de N.
Soit U un domain de W. Si uu e N(U), soit u e N telle que ulu = uu ; l'endomorphisme local T de N induit sur U par Tu(uu) = T(u) Iu un endomorphisme Tu de N(U) tel que 8Tu soit une 2-cochaine d-differentiable de N(U). D'apres la proposition precedente, it existe sur N(U) un operateur differentiel
Pu d'ordre d tel que Tu = Pu. Introduisons un recouvrement localement fini {Uj de W par des domains de cartes canoniques. On pose T. = Tu., P, = Puy. Pour u e N, on a pour
xEU, (PXuIu'))(x)
Pour x e U, (1 U,., it vient
LES VARIETES DE POISSON
267
(Pv(u1u,))(x) = (PXuIU))(x) = (T(u))(x)
Il en resulte que les P. definissent sur N un operateur differentiel P d'ordre d tel que pour tout u e N, on ait T(u) = Pu. Nous enongons Theoreme. Si T est une 1-cochaine locale de N telle que aT soit une 2cochaine d-differentiable (d > 1), T est elle-meme d-differentiable. Toute derivation locale de N etant un 1-cocycle, on peut lui appliquer le theoreme precedent avec d = 1. 11 vient Corollaire. Tout derivation locale de l'algebre de Lie Nest 1-differentiable.
IV.
ALGEBRES DE LIE ATTACHEES A UNE VARIETE DE POISSON DERIVATIONS 8.
Les algebres de Lie LG, L, L*
(a) Soit (W, G) une variete de Poisson. Etudions les espaces de cohomologie H°(W ; G) et H'(W; G) correspondant a la G-cohomologie. Pour que a (0-tenseur) element de N definisse un 0-cocycle, i1 faut et it suffit que (8.1)
[G, a] = 0 .
Les fonctions a satisfaisant (8.1) definisse un anneau sl et H°(W G)' est iso-
morphe a sl.
Pour que le vecteur X definisse un 1-cocycle, it faut et it suffit que l'on ait [G, X] .= 0
P(X)G = 0
,
.
Un tel champ de vecteurs definit un automorphisme infinitesimal de la variete
de Poisson (W, G). L'algebre de Lie des automorphismes infinitesimaux de (W, G) sera note LG, celle des automorphismes infinitesimaux a supports compacts (LG)0 ; LG et (LG)0 sont des s/-modules. Un element X de LG sera encore dit une transformation infinitesimale (t.i.)_ de Poisson. Soit {xa, xp} une carte canonique de (W, G) de domain U. Si X E LG, it
resulte de (8.2) d'une part que
apxa - 0, et Xa appartient a sl(U), d'autre part que
ANDRE LICHNEROWICZ
268
GTParXq
- GT46,XP = 0 ,
et, d'apres 1'etude du § 3, it existe une fonction ua e N(U) telle que XP = [GIa, ua]P .
(8.4)
(b) Soit L (resp. Lo) le sous-espace de LG (resp. (Lo)o) defini par les elements tangents au feuilletage; L et L, sont encore des d-modules. Si X e LG, y e L, it resulte du a que l'on a sur le domaine U
[X, Y]° = XAaAY° - YAaAX° = 0 ,
puisque Ya = 0 et [L0, L] c L. Ainsi l'algebre de Lie L (resp. L) des t.i. de Poisson tangentes au feuilletage (resp. a supports compacts) est un ideal de LG.
Soit I une feuille connexe du feuilletage. Si X, est le champ de vecteurs induit sur I par X e L, le 2-tenseur G1 definissant la structure symplectique de I verifie 92(X2)G2 = 0 .
Ainsi, si X est une t.i. de Poisson tangente au feuilletage, elle induit sur chaque feuille de (W, G) une t.i. symplectique. Si p, est l'isomorphisme entre vecteurs et 1-formes de I determine par G, de 1-formes fermees 1'element X de L definit une famille reguliere e _
, = pE (X') de 1. Nous notons ce fait
d=o,
(8.5)
ou d est la differentielle exterieure la long des feuilles. les deux familles de 1-formes correspondantes (c) Soit X X2 e L, 1, 2
verifiant (8.6)
0.
0,
Dans la carte canonique {xa, xP} de domaine U, nous avons
Xl =
X2 =
On en deduit [XI, X2]P = Gr4eiraq(GTPe2a) -
ce qui peut s'ecrire, compte-tenu de (8.6), [X 1, X 2]P =
Or
LES VARIETES DE POISSON
269
wU = G4TS14ezr
estra restriction a U d'une fonction w E N notee (8.7)
w = i(G)(E, A E )
Si F1 est la 2-forme de I define par G1., on a pour tout x E W w(x) = i(Xl(x) A X2(x))FI(X)
On a donc (8.8)
[XI, X2] = [G, w] ,
(w E N) .
Soit N. le sous-espace de N defini par les fonctions a supports compacts. Nous sommes conduits a introduire l'espace L* (resp. Lo) des champs de vecteurs X definis par (8.9)
X = [G, u]
oiI u est un element arbitraire de N (resp. No) ; L* et Lo sont des ,d-modules. Si Z E LG et X E L*, on a
[Z, X] = ' F(Z)X = 2(Z)[G, u] _ [G, Y(Z)u] , et [LG, L*] C L*, [LG, L0 *1 C Lo*. Il resulte d'autre part de (8.9) que
[L, L] C L*
,
[L, L0] C Lo
.
Ainsi Proposition. (1) L* est un ideal de LG et L/L* est abelien. (2) On a [LG, L0*] C Lo*, [L, Lo] C Lo ; Lo est un ideal de LG et LO/Lo est abelien. Notons que, pour que le 1-cocycle defini par X E LG soit exact, it faut et it suffit qu'il existe u E N tel que X = [G, u], c'est-a-dire que X appartienne a
L*. Ainsi H'(W ; G) est isomorphe a LG/L*. (d) Soit N 1'espace des classes de fonctions de N, modulo les fonctions additives appartenant a d. C'est encore un d-module. Nous notons z: u E N -4 7r(u) = it E N la projection canonique de N sur N. Si U E N, sa differentielle du tangente aux feuilles ne depend que de la classe u de u ; nous la notons eventuellement du. Si u, v E N, leur crochet
{u,v}=i(G)(duAdv) ne depend que des classes u, v E N de u, v et it induit par suite sur N une structure d'algebre de Lie. L'isomorphisme naturel du d-module L* sur le d-module N est, d'apres (8.7), (8.8), un isomorphisme d'algebres de Lie.
ANDRE LICHNEROWICZ
270
9.
Derivations locales de N et algebre de Lie L°
(a) Soit -9 une derivation locale de N, c'est-a-dire un endomorphisme local de N verifiant pour tout u, v E N la condition
-9{u, v} = {-9u, v} + {u, -9v} ,
(9.1)
qui exprime que 8-9 = 0. D' apres le corollaire du § 7, -9 est necessairement un cocycle 1-differentiable : si nous posons -9 = (X, a), ou X est un vecteur et a un scalaire, on a pour u E N
-qu ='(X)u + au.
(9.2)
Pour que (9.1) soit satisfaite, it faut et it suftit,d'apres (5.1) que
8-9 _ (-[G, X] + aG, [G, a]) = 0 , c'est-a-dire que
Y(X)G = aG ,
[G, a] = 0 .
Il est clair que les derivations interieures de N sont donnees par - _ (X, 0)
ouXEL*.
Nous dirons que le vecteur X definit une t.i. conforme de Poisson s'il existe ax E d telle que (b)
'(X)G = axG .
(9.3)
Soit
{Xa, XP} une carte canonique de (W, G) de domaine U. Si X est une t.i.
conforme de Poisson, on deduit de (9.3) comme au § 8, a, que 8PX° = 0 et que, par suite, les composantes Xa appartiennent a d(U). Il en resulte que si
bEd,ona £(X)b E d .
(9.4) "`°-
Si X, Y sont deux t.i. conformes de Poisson:
2(X)G = axG ,
2(Y)G = ayG ,
of ax, ay E d. On en deduit
9([X, Y])G = (1(X)2'(Y) - 2'(Y)c(X))G = £(X)(ayG) - 2'(Y)axG) soit, apres simplifications : (9.5)
ou
Y([X, Y])G = (Y(X)ay - 2'(Y)ax)G ,
LES VARIETES DE POISSON
Y(X)a, - 2(Y)ax E .sad
271
.
Les t.i. conformes de Poisson definissent ainsi, pour le crochet naturel, uhe algebre de Lie notee Lc. Nous avons etabli Proposition 1. L'algebre de Lie naturelle des derivations locales de N est isomorphe a 1'algebre de Lie L° des transformations infinitesimales conformes de Poisson pour l'isomorphisme defini de la maniere suivante: si X E Lc, on a
2(X)G = a.,G (avec a., e a) et X donne la derivation (9.6)
_9g = 2(X) + ax .
On note qu'avec les notations du § 5, H'(N) est isomorphe a L°/L*. Designons par .4 le sous-anneau de a (qui est un a-module) defini par les fonctions b de a telles que bG soit exact. D'apres le theoreme du § 5, H'(N) est isomorphe a R Q LG/L*. (c) Pour X E L°, Y E L, on a d'apres (9.5)
2([X, Y])G = 0
.
D'autre part, dans une carte canonique {xa, xP} de domaine U, it vient [X, Y]b = XABAYb - YAaAXb = 0
et [X, Y], tangent au feuilletage, appartient a L. Ainsi les algebres de Lie L et Lo sont des ideaux de L°. Pour X E L°, Y = [G, v] E L* (avec v (= N), on a
[X, Y] _ 2(X)Y = 2(X)[G, v] _ [a1G, v] + [G, 2(X)v]
.
soit
(9.7)
[X, Y] = [G, 2(X)v + agv]
oil 2(X)v + axv r= N. On en deduit que L* et Lo sont des ideaux de L°. Proposition 2. Si L° est l'algebre de Lie des transformations infinitesimales conformes de Poisson de la variete (W, G), les algebres de Lie L, Lo, L*, L°* sont des ideaux de L°. 10.
Caractere local des derivations de L°, L, L*
Nous nous proposons, dans la suite, de determiner les derivations des algebres de Lie Lc, L, L* et nous voulons d'abord etablir le caractere local des "derivations de ces algebres. Une derivation de l'algebre de Lie L° est un endomorphisme Dc: L° -f L° tel que pour tout X, Y E Lc, on ait
272
(10.1)
ANDRE LICHNEROWICZ
D`[X, Y] = [D°X, Y] + [X, DcY] .
1Vlemes definitions pour une derivation D de L, D* de L*, -9 de N, de N. Soit De une derivation de L`, X un element de L° tel que X jv = 0 pour un domaine U de W. Donnons-nous un element Y de Lb* a support S(Y) C U. On a [X, Y] = 0 et [X, D°Y] J u = 0. 11 resulte de (10.1) que l'on a (10.2).
[D°X, Y] = 0 .
Soit x un point de U. Donnons-nous une carte canonique {Xa, xP} de domaine V tel que x e V C U pour laquelle x admet des coordonnees nulles. Prenons
Y = [G, v] oii v e No est a support S(v) C U et est tel que (dv)(x) = 0. On a alors Y(x) = 0 et (10.2) peut s'ecrire en x ((D`X)AaAYP)(x) = 0
Or on a aAYP = G'Pa,Av .
La relation precedente s'ecrit donc (10.3)
((DcX)Aa,Av)(x) = 0
Choisissons pour v une fonction a support compact S(v) C U et qui daps le voisinage V de x s'ecrSive v 1, s= xBxP. On a sur Vsq aAv = (UAxP + uAxB) ,
sq
qq
arAV = (vAUP + SABB)
11 vient en x (DCX)A(x)(SASP + SAoB) = 0 soft
(10.4)
(DcX)B(x) SP + (D,X)P(x) SB = 0
Prenons B = r # p ; it vient (D°X)P(x) = 0 et (10.4) se reduit a (D`X)b(x) 8P = 0 11 vient (DCX)b(x) = 0. Ainsi (DcX)(x) = 0 et par suite DcX Iv = 0. Le meme raisonnement s'applique aux derivations de LG, L, V. Nous avons etabli Proposition. Toute derivation de L`, LG, L, L* est un operateur local. de N, nous obtenons seulement par ce Pour une derivation -9 de N ou type de raisonnement (S support) (10.5)
S(d-9u) C S(du) ,
S(d-9u) C S(du) ,
ou u e N, u e N. En particulier si a e d, on a -9a e d.
273
LES VARIETES DE POISSON
11.
Etude de N(V)
Le caractere local des derivations que nous voulons determiner nous conduit a proceder a une etude purement locale. Soit {y¢, yP} une carte canonique de domaine V ; nous supposons que sur V, {y¢} decrit un pave Ih de Rh et que {yP} decrit un domain contractile V_, de R2n. La domaine V Ih X V, sera dit un domaine contractile produit de W. Considerant (V, Glv) comme une variete de Poisson, nous nous proposons essentiellement d'etudier son algebre de Lie dynamique N(V) et l'algebre de Lie L(V) = L*(V) des t.i. de Poisson tangentes, an feuilletage. Nous noterons LG(V) (resp. LC(V)) 1'algebre de Lie des t.i. (resp. conformes) de Poisson de (V, Giv) (a) Introduisons sur V 1'element auxiliaire defini par la h-forme
fjdy¢ =1
Nous notons I. 1'ideal de l'algebre exterieure des formes cP sur V admettant en
facteur la forme ir, c'est-a-dire telles qu'il existe une forme i de V pour laquelle (11.2)
cp=irA
.
Nous allons montrer que si cp est une (h + i)-forme f ermee de I., elle est la differentielle d'une (h + i - 1)-forme de I.. En effet on peut supposer dans (11.2) / de type (0, i) par rapport a la carte = /(Ya, YP)g,...gidyg' A ... A dyg= ,
cp etant fermee, on a d* = 0 et it existe une (i - 1)-forme x de V de type (0, i - 1) telle que w = dx. Il vient
cp=irAdx=7rAdx, et par suite cP=(-1)ld(,rAx)
ce qui demontre la propriete. (b) Introduisons sur V la m-forme element de volume ;2v define par liv(x) = 7r(x) A riE(x)
(x E V)
,
of i . = FE / n ! est 1'element de volume symplectique de la feuille z. On sait que la donnee d'un element de volume riv definit un isomorphisme *: A i(A)riv de 1'espace des i-tenseurs de V sur 1'espace des (m - i)-formes; 8 = (-1)1*-'d* definit alors 1'operateur divergence sur les i-tenseurs (a' = 0). En coordonnees canoniques, on a
274
(11.3)
ANDRE LICHNEROWICZ
(3A)B2...Bi = -aRARB2...st
Toute m-forme de V peut s'ecrire u7 v, ou u e N(V) ; elle est fermee et appalIl existe par suite une (m - 1)-forme r de V telle que tient a (11.4)
uri = d*
(11.5)AX avec
Posons Z
(11.4) peut s'ecrire
u = -*-'d*Z , soit
(11.6)
u=BZ.
Il resulte de (11.5) que dyd A = 0 pour tout a, soit i(Z)dyd = 0 et Z est tangent an feuilletage. Nous avons Lemme 1. Pour l'element de volume )7v, tout element u de N(V) peut s' ecrire
u=BZ, of Z est un champ de vecteurs sur V tangent au feuilletage. En coordonnees canoniques (11.7)
u = -aPZP
L'element de volume symplectique )7E definit sur I une divergence 8E sur les i-tenseurs de E. Si ZE est le champ de vecteurs induit par Z sur E, it resulte de (11.7) que (11.8)
u I E = BEZE
(c) Si X E L(V), on a 2(X)7r = 0 et par suite )7Y est invariant par X, ce qui se traduit par (11.9)
ax = 0
Pour X, Y E L(V), on a X = [G, u], Y = [G, v],, avec u, v e N(V) et la fonction {u, v} associee an crochet [X, Y] pent s'ecrire (11.10)
{u, v} = 2(X)v = -8(vX)
Si N0(V) est le sous-espace de N(V) defini par les fonctions a supports compacts, nous sommes conduits a la definition suivante,
LES VARIETES DE POISSON
Definition.
275
On note N1(V) le sous-espace de N0(V) defini par les fonctions
u pour lesquelles it existe un vecteur Z a support compact S(Z) c V tangent au feuilletage tel que
u = OZ.
(11.11)
11 resulte de (11.10) que l'on a (11.12)
{N(V), N0(V)} C N,(V)
La relation (11.11) peut s'ecrire
u it = orZr oii Z, est a support compact et it vient (11.13)
f
ulo" = 0 .
Inversement si u c: N(V) verifie (11.13) pour toute feuille, u appartient a N1(V).
Cela pose, on etablit exactement comme dans [4, § 8], a partir de la caracterisation (11.13) des elements de N1(V), le lemme suivant qui est un instrument important. Lemme principal 2. Soit U, U' deux sous-domaines contractiles produits du domaine V, avec U' C U. Donnons-nous 2n fonctions w(P) E N1(V), a supports S(w(P)) C U telles que {y', xP = w(P)u,} definisse une carte locale de domaine V. Si u est un element de N1(V) tel que S(u) C U', it existe 2n fonctions v(P) e N,(V), a supports S(v(P)) C U telles que (11.14)
u = E {v(P), W(P)} . P
En particulier, si U est un sous-domaine contractile produit de V et si u est un element de N1(V) tel que S(u) c U, on peut trouver 2n couples (V(P), w(P)) d'elements de N1(V) a supports dans U, tels que (11.14) soit satisfaite. (d) Donnons-nous un recouvrement {U,},,, de V par des domains contractiles produits verifiant la condition suivante (recouvrement de Palais) : it
existe une partition de I en une collection finie de sous ensembles I, (u = 1, , k) telle que, pour chaque i, les domaines pour lesquels v E I, soient deux a deux disjoints. Soit {cps} une partition differentiable de l'unite subordonnee au recouvrement ; nous posons vEIP
cv
Si u e N(V), it existe sur V, d'apres le lemme 1, un champ de vecteurs Z tangent au feuilletage tel que u = 8Z. Posons
276
ANDRE LICHNEROWICZ
Zp,=2pZ,
uP=BZP .
Considerons, pour p fixe, les domaines {Uv},,EI" deux a deux disjoints ; en appli-
quant a u, ju, le lemme principal, on voit que u,, est la somme des crochets de 2n couples (v(P), w°') d'element de N(V). Ainsi u, e {N(V), N(V)} et u, some finie d'elements de {N(V), N(V)} appartient a {N(V), N(V)}. On en deduit {N(V), N(V)} = N(V) ,
(11.15)
[L(V), L(V)] = L(V) .
Proposition. Pour un domaine contractile produit V d'une variete de Poisson, N(V) et L(V) coincident avec leurs ideaux derives. 12. (a)
Derivations de L(V)
Etudions d'abord les derivations 9, de N1(V). Soit U un sous-domaine
contractile produit du domaine V. Si u E N1(V) est a support S(u) C U, it resulte du lemme principal (§ 11) qu'il existe 2n couples (v(P), w(P) d'elements de N,(v) a supports dans U tels que (12.1)
u = E {v(P), W(P)} P
On en deduit
lu = EP {
lv(P), w(P)1 + E {v(P), -91w(P)} P
De 1'expression du second membre, it resulte que S(u) C U implique S(-9,u) C U. Cela pose, soit u un element arbitraire de N1(V). 11 existe un champ de vecteurs Z, a support compact S(Z), tangent au feuilletage, tel que u = 8Z. Introduisons un recouvrement fini {U.}.,1 d'un voisinage ouvert de S(Z) par des domaines contractiles produits et soit {(p.} une partition de l'unite subordonnee. Posons Z. = cp,Z, u. = BZ,. On a -9,u = -9,u,, oii S(.9,u,) C U,,. Il en resulte (12.2)
S(-q,u) C S(Z) .
Soit U un domaine contractile produit tel que ul;, = 0. On a 5ZI;, = 8(ZIU) = 0. Il existe sur U un 2-tenseur AU tangent au feuilles et tel que ZIc, = JA t. Choisissons sur V un 2-tenseur B a support compact, tangent aux feuilles et
tel que B It = AU. Nous pouvons substituer a Z le vecteur Z = Z - 8B, a support compact, tangent aux feuilles et tel que u = 8Z, Z1U = 0. 11 resulte de (12.2) applique a Z que -9,u I U = 0. Ainsi 9, est un operateur local. (b) Soit -9 une derivation de N(V) et etudions sa restriction a N1(V). Soit U C V un domaine contractile produit ; si u e N,(v) est a support S(u) C U, on a (12.1) et par suite
LES VARIEi ES DE POISSON
-9u = E
277
wcP>} + E {v(P , -9wIP'}
et 1'on voit que clu e NI(V). Si u est un element arbitraire de NI(V), on voit a partir d'un recouvrement fini {U.} semblable a celui du a, que clu est Somme finie d'elements 1u, appartenant a NI(V), d'apres ce qui precede, done que .2u e NI(V). Ainsi la restriction de -9 a NI(V) est une derivation -91 de NI(V). Inversement soit -91 la derivation de NI(V). Si u e N(V), introduisons uI e NI(V) telle que pour un domain U, on ait u l t, = uj ju. En posant -9u ju = 1Iullu on definit, d'apres le caractere local de l1, une derivation necessairement locale -9 de N(V), dont la restriction a NI(V) coincide avec l1. Une telle derivation locale de N(V) est manifestement unique. Proposition. Toute derivation -9, de NI(V) est necessairement locale.
L,espace des derivations de NI(V) est l'espace des restrictions a NI(V) des derivations de N(V). Il est isomorphe a l'espace des derivations locales de N(V). (c)
Soit -9une derivation de N(V). A partir de -9et compte-tenu de (10.5),
on peut definir par
Yu une derivation - de N(V) telle que
o 2r =
n o -9. Inversement, donnons-nous une derivation de N(V). Cherchons un endomorphisme -91 de N1(V) tel que, pour tout uI e NI, on ait 21u, = Jul. Un tel endomorphisme est unique : si d9Iul = 0, on a sur une feuille X de v
-9,u,1, = C = const. D'apres (11.13), f -91u1 Ix 7)E = 0
et par suite 11u1 = 0. D'autre part
etant une derivation de N(V), on a de
meme
-9I{u,v,} - {-9Iu v,} - {u1, -91v,} Is = const. = 0
,
pour tout u v, e N1(V). Ainsi -91, si elle existe, est une derivation bien determine de NI(V). Soit U e V un domain contractile produit. Si u e NI(V) est a support S(u) e U, on a (12.1) et on peut poser par definition: _9,u = E {19; gyp), wIP)} + E {v(P>> -wIP)} e PLl P
p
oii 1,u E NI(V). Soit u un element arbitraire de NI(V). Avec le meme recouvrement qu'au
A, on pose 1,u = Y -9Iu, oii les 1,u, sont definis comme ci-dessus. On a i,u e N1(V) et
ANDRE LICHNEROWICZ
278
2ilu = L u = La derivation -91 de N1(V) ainsi construite est la restriction a N1(V) d'une derivation locale -9 de N(V). Si u E N(V), introduisons ul E N1(V) telle que, pour un domain U de V, on ait u 1 u = ul l u. On a -9u Ju = 11u11 u et, d' apres Ainsi -9u = -5M et -q verifie (10.5), d-'Puju = 27 0 JJ = -9 0 7r
(12.3)
.
La derivation locale -q verifiant (12.3) est manifestement unique. On a 9 _ 2(X) + ax, oii X E Lc(V) avec 2(X)G - axG = 0. On pent poser
2(X)u = Y(X)U'
axu=axu.
Il resulte de (12.3) qu'avec ces notations, toute derivation -9 de N(V) peut s'ecrire
= 2(X) + ax
(12.4)
(X E L0(V))
On deduit de (9.7) et de l'isomorphisme entre N(V) et L(V) que l'on a Proposition. Toute derivation de L(V) est donnee par Y E L(V) -* [X, Y] E L(V), oh X E L'(V). 13.
Determination des derivations de L, L*, Lc
(a) Etant donne un domain contractile produit V de W, tout element Y. de L(V) peut etre prolonge en un vecteur Y de L et, en particulier, en un vecteur de L*. Cela pose, soit D une derivation de L dont nous savons qu'elle est necessairement locale. Donnons-nous un recouvrement {Uv},,, de W par des domains contractiles produits et designons par l'indice v les elements relatifs a U. Si Yv E L(U), it existe Y E L tel que Yuv = Y,. Compte-tenu du caractere local de D, en posant
DvY, = DY I u,
on definit une derivation D, de L(U.). Il resulte de la proposition precedente (§ 12) qu'il existe X. E Lc(U.) tel que D. soit defini par DvY, _ [X,, Y,]
Si Y est un element de L, on a done
DYlu. = [X., Yluj Pour x e Uv fl U,,, it vient
LES VARIETES DE POISSON
279
(DY)(x) = [Xv, YIUJ](x) = [Xv,, YIuj(x)
Pour tout Y e L et tout x e u, n U, on a donc (13.1)
[Xv- - Xv, Y](x) = 0
De la relation (13.1), on deduit par un raisonnement identique a celui du § 10 Xy,(x) = X,(x)
,
x e Uy n Uy, .
On voit qu'il existe sur W un champ de vecteurs X unique, element de Lc, tel que X. = X I U,. On en deduit, le meme raisonnement etant valable pour L*. Theoreme 1. Toute derivation de L (resp. L*) est donnee par Y - * [X, Y],
ouXeL (b)
Soit maintenant Dc une derivation, necessairement locale, de L La
restriction D de Dc a L definit une application lineaire locale de L dans L' que nous allons etudier. Soit V un domain contractile produit arbitraire de W. Si Yv E L(V), it existe
Y e L tel que YI, = Y. Compte-renu du caractere local de D, en posant (13.2)
DvYv = DYIV,
on definit une application lineaire Dv de L(V) dans Lc(V) qui, pour tout couple Yv, Zv e L(V) verifie la relation (13.3)
DV[Yv, Zv] = [DvYv, Zv] + [Yv, DvZv]
Le second membre appartient necessairement a L(V), ideal de Lc(V). Ainsi
Dv[L(V), L(V)] c L(V)
.
Comme [L(V), L(V)] = L(V) (§ 11, d) on voit que Dv est un endomorphisme de L(V) verifiant (13.3), c'est-a-dire une derivation de L(V). Il en resulte d'apres (13.2) que, pour tout Y e L, DY laisse G invariant et est tangent au feuilletage, c'est-a-dire est un element de L. Ainsi la restriction D de D' a L est une derivation de L. D'apres le theoreme 1, nous pouvons la definir par faction d'un element X de Lc. Pour Y e Lc, Z E L, on a avec nos notations
D°[Y, Z] = [D°Y, Z] + [Y, D°Z]
,
soit
2(X)[Y, Z] = [DcY, Z] + [Y, 2(X)Z] . Or '(X) definissant une derivation interieure de Lc:
280
ANDRE LICHNEROWICZ
2'(X)[Y, Z] = [2'(X)Y, Z] + [Y, 2'(X)Z] Il en resulte par difference que, pour tout Z E L,
[(DO - 2'(X))Y, Z] = 0 . On en deduit par un raisonnement encore identique a celui du § 10
D°Y = 2'(X)Y , et D° est une derivation interieure de Li'. On a Theoreme 2. Toute derivation de l'algebre de Lie L° est interieure. 14.
Derivations de LG
(a) La determination des derivations de LG peut proceder comme celle des derivations de L Soit D la restriction a L d'une derivation DG de LG. On etablit comme au § 13, b que D est necessairement une derivation de L ; elle peut done etre definie par Faction d'un element convenable X de Lc. On etablit alors comme precedemment que l'on a necessairement
(14.1)
DG = 2'(X) .
Mais on doit noter que LG n'est pas un ideal de L° en general. Pour que 2'(X) (avec X E Lc) definisse une derivation de LG, it faut et it suffit que ce soit un endomorphisme de LG, c'est-a-dire que 2'(X)Y E LG pour tout Y E LG. II en resulte Theoreme 3. Toute derivation de LG est donnee par Y E LG -> [X, Y] E LG, ou X appartient au normalisateur .K (LG; Le) de LG dans Le. Pour que X E L° appartienne a .K (LG; Le), it faut et it suffit que 1'on ait, avec les notations du § 9 (14.2)
2'(Y)ax = 0 ,
pour tout Y de LG. (b) Interessons-nous aux derivations de LG(V), ou V est un domaine contractile produit de W ; (14.2) s'ecrit
Ybabax = 0 ,
et doit etre verifie pour tout Yb e d(V). Il en resulte ax = Kx = const. Dans ce cas, .K (LG(V) ; Lc(V)) est donne par les vecteurs X de V verifiant (14.3)
2'(X)G = K1G .
Pour une variete de Poisson (W, G) arbitraire, .K (LG ; LC) contient l'algebre de Lie Li des vecteurs X verifiant (14.3).
LES VARIETES DE POISSON
15.
281
Etude des derivations de I'algebre de Lie N
En ce qui concerne les derivations de N, les resultats concemant les varietes canoniques se transposent sans difiicultes. Soit -9 une derivation de N ; nous avons vu (§ 12, c) qu'il lui correspond une derivation unique 2 de N telle que 7C0-9 _ -9 0 7r.
Soit X 1'element de Le definissant 2, ax 1'element de a correspondant. Pour la derivation locale de N donnee par
-9'=2(X)+a,, on a n o (-9
0, c'est-a-dire -9u - -9'u E a pour tout element u de N. Ainsi, etant donnee une derivation -9 de N, it existe X E Lc et une application lineaire A non locale de N dans a telle que
1=8(X)+a,+A.
(15.1)
Pour que -9 donnee par (15.1) soit une derivation de N, it faut et it suffit qu'il en soit de meme pour A, c'est-a-dire que pour tout u, v E N A{u, v} = {Au, v} + {u, Av}
,
oii le second membre est nul. Ainsi it faut et it suffit que A soit nul sur IN, N}. Theoreme. Toute derivation de N peut s'ecrire d'une maniere unique
l=2(X)+a.+A, oil X est un element de Lc et oil A est une application lineaire non locale de N dans a, nulle sur IN, N}. En general IN, N} differe de N (comme le montre 1' exemple des varietes symplectiques compactes). Si V est un domaine contractile produit de W, on a {N(V), N(V)} = N(V). Par suite toute derivation de N(V) est de la forme
-9 _ £(X) + ax, Oil X E Lc(V)
V. DEFORMATIONS DE L'ALGEBRE DE LIE N 16. (a)
Deformations formelles 1-differentiables de N
Soit E(N; 2) 1'espace des fonctions formelles en 2 a coefficients dans
N. Considerons une application bilineaire altemee N X N --> E(N; 2) qui donne une serie formelle en 2: (16.1)
[u, v]2 = {u, v} +
2rCr(u, v) r=1
ANDRE LICHNEROWICZ
282
ou les C,r(u, v) sont des 2-cochaines sur N qui s'etendent naturellement a E(N; 2); (16.1) definit une deformation formelle de l'algebre de Lie N si l'identite de Jacobi est formellement satisfaite (16.2)
S[[u, v] w], = 0 ,
oil S est la sommation apres permutation circulaire. On sait, d'apres Gerstenhaber [17], que (16.2) peut etre traduit par (16.3)
8C,=Et
(t =
oil (16.4)
E,(u, v, w) _
SC,(Cr(u, v), w) r+S=L
r.SZ1
Si (16.3) est satisfaite pour t = 1, . , q - 1, on a 8E, = 0 et Eq est un 3cocycle de N. On peut trouver une 2-cochaine Cq verifiant (16.3) pour t = q
si et seulement si le 3-cocycle Eq est exact. La classe define par Eq est l'obstruction a l'ordre q a la construction d'une deformation formelle de N [17].
On dit que (16.5)
[u, v], = {u, v} + 2C(u, v)
definit une deformation infinitesimale de N si l'identite de Jacobi correspondante est satisfaite a l'ordre 2, c'est-a-dire si C est un 2-cocycle de N. Une deformation formelle (resp. infinitesimale) de N est 1-differentiable si les 2-cochaines Cr de (16.1) (resp. C de (16.5)) sont supposees 1-differentiables. 11 resulte de (16.4) et du lemme suivant que cette restriction fournit un cadre coherent pour les deformations. Lemme. Si C, C' sont des 2-cochaines 1-differentiables sur N, la 3-cochaine
D define par 2D(u, v, w) = SC(C'(u, v), w) + SC'(C(u, v), w)
est 1-differentiable. De plus si C' = C = (A, B), on a (16.6)
D = (2 [A, A] - B A A, - [B, A])
La demonstration de ce lemme est identique a celle donnee dans [8, § 81. Si (16.3) est satisfaire pour t = 1, , q - 1 par des 2-cochaines 1-differentiables, it resulte du lemme que Eq est un 3-cocycle 1-differentiable sur N. L'element de H3(N) defini par Eq est l'obstruction a l'ordre q a la construction d'une deformation formelle 1-differentiable de N. (b) Consid&ons une s&ie formelle en 2
LES VARIETES DE POISSON
(16.7)
T, = Id +
283
28T, S=1
oii les T, sont des operateurs differentiels d'ordre s sur N, T, opere naturellement sur E(N; 2). Nous dirons que (16.1) est une deformation formelle triviale de N s'il existe (16.7) tel que l'identite (16.8)
T,[u, v], - {T,u, T,v} = 0
soit formellement satisfaite. On deduit du theoreme du § 7 par un raisonnement identique a celui de [8, § 9] la coherence de cette definition. La deformation infinitesimale.(16.5) est dite triviale s'il existe une 1-coehaine 1-differentiable T telle que (16.9)
T, = Id + AT
verifie (16.8) a dordre 2. Pour qu'il en soit ainsi, it faut et it suMt qu'il existe T telle que C = aT, c'est-a-dire que le 2-cocyle C soit exact dans la cohomologie 1-differentiable de N. La trivialite definit sur les deformations infinitesimales
une relation d'equivalence et l'on a Proposition. L'espace des deformations infinitesimales 1-dif}erentiables de N, modulo les deformations triviales, est isomorphe a HZ(N), soit P'(W; G) O+ HZ(W ; G)/QZ(W ; G)
17. (a) (17.1)
.
Deformations formelles et infinitesimales inessentielles
Considerons. une serie formelle en 2
G,=G+ E2Gr, r=1
of les G. sont des 2-tenseurs de W tels que l'identite (17.2)
[G GA] = 0
soit formellement satisfaite. Il est equivalent de dire que le crochet (17.3)
{u, v},, = i(G,)(du A dv)
,
(u, v r= N) ,
satisfait formellement 1'identite de Jacobi ; (17.3) definit aussi une deformation formelle 1-differentiable de N qui se deduit Tune deformation formelle de la structure geometrique de variete de Poisson; (17.3) est ,dite une deformation formelle de Poisson de N. Une deformation formelle 1-differentiable [u, v], de N est dite inessentielle s'il existe G, et T, tels que
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284
T,[u, v], - {T,u, T,v}G, = 0 .
(17.4)
(b) Soit G1 un 2-cocycle pur sur N et Tune 1-cochaine 1-differentiable tels que pour une deformation infinitesimale 1-differentiable (16.5),
G,=G+AG1,
T,=Id+2T
verifient (17.4) a dordre 2. La deformation infinitesimale (16.5) est alors dite inessentielle. Pour qu'il en soit ainsi, it faut et it suffit que C soit homologue dans H3(N) an 2-cocycle pur (G1, 0). L'inessentialite definit sur les deformations infinitesimales une relation d'equivalence et l'on etablit comme dans [8, § 10] Theoreme. L'espace des deformations infinitesimales 1-differentiables de N, modulo les deformations inessentielles, est isomorphe a P'(W ; G). Pour qu'une deformation formelle 1-differentiable de N soit inessentielle, it est necessaire que la deformation infinitesimale definie par sa partie d'ordre 1 le soit. (c) Supposons G exact dans la G-cohomologie. D'apres une remarque du
§ 5, b,,ona
H'(N) = H'(W; G)H'(W; G) , H3(N) = H3(W ; G)
H3(W ; G) ,
oii H'(W ; G) est isomorphe a LG/L*. On deduit du theoreme precedent Corollaire. Soit (W, G) une variete de Poisson telle que G soit exact dans
la G-cohomologie (variete de Poisson exacte). Si LGIL* est # {0} et si HZ(W ; G) = H3(W ; G) = {0}, l'algebre de Lie dynamique N admet des deformations formelles 1-differentiables essentielles (et en particulier non triviales)
VI.
VARIETE SYMPLECTIQUE ET DYNAMIQUE ASSOCIEE A UNE SOUS-VARIETE 18.
Sous-variete symplectiquement reguliere
Dans un article classique [7], Dirac a ete amene a etudier la dynamique analytique associee a une sous-variete d'une variete symplectique dans un contexte local et non invariant. Sniatycki [16] et Tulczyjew ont etudie recemment la geometrie globale sous-jacente. Nous nous proposons ici de reprendre cette etude en la precisant et de determiner la dynamique associee a cette geometrie. Nous preservons autant que possible la terminologie initiale de Dirac. Soit (W, F) une variete symplectique de dimension 2n, de 2-forme fondamentale F; nous posons encore G = lr1(F) (notations du § 1, c). Nous nous
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donnons une sous-variete reguliere fermee M de W de codimension h (variete des etats permis). Nous nous proposons d'etudier la dynamique analytique correspondant a M et define a partir d'un hamiltonien H e N = C°°(W ; R). Nous analysons d'abord la situation geometrique. (a) Soit U un domain contractile de W tel que M n U 0. On note WU (espace des contraintes pour U) le sous-espace de C°°(U ; R) defini par les fonctions f telle que fMn( = const. ; Cu est ici l'espace des champs de vecteurs hamiltoniens X = pUl(df), definis sur U, associees aux f e VU. On peut trouver sur U des systemes {x¢}, a = 1, , h, de h fonctions independantes de rU tels que M soit define sur U par x¢ = 0; it existe alors des cartes locales de W de domaine U de la forme {xa, xi}, i = h + 1, , 2n. Si f e WU, on deduit de df !MnU = 0 que le vecteur x = pU1(df) e Cu est tel qu'en x e m fl U on a i(X,)FPM = 0. Par suite X2 appartient a l'espace
vectoriel
K, = {V e T2(W) ; i(V)FIM (x) = 0}
(x e M).
Inversement si V e K., on peut trouver, pour un domain convenable U tel que x e M fl U, un element f e W u tel que pour le vecteur correspondant X., = V. Nous sommes conduits a introduire l'ensemble
qui admet une structure naturelle de fibre vectoriel sur M et dont la fibre Kx est de dimension h. On note 1U (espace des fonctions de premiere classe pour U) le sous-espace de C`°(U; R) defini par les fonctions f telles que, pour tout element g de rU, on ait If, g}MnU = 0; BU est l'espace des champs de vecteurs hamiltoniens X = pul(df) associes aux f e .qU. La relation de definition de RU exprime que, pour tout X e RU, X 1,,u est tangent a M. On note du (espace des contraintes de premiere classe pour U) l'intersection
dU = au fl WU ; AU est l'intersection BU fl CU. Le lemme suivant est immediat. Lemme. (1°) Si f e RU, g e WU, alors If, g} e VU . (2°) Le crochet de Poisson munit RU d'une structure d'algebre de Lie.
(3°) du est un ideal de l'algebre de Lie RU. Le 3° resulte des 1° et 2°. Il suffit d'etablir le 2° qui est une consequence immediate de 1'identite de Jacobi pour les crochets de Poisson, appliquee a deux elements de RU et un element de VU. Ainsi BU est une sous-algebre de l'algebre des champs hamiltoniens sur U et AU est un ideal de BU. (b) Nous faisons dans la suite de cette section, 1'hypothese suivante. Hypothese (H). La 2-forme fermee FM induite sur M par F est de rang
fixe (2n - h - k).
h + k (ou h - k) est ainsi pair. Pour que V e T.,(M) annule FM en x e M,
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286
it faut et it suffit que V appartienne a Qy = {V E T.x(M) ; i(V)FIM (x) = 0} = T.,(M) n K.,
.
Il en resulte que sous l'hypothese (H), Q,,, a la dimension k et Q = T(M) n K est un fibre vectoriel sur M. S'il en est ainsi, nous pouvons, introduire le fibre vectoriel
P=T(M)+K, dont la fibre est de dimension (2n - k). Q definit sur M un champ-encore note Q-de k-plans Q. Si X, Y sonf des
sections locales de Q, on a i(X)FM
0, i(Y)FM = 0 et on en deduit
i([X, Y])FM = 0 puisque FM est fermee. Il resulte du theoreme de Frobenius que le champ Q est integrable et definit un feuilletage de M en sous-varietes integrales maximales. Si R est la relation d'equivalence define sur M par le feuilletage precedent, les points equivalents de M decrivent un meme etat dynamique (avec "changement de jauge"). Soit M = M/R 1'espace quotient; p: M M est la projection correspondante. Nous supposons Hypothese (H). La projection p munit M d'une structure de variete differentiable de dimension (2n - h - k) telle que p soit elle-meme de rang
(2n - h - k) (submersion). Nous posons [13] Definition. La sous-variete M de (W, F) est dite symplectiquement reguliere si les hypotheses (Hl) et (H) sont satisfaites. Supposons qu'il en soit ainsi ; si X est une section locale de Q, on a i(X)FM = 0, dFM = 0. Il existe par suite une 2-forme fermee P de M, de rang (2n -
h - k), telle que FM-p*F.
(18.1)
La variete symplectique (14,P) est la variete des etats dynamiques definis par M. (c) Si f e au, le vecteur X = p-'(df) est tel qu'en x e M (1 U, on ait Xz E T.(M) (1 K,,, = Q.. Sous l'hypothese (H1), on peut trouver une carte locale de W de domaine U , 2n; ,i = , h ; i = h + 1, de la forme {xd, xi} = {xd, x', x"}, (a = 1 ,
telle que sur
la variete M (qui est define par xa = 0 dans U), le feuilletage soit defini dans
U par x' = const. Il en resulte Fli = o .
(18.2)
La matrice (F,,,) est de rang k puisque VF,,, = 0, equivalent a V'F,A = 0 implique V'
0.
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Pour 2 fixe, considerons une fonction y' telle que sur M fl u Y'(0, XI) = 0
,
(aay')(0, xI) = Faa(x')
Nous pouvons poser par exemple sur U : (18.3)
Y'(xa, x1) = xaFza(X')
La fonction y' s'annule sur M n U et est telle que le champ XM = ual(dy') est tangent a M sur M n U puisqu'il admet les composantes (X('Ia = 0, X(')' = 1, X«:N = 0 pour p * 2, X(1 2' = 0). Ainsi y' appartient a slU. Pour x e
MfU
oil (ay'/axa) est de rang k. On pent construire une carte de W, de domaine U an besoin reduit, {ya, xi} oil {ya} est de la forme {y', y°}, (2 = 1, ,k;a= k + 1, , h). On en deduit en particulier le lemme suivant. Lemme. Si V e Qy (avec x E M fl U), on peut trouver une fonction f e .ABU telle que le champ de vecteurs X = is-'(df) verife X,, = V. 19. (a)
Hamiltonien admissible et dynamique correspondant a M
Etant donnee une sous-variete symplectiquement reguliere M de (W, F,
considerons une fonction H E N = C`°(W ; R) et designons par Z = u-1(dH) le champ hamiltonien correspondant. Nous introduisons la definition suivante [13].
Definition. H est dit un hamiltonien admissible pour M si ZIM est une section du fibre vectoriel P. Supposons qu'il en soit ainsi et soit U un domain de W tel que m fl u soit
*0; H etant admissible, Z admet des decomposition de la forme (19.1)
ZIMnu = ZMnU + XMnU
,
ou ZMnU et XMnU sont des sections locales respectivement de T(M) et de K. Les decompositions (19.1) nous conduisent an lemme suivant. Lemme. Si H est un hamiltonien admissible pour M, H I U admet des decom-
position de la forme (19.2)
HIU=H U + fU,
ou HU E RU, fu E VU. Une telle decomposition est definie a HU -> HU + gU, fu - fu - gu pres, oh gU e slU. Elle definit par passage aux champs hamiltoniens correspondants et restriction a M une decomposition (19.1). Inversement toute decomposition (19.1) de ZMnu peut titre ainsi definie.
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ANDRE LICHNEROWICZ
En effet partons d'une decomposition (19.1) de Zl,nu. Introduisons une carte locale {xa, xi} de W de domaine U telle que M fl U soit define par xa = 0, a = 1, , h. Sur M fl U, ,u,(X,,U) definit une 1-forme de composantes ta(xi) ; considerons sur U la fonction f, E `e, donnee par Wxi)xa
fu =
Soit x un point de M fl U. E/ n ce point xa(x) _ /0 et 1'on a
dfu(x) = a(xi)dxa = c(x) On en deduit XMnu(x) = (pj1(dfu))(x)
Pour cette fonction fu, posons sur U
HIU=HU+fU, ZU=pal(dHU), Xu=pU1(dfu) de telle sorte que Zlu = ZU + X. Sur M fl U, XU se reduit a XasnU et par suite ZU se reduit a Zmnu tangent a M. Nous avons bien mis en evidence une decomposition de HIU en somme d'un element HU de RU et d'un element fu de VU, decomposition (19.2) qui done naissance a la decomposition (19.1) donee de ZImnu. Notre lemme est etabli. (b) En dynamique analytique classique, sous sa forme elementaire locale,
un mouvement dans (U, Flu), soumis aux contraintes xa = 0 et associe a l'hamiltonien HIU, s'obtient a partir d'un hamiltonien sur U de la forme (19.3)
HU=HIU-Z'2ax", a
of les A. sont les multiplicateurs associes aux xa. Sur M fl U, on a ZUImnU = ZMnU = Zlarnu - Z a2apU1(dxa)
ob les .2a sont choisis de facon que Zmnu soit tangent a M; (19.3) fournit une decomposition de Hlu du type (19.2), puisque Flu e °Mu, fu = Z 2a xa E VU (c) Dans le contexte et avec les notations du lemme du a, nous sommes ainsi conduits a nous interesser aux trajectoires de Z,,,u, tangent a M, defini a partir de HU E Ru ; ZU E BU est defini modulo un element de AU et determine une classe element de BU/AU. Si YU est un element de AU, on a d'apres le lemme du § 18, a (19.4)
[Zr, YUl E AU
Il resulte de (19.4) et de 1'etude du § 18, c que les ZMnU passent au quotient par R et determine sur NI un champ global Z.
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Soit gU une fonction locale de W telle que {gu, hU} I m = 0 pour tout hU E f u. La restriction a M d'une telle fonction est, d'apres § 18, c, l'image reciproque par p d'une fonction locale de M. Il en est en particulier ainsi pour tout element de RU, donc pour Hu. Il existe une fonction locale HU de M telle que HUIM = p*Hu et les dHU definissent sur M une 1-forme fermee globale cp telle que 1(cp) = Z, oiI 2 est l'isomorphisme defini par la structure symplectique de (M, F). Nous enoncons Theoreme. Soit M une sous-variete symplectiquement reguliere de la variete symplectique (W, F) et soit H un hamiltonien admissible pour M. Le champ hamiltonien Z = p-'(dH) determine sur la variete (M, )) des etats dynamiques un champ global unique Z, localement hamiltonien dont les trajectoires definissent le mouvement dans la variete des etats dynamiques ; Z est dit le champ dynamique.
20.
Sous-variete de seconde classe et crochets de Dirac
Une sous-variete M de la variete symplectique (W, F) est dite de premiere classe si Q = K, c'est-a-dire si k = h. Elle est dite de seconde classe si Q est de fibre nulle, c'est-a-dire si k = 0; h est alors pair et nous posons h = 2h'. Si M est de seconde classe, la forme FM induite par F sur M est de rang 2(n - h') et la variete (M, FM) est symplectique. (a) Sniatycki [16] a etabli substantiellement la proposition suivante. Proposition. Soit M une sous-variete symplectiquement reguliere de (W, F). Il existe des sous-varietes symplectiques (W, F) de seconde classe de (W, F) telles que M soit une sous-variete de premiere classe de Nous affectons d'un - les elements relatifs a W ; F est ici la 2-forme induite
par F sur W et G =
1(F).
La variete symplectique (W, F) et la sous-variete M symplectiquement reguliere etant donnees, nous allons montrer que, malgre l'arbitraire sur he choix de Jr", on peut utiliser la variete (W, F) comme intermediaire pour obtenir la dynamique relative a M donnee par he theoreme du § 19.
En effet considerons W comme sous-variete de (W, F) au sens du § 18 ; comme k = 0, it n'y a pas de passage au quotient et (W, F) est sa propre variete des etats dynamiques. Un hamiltonien H arbitraire de (W, F) est toujours admissible pour W. On peut ecrire d'une maniere et d'une seule (20.1)
ZIw = Z -I- X ,
oil, pour . E W, on a Z(.) E T,(W), !(9) E K, Soit U un domain de W tel que W fl U # 0; -9u se compose de constantes. Nous notons {x-4} _ {xa, x1} une carte locale de W de domain U telle que W soit define par Xa = 0. La 2-forme F, restriction de F a W admet sur W n u les composantes F,; = Fij, (i, j = h + 1, , 2n). D'apres le raisonnement
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ANDRE LICHNEROWICZ
du § 19, HIu admet une decomposition de la forme (19.2), relativement a W, soit (20.2)
Hju=Flu +fu,
ou les fonctions Flu E -Vu, f u e 'u sont defines a une constante additive pres. Les vecteurs bien definis sur U
Zu = p-1(dHu)
,
Xu = p-'(dfu)
et X Iu. D'apres la definition de se reduisent sur W n U respectivement a Zu, p(Z) J u admet en . E W fl U les composantes
p(Z)A(x) = (Z'FAJ)(x) = (aAHu)(x) Il en resulte que p(Z) lu admet en z les composantes
p(Z)s(x) _ (Z'Fzj)(x) _ (Z'Fz. )x = (azHu)(x) _ (a1H)(x) Si k est la restriction de H a W, on obtient ainsi sur W (20.3)
Z = ,u-1(dH) .
Le champ hamiltonien 2 est le champ dynamique pour (W, P) consideree comme variete des etats. (b) Cela pose, revenons a M, sous-variete symplectiquement reguliere de (W, F) et sous-variete de premiere classe de (W, P). La forme FM peut etre consideree comme induite sur M par P et M est symplectiquement reguliere dans (W, F). Il resulte de plus de 1'etude de Z au a que (20.4)
ZiM = ZJM + Y .
ou Y(x) E K., pour tout x de M. On en deduit que pour que l'hamiltonien H de (W, F) soit admissible pour M, it faut et it suffit que sa restriction k a W soit admissible pour M consideree comme sous-variete de (W, F. D'autre part, d'apres (20.4), le champ dynamique Z sur M peut etre deduit de Z conformement au § 19. Nous enoncons Theoreme. Sous les hypotheses du theoreme du § 19, soil (W, P) une sousvariete de seconde classe de (W, F) telle que M soil une sous-variete de premiere
classe de (W, F). La variete (M, F) des etats dynamiques peut etre define en considerant M comme sous-variete 'symplectiquement reguliere de (W, P). Si H est un hamiltonien sur (W, F) admissible pour M, le champ dynamique 2 de M pent etre defini a partir de la restriction k de H a W. Ainsi, pour tout choix de W, la dynamique associee a M, sous-variete de (W, P), et a I'hamiltonien k coincide avec la dynamique associee a M, sous-variete de (W, F), et a I'hamiltonien H.
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(c) L'etude precedente montre l'interet des sous-varietes de seconde classe qui determinent en fait l'essentiel de la dynamique envisagee, puisque l'introduction des contraintes de premiere classe ne se traduit que par un passage au quotient. Dans la suite, nous nous limit ons a 1'etude de sous-variete de seconde classe (W, F), de codimension h = 2h', de la variete symplectique donnee (W, F), de dimension 2n, de 2-tenseur fondamental G = r-'(F). Nous adoptons pour W des notations identiques a celles relatives a M. Il vient, W etant de seconde
classe
T(W) 1* = T(W) E K .
(20.5)
Nous notons H : V E T(W) IG , V e T(W) le projecteur defini par la decomposition (20.5). Ce projecteur s'etend naturellement aux 2-tenseurs. Nous montrerons que le 2-tenseur HG de W n'est autre que le 2-tenseur fondamental
G = p-'(P) de la variete symplectique (W, F) ; it en resulte que ce qu'on nomme crochet de Dirac est directement defini a partir du crochet ordinaire de Poisson relatif a (W, P). 21. (a)
Une etude locale
U etant un domain contractile de (W, F), considerons la variete
symplectique (U, FLU) que nous notons dans ce paragraphe, par abus de notation, (U, F). Soit U une sous-variete de seconde classe de (U, F) define par , h, sont xa = 0, oil les h fonctions de contraintes xa E C°°(U ; R), a = 1, independantes. Nous introduisons sur U les h vecteurs (21.1)
P(a) = u-'(dxa) .
, = 1, Il existe sur U des cartes {xA} de la forme {xa, xi} (avec A, B, 2n). Dans une telle carte, , = h + 1, = 1, - , h ; i, j, 2n ; a, b, les P(a) ont pour composantes
P(a)B = GaB = {Xa XB} .
En particulier, (21.2)
P(a)b = Gab = {Xa, Xb}
Pour x c U, les P(a)(x) definissent une base de l'espace K,x relatif a U. Pour V E T,x(U), on peut, d'apres (20.5), ecrire d'une maniere unique (21.3)
V = 1 + z 2ap(a)(X) a
oil V E T.(U), c'est-a-dire admet des composantes {Vb = 0, Vi} dans la carte
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envisagee. 11 en resulte que pour tout ensemble {Vb}, le systeme de h equations lineaires aux h inconnues Aa 2ap(a)b(x) = Vb
(21.4)
admet une solution unique. Ainsi la matrice h x h define par (21.2) est inversible sur (T, donc sur U, au besoin reduit. L'inversibilite sur U de la matrice ({xa, xb}) traduit ainsi le caractere de sous-variete de seconde classe de U. (b) Nous sommes conduits a considerer le feuilletage de U de codimension h defini par xa = const., a = 1, , h. Il resulte des considerations precedentes que la feuille U(x) passant par x E U de ce feuilletage est une sous-variete de seconde classe de (U, F). C'est a ce feuilletage que nous nous interessons maintenant; le meme feuilletage peut titre defini, en substituant a 1'ensemble {xa} des fonctions de contraintes, un autre ensemble {xb' = xb'(xa)} de fonctions de contraintes independantes, a jacobien non nul.
Soit C = (Cab) la matrice inverse de (21.2). On a (21.5)
CaCGbC
= Cae{xb, xb} = 6¢
Si X est un champ de vecteurs sur U, on ecrit d'une maniere et d'une seule
X=X+T2aP(a) a ou X est tangent au feuilletage, E U 'al est une section du fibre ' sur U, defini par les differents K et ou as = Cabi(X)dxb .
Sur U est defini le projecteur H : X - X ne dependant que du feuilletage, qui s'exprime par
H : X -p X = X - CabP(a) (i(X)dxb)
,
(sommation en a, b)
Dans une carte locale {xA} arbitraire, H admet pour composantes
HC = aC -
CabP(a)AaCxb
Evaluons le 2-tenseur HG tangent aux feuilles et qui ne depend, d'apres sa
definition, que du feuilletage. 11 vient dans la carte {x Al (HG)AB = (a& - CabP(a)AaCxb)(&BD - CCdP(C)BaDxd)GCD
En developpant et simplifiant, compte-tenu de (21.5), on obtient (21.6)
(HG)AB = GAB - CabP(a)AP(b)B
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(21.6) est, aux notations pres, la formule ecrite par Dirac donnant le 2-tenseur definissant son crochet. Si nous posons
T=
(21.7)
2CabP(a) A p(b)
Il vient
17G=G-P,
(21.8)
ou 1', comme 17G ne depend que du feuilletage. (c) Soit F la 2-forme induite par F sur une feuille U. Dans une carte locale adaptee au feuilletage, HG verifie
(HG)aa = 0
(21.9)
et l'on a (17G)ikFjk
= (Gik -
CabP(a)ip(b)k)Ffk
soit
(11 G)ibPfk = GiAFIA - GiaFia - CabGai(GbAFjA - Gb'F,C)
Il en resulte (17G)ik'Pjk = a; - GiaF1a + GiaFga c'est-a-dire
(21,10)
(HGik)F',k = a;
On a montre que la restriction de HG a chaque feuille coincide avec le 2tenseur fondamental ir'(F) de la feuille. Nous noterons dans la suite a le 2tenseur HG de U. 11 resulte des considerations precedentes que sur U (21.11)
[6,G]=0,
et que pour toute fonction de contrainte (21.12)
[G, xa] = 0 .
Ainsi G definit sur U une structure de Poisson, dont le feuilletage associe est le feuilletage donne de U en sous-varietes de seconde classe. Le crochet de Dirac n'est autre que le crochet defini par cette structure : si u, v e C`°(U ; R), le crochet de Dirac {u, v}D est donne par {u, v}D = i(G)(du A dv) .
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(d) L'etude precedente entrain, avec les notations du § 20, c. Proposition. Si W est une sous-variete de seconde classe de la variete symplectique (W, F), le 2 tenseur HG de 1 " n'est autre que le 2-tenseur fondamental de la variete symplectique (1%', P). Son expression locale est donne par (21.6) ou (21.8) et ne depend que de W.
22.
Cas d'un feuilletage donne en sons-variete de seconde classe
(a) Soit (W, G) une variete symplectique de dimension 2n. Supposons donne un feuilletage de W de codimension h = 2h' en sous-varietes W(x) de seconde classe. Du raisonnement du § 21, it resulte que G et le feuilletage definissent
sur W un 2-tenseur G' = G - r de rang (2n - 2h'), verifiant (22.1)
[G, 6]=[G-r,G-r]=0,
c'est-a-dire une structure de Poisson dont le feuilletage associe est le feuilletage de W en sous-varietes de seconde classe donne ; (22.1) peut s'ecrire
2[G, r] = [r, r]
(22.2)
.
Introduisons le 2-tenseur
G,=G-dr, qui se reduit a G pour d = 0, a G pour d = 1. Si l'on a (22.3)
[G G,] = 0 ,
le crochet {u, v}G, = i(G)(du A dv) definit une deformation rigoureuse (inessentielle) de l'algebre de Lie dynamique de la variete symplectique (W, G) en 1'algebre de Lie dynamique de la variete de Poisson (W, G). Pour que (22.3) soft satisfaite, it faut et it suffit que (22.4)
[G, r] = 0,
[r, r] = 0
.
D'apres (22.2) l'une de ces conditions entrain 1'autre; en particulier it faut et it suffit que r soit un 2-cocycle. Le § 18, a nous conduit a considerer le 2h'-plan K, defini en chaque point x de W par Kx = {V E T,(W) ; i(V)FjW(X,(x) = 0}
.
Nous avons ainsi defini sur la variete W un champ K de 2h'-plans. Nous allons etablir Theoreme. Pour que G, = (G - dr) definisse une deformation rigoureuse du crochet de Poisson en le crochet de Dirac, it faut et it suff"it que le champ K soit un champ integrable
LES VARIETES DE POISSON
295
En effet supposons le champ K integrable. Sur un domaine U de W, on peut , 2h', = 1, definir les feuilles du feuilletage par xa = const., a, b, c, , 2n), {x-4} = les integrales maximales de K par xi = const. (i = 2h' + 1, {xa, xi} definissant une carte locale de domaine U. Le champ K restreint a U est engendre par les champs de vecteurs P(a' et l'on a par suite
p(a)i = Gal = 0 Le 2-tenseur r a pour composantes Fed = CabGacGbd = Ged
La relation [G, G] = 0 s'ecrit sur U SGdaadGbe + SGiaaiGbe = 0
ou S est la sommation apres permutation circulaire sur (a, b, c), soit Srdaadrbe = 0
ce qui exprime que [F, r] = 0 et (22.4) est satisfaite. Inversement, supposons (22.4) satisfaite. Il est equivalent, d'apres (1.8), de dire que la 2-forme Q = p(r)
est f ermee.
Introduisons une carte locale {xa, xi} de domaine U telle que les feuilles du , 2h'; i = 2h' + 1, = 1, feuilletage soient definis par Xa = const., (a, b,
, 2n). On a sur U 011 = 2Cabdxa A dxb ;
0 etant fermee, it vient aiCab = 0 et par suite, d'apres (21.5), aiGab = 0, soit (22.5)
ai{xa, xb} = 0 .
D'autre part, pour que K restreint a U soit integrable, it faut et it suffit d'apres le theoreme de Frobenius que [P(a', Pub'] snit une combinaison lineaire des P(e', c'est-a-dire, par image par p, que d{i(G)(dxa A dxb)} = dfxa, xb}
soit une combinaison lineaire des dxc, ce qui est equivalent a (22.5). Notre theoreme est etabli.
298
ANDRE LICHNEROWICZ
Soit K2 le champ de 4-plans defini sur U par les quatre champs de vecteurs independants Pct), P(l) pc2> p(2)On verifie immediatement que [P (a), p(b) l
=0,
(a, b = 1, 1, 2, 2)
Le champ K2 est done integrable et on pent trouver sur U une carte {x1, x', x2, x2, xi2} telle que les integrales du champ K2 soient defines par x12 = const. ; les seules composantes non nulles des quatre vecteurs envisages sont alors : P(I)'
=1
,
P(1)1 = -1
,
=1
Pc2>2
,
p(2)2
= -1
La forme F s'ecrit dans la carte envisagee : F = dx1 A dxI + dx2 A dx2 + 2Fi272dxi2 A dxJ2
(23.10)
Introduisons sur U le. 2-tenseur r2 = Pc2> A p(2)
(23.11)
Ce 2-tenseur verifie par construction (23.12)
[G, r2l = 0 ,
[r2, r2l = 0
et de plus
[r r2l = 0
(23.13)
11 en resulte que, pour toute valeur des parametres 2, 2, le 2-tenseur
G,,,2=G-2,r,-22x2
(23.14) verifie
(23.15)
[G,,,,, G,,,21 = 0 .
On a ainsi mis en evidence un feuilletage de U de codimension 4, defini par x1 = const., x' = const., x2 = const., x2 = const., admettant S2 (contenant U) comme feuille et tel que G,,,2 satisfasse (23.15). (c) En poursuivant le processus, on definit sur U une carte {x1, x1, , xh', xh', xi} jouissant de la propriete suivante : si nous posons
P"> = u-'(dx')
,
P(2) = u-'(dxl)
(2 = 1, ... , h')
,
et
ra = P(l) A Pux> le 2-tenseur
,
,
LES VARIETES DE POISSON
299
=G-2,I''- ... -2",I'", verifie (23.16)
et le feuilletage de U, de codimension 2h', defini par x' = const., x' = const. (2 = 1, , h') admet U comme feuille. Pour 21 = _ 2n,. = 1, la restriction de a U definit la structure symplectique de cette sous-variete de seconde classe. Nous avons Theoreme. Etant donnee, dans la variete symplectique (U, F) une sousvariete U de seconde classe, on peut definir un feuilletage de U, de codimension
2h', en sous-varietes de seconde classe, admettant U pour feuille, tel que si G = G - F definit la structure de Poisson correspondante de U, le 2-tenseur G; = G - 21' verifie pour tout 2 [G Gz 1 = 0
.
On a meme montre qu'il existe sur U un 2-tenseur G,,...,n, dependant lineairement de h' parametres, verifiant (23.16) et se reduisant a G, pour 21 = = 2, = 2. On peut donc relier G a G par une famille lineaire a h' parametres de structures de Poisson. References
[1] [2] [3]
[4] [5] [6]
[7] [8]
[9]
R. Abraham & J. Marsden, Foundations of mechanics, Benjamin, New York, 1967. V. I. Arnold, One-dimensional cohomologies of Lie algebras of nondivergent vector fields and rotation numbers of dynamic systems, Functional Anal. Appl. 3 (1969) 319-321. A. Avez & A. Lichnerowicz, Derivations et premier groupe de cohomnologie pour des algebres de Lie attachees a une variete symplectique, C. R. Acad. Sci. Paris Ser. A, 275 (1972) 113-118. A. Avez, A. Lichnerowicz & A. Diaz-Miranda, Sur l'algebre des automnorphismes infinitesimnaux d'une variete symplectique, J. Differential Geometry 9 (1974) 1-40. E. Calabi, On the group of automnorphismns of a symnplectic manifold, Problems in Analysis, A sympos. in honor of S. Bochner, Princeton University Press, Princeton, 1970, 1-26. C. Chevalley & S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948) 85-124. P. A. Dirac, Generalized Hamniltonian dynamics, Canad. J. Math. 2 (1959) 129148; Lectures on quantum mechanics, Yeshiva University, New York, 1964. M. Flato, A. Lichnerowicz & D. Sternheimer, Deformations 1-dif}erentiables d'algebres de Lie attachees a une variete symnplectique ou de contact, C. R. Acad. Sci. Paris Ser. A, 279 (1974) 877-881; Deformations 1-differentiables des algebres
de Lie attachees a une variete symplectique ou de contact, Compositio Math. 31 (1975) 47-82. , Aggebres de Lie attachees a une variete canonique, J. Math. Pures Appl. 54 (1975) 445-480.
ANDER LICHNEROWICZ
300
[10]
C. Godbillon, Geometrie differentable et mecanique analytique, Hermann, Paris,
[11]
A. Lichnerowicz, Cohomogie 1-differentiable d'algebres de Lie associees a une
1969.
variete symplectique, C. R. Acad. Sci. Paris Ser. A, 277 (1973) 215-219;
[12]
Cohomologie 1-differentiable des algebres attachees a une variete symplectique ou de contact, J. Math. Pures Appl. 53 (1974) 459-484. , Varietes canoniques et transformations canoniques, C. R. Acad. Sci. Paris Ser. A, 280 (1975) 37-40; Varietes symplectiques, varietes canoniques et systemes dynamiques, Topics in differential geometry, Academic Press, New York, 1976, 51-84; Cohomogie 1-differentiable et deformations de l'algebre de
Lie dynamique d'une variete canonique, C. R. Acad. Sci. Paris Ser. A, 280
[13] [14]
[15] [16] [17]
(1975) 1217-1220. , Variete symplectique et dynamique associee a une sous-variete, C. R. Acad. Sci. Paris Ser. A, 280 (1975) 523-527. A. Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields. I, Indag. Math. 17 (1955) 390-403. J. A. Schouten, On the differential operators of first order in tensor calculus, Convengo Intern. Geometria Differenziale Italia, 1953, Ed. Cremonese, Roma, 1954, 1-7. J. Sniatycki, Dirac brackets in geometric dynamics, Ann. Inst. H. Poincare Sect. A, 20 (1974) 365-372.
M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79 (1964) 59-103.
COLLEGE DE FRANCE, PARIS
J. DIFFERENTIAL GEOMETRY 12 (1977) 301-317
VARIETES DE TYPE SURJECTIF ET VARIETES PARTIELLEMENT PARALLELISABLES PAUL GAUDUCHON
INTRODUCTION
Au paragraphe 13 de [7], A. Lichnerowicz demontre qu'une "variete kahlerienne compacte a premiere classe de chern non-negative est fibree analytiquement au-dessus de sa variete d'Albanese."
Un des buts du present article est d'etendre ce resultat en supprimant l'hypothese kahlerienne. Plus precisement, nous attachons a une variete hermitienne compacte (M, g) sa connexion canonique ou connexion de chern,
a partir de laquelle est construite le tenseur de Ricci hermitien. Ceci etant pose, 1'extension proposee est possible (Theoreme de fibration du § 6) au prix des pertes suivantes: (a) Le groupe structural de la fibration analytique ne peut plus, en general,
etre reduit a un sous-groupe discret de la composante connexe de l'identite du groupe des automorphismes analytiques de M. (b) L'hypothese : "a premiere classe de chern non-negative" doit etre remplacee par celle-ci : "la variete admet une metrique hermitienne a tenseur de Ricci hermitien non-negatif". Si (M, g) est kahlerienne, le tenseur de Ricci hermitien coincide avec le tenseur de Ricci usuel (riemannien) ; la seconde hypothese dans ce cas implique la premiere, mais l'inverse nest pas vrai car un representant de la premiere classe de chern peut etre nonnegatif qui n'est le tenseur de Ricci d'aucune metrique hermitienne sur M. Ainsi, sur ces deux points, la situation kahlerienne reste plus riche que le cas hermitien general. Par ailleurs, les varietes hermitiennes (compactes) a tenseur de Ricci nonnegatif appartiennent a une famille de varietes plus vaste pour lesquelles le theoreme de fibration analytique reste vrai; c'est d'ailleurs dans ce cadre que
nous le demontrerons au § 6. Ces varietes que nous appelerons varietes de type surjectif, comptent dans leur rang, outre celles que nous avons deja mentionnees, les varietes complexes compactes homogenes.
A leur tour, les varietes de type surjectif entrent dans le cadre plus vaste des varietes partiellement parallelisables, qui sont des varietes M dont le fibre tangent holomorphe est partiellement trivialise par des champs de vecteurs a derivee covariante nulle pour une certain structure hermitienne sur M ; la Communicated by A. Lichnerowicz, May 24, 1975, and, in revised form, July 8, 1976.
302
PAUL GAUDUCHON
situation est decrite par le theoreme de parallelisme partiel du § 8. Les varietes compactes completement parallelisables analytiquement sont des cas-limites dans cette famille et quelques unes de leurs proprietes sont redemontrees de
fagon nouvelle aux paragraphes 7 et 8 comme applications des theoremes generaux portant sur les varietes partiellement parallelisees ou sur les varietes de type surjectif.
Dans tout le texte, M est une variete complexe, compacte, connexe, de dimension (complexe) n, munie d'un point-base z, fixe une fois pour toute.
Le fibre tangent holomorphe est note T(M) (ou
lorsqu'il n'y a pas
d'ambiguite), de fibre YZ(M) ou -Z au point z de M; Z est l'espace vectoriel complexe des vecteurs de type (1, 0) en z. Le fibre cotangent holomorphe et ses elements sont notes de fagon analogue en remplagant - par T ; TZ est l'espace vectoriel (complexe) des 1-formes de type (1,0) en z. Un champ de vecteurs holomorphe est une section holomorphe globale de , une 1-forme holomorphe une section holomorphe globale de T. Les deux adjectifs "holomorphes" et "analytique" sont synonymes.
1.
1.
LES FORMES D'ALBANESE D'UNE VARIETE COMPLEXE COMPACTE Rappels concernant la construction du tore d'Albanese
Nous rappelons brievement la construction du tore (ou variete) d'Albanese de M; cf. [1] ou [7]. Dans toute la suite du texte, H design l'espace des 1-formes holomorphes de M, B celui des 1-formes holomorphes fermees, de dimension complexe p.
A tout chemin r (que nous pouvons supposer C°° par morceaux) de M, d'origine z, et d'extremite z, nous associons 1'element 7 de B*, dual complexe de B, defini par
(1)
7(/3)=f
V(eB.
S.
Comme j9 est fermee, 7 ne depend que de la classe d'homotopie [r] de r (a extremites liees) ; en particulier, nous definissons un homomorphisme p du premier groupe d'homotopie 7r,(M, z0) dans B*, d'image d ; comme B* est commutatif, p induit un homomorphisme j du premier groupe d'homologie entiere H1(M, Z) dans B*, de meme image J. Deux chemins quelconques de zo a z induisent le meme element de B*/d; nous notons Jl l'application ainsi construite de M dans B*/J. L'espace vectoriel reel engendre par d dans B coincide avec B luimeme ;
VARIETES DE TYPE SURJECTIF
303
pour la demonstration de ce fait nous renvoyons a [1] : it revient a dire pour 1'essentiel que la partie reelle d'une 1-forme holomorphe fermee, bien que non harmonique, en general, au sens riemannien, est neanmoins caracterisee pour ses periodes ; it en resulte aisement la propriete qui vient d'etre dite ainsi que l'inegalite
2p induit par g (cf. comme dg(p) e L, dans la situation en[3]), an sous-espace L visagee, it est nul, ainsi que p. Nous avons donc montre Proposition 5. Une variete complexe compacte admettant une structure hermitienne a tenseur de Ricci hermitien non-negatif est de type surjeetif (et, mime, partiellement parallelise relativement a H).
310
PAUL GAUDUCHON
Remarque 1.
Pour une telle variete (M, g) a R non-negatif, it est possible
de choisir pour L1, l'espace a '(F) C L, dont tous les elements sont, avonsnous rappele, a derivee covariante nulle pour la connexion de Chern. Si (M, g) est kahlerienne, la connexion de Chern coincide avec la connexion riemanienne, i.e., n'a pas de torsion; autrement dit le crochet de deux champs de vecteurs a derivee covariante nulle est nul et Ll est une sousalgebre de Lie abelienne de L; le tenseur de Ricci hermitien coincide ici avec le tenseur de Ricci usuel (riemannien) : une variete kahlerienne compacte a tenseur de Ricci non-negatif entre donc dans le cadre plus large des varietes kahlerienne a premiere classe de Chern non-negative etudiees en [7] ; pour ces varietes it est encore possible de faire choix d'un Ll abelien qui ne peut toutefois s'ecrire, en general, sous la forme '(B) pour aucune metrique g, kahlerienne ou non, de M. Remarque 2. Si la caracteristique d'Euler-Poincare X(M) d'une variete compacte M n'est pas nulle, M n'admet aucun champ de vecteur continu-et
a fortiori, holomorphe-depourvu de zero sur M ; dans cc cas, pour tout sous-espace Q de H, 1'annulateur Qo de L daps Q coincide avec Q lui-meme ; nous avons donc, compte-tenu des propositions 4 et 5, Proposition 6. Soit M une variete complexe, compacte, de caracteristique d'Euler-Poincare non nulle. (a)
Si M est de type surjectif, l'unique forme d'Albanese sur M est la
forme nulle. (b) Si M est homogene ou si M admet une structure hermitienne a tenseur de Ricci non-negatif elle ne possede aucune 1-forme holomorphe en dehors de la forme nulle. Bien entendu le resultat est tout autre si X(M) est nul ; les tores complexes
soot des exemples de varietes qui entrent a la fois dans la categorie A et la categorie B et pour qui les espaces F, B et H coincident et different de {0}. La proposition 6 montre a contrario qu'une surface de Riemann de genre superieur a 1 n'est pas de type surjectif, contrairement a celles de genre 0 et 1. 6.
Le theoreme de fibration analytique
Theoreme de fibration analytique.
Une variete de type surjectif M est fibree analytiquement, par l'application de Jacobi, au-dessus de son tore d'Albanese. Le groupe structural de la fibration est un sous-groupe du noyau F de j dans G. La fibre est une sous-variete complexe, compacte, connexe, de M. La demonstration de ce theoreme se fait au moyen de trois lemmes. Lemme 1. J est de rang constant 2m. Nous rappelons que m est la dimension complexe du tore d'Albanese A(M) ; comme M est de type surjectif it existe, sur tout supplementaire L, de K, une base {XA} unique, duale d'une base donnee {bA} de F (proposition 3) ; l'espace L, et la base {XA}, A = 1, , m, seront fixes une fois pour toute.
VARIETES DE TYPE SURJECTIF
311
Pour z e M quelconque, le sous-espace L,(z) de , des valeurs en z des champs de L, est de dimension m et a tout X, e L,(z) correspond un champ unique X de L, tel que X(z) = X, (cf. remarque 1 du § 4) ; J,x(X,) est done nul, dans le cas seul oii le champ uniforme J,e(X) sur A(M) est nul, soit encore, en vertu de (5), si X est un element de K ; comme K fl L, = {0}, X est alors nul, ainsi que X, = X(z). La restriction de J,k a L,(z) est donc injective ; c'est donc, pour des raisons de dimension, un isomorphisme de L,(z) sur J(2)(A(M)) ce qui acheve la demonstration du lemme 1. Lemme 2. J est surjective.
De la demonstration du lemme 1 et de la Remarque importante du § 3, nous deduisons immediatement que J* : L -* LA est surjective ainsi donc que
J : G - GA. GA est transitif sur A (M) : pour dC e A (M), it existe un r e GA unique tel que _ r + C,; si g e G est tel que J(g) = r, on a, grace a (4) J(g z,) = J(g) + J(zo) = r + Co = C
qui demontre le lemme 2. Lemme 3. J possede un relevement analytique local a au voisinage de 0EGA. Considerons, dans GA, un voisinage U de zero, analytiquement isomorphe, par exp;1, a un voisinage 0& de zero dans LA ; nous pouvons supposer
symetrique i.e., tel que -0I coincide avec 0I et donc -U avec U. La restriction de J* a L, est un isomorphisme d'espaces vectoriels complexes de L, sur LA que nous noterons I. L'application a = exp o h1 o expA-' la de U dans G est analytique et verifie
(9)
fo6(r)=r
`V-EU,
en raison de la relation de commutation
Joexp = expAaJ* qui lie J et le morphisme d' algebres de Lie J* derive. Ceci acheve la demonstration du lemme 3. Dans le cas general oii L, n'est pas une sous-algebre de Lie complexe de L et oil, par consequent, exp (L) = G, n'est pas un sous-groupe de Lie de G, le relevement local a ne respecte pas les structures de groupe, mais it verifie neanmoins, comme on le voit immediatement sur la formule de definition, la relation (10)
G(-r) = 6-'(r)
Demonstration du theoreme.
Yr E U .
Soit Cz un point quelconque de A(M), et con-
312
PAUL GAUDUCHON
siderons le voisinage UI = U + CI de Cz ou U est le voisinage de 0 dans GA qui apparait dans la demonstration du lemme 3. Soit li 1'application de J'(UI) dans UI X J(CI)' construite de la facon suivante : pour z E J'(UI) nous posons 0I(z) = (yyb, Q'(r) z)
dz E J'(UI) ,
ou C E UI est la projection J(z) de z sur UI C A(M) et our est 1'element unique de U tel que (11)
C=r+CI
Il resulte immediatement de (9) et de (10) que vl(r) z appartient effectivement a J'(CI). L'application 0I, produit de deux applications analytiques, est elle-meme analytique, de meme que 1'application lI de UI X J'(CI) dans J'(UI) define par
T'I (b, z) = a(r) z
VC E UI , dz E P(CI) ,
our est 1'element unique de U defini par (11). On voit grace a (9) que appartient a J'(C) et que TI est 1'inverse de 0I. 0I est ainsi un isomorphisme .analytique de J'(UI) sur UI X Jl(CI) Comme A(M) est compact, le recouvrement {UI}, indexe sur A(M) luimeme, peut titre ramene a un sous-recouvrement fini {Ui}, i = 1, , N. Par ailleurs, it resulte de (4) que tout element g E G envoie une fibre de J sur une autre ;
comme J est surjectif, G operant de fibres a fibres est donc transitif : pour , N, nous pouvons faire choix d'un gi E G, non uniquement
tout i = 1,
determine, qui envoie J'(C) sur M. = J'(C.). Nous obtenons ainsi un recouvre-
ment fini de A(M) par des ouverts Ui, i = 1,
, N avec, pour chaque
ouvert, un isomorphisme analytique hi = (Id x gi) o 0I de J'(Ui) sur Ui X Mo ; J est donc une fibration analytique, de fibre type Mo = Si Ui fl U; # 0, h1 o h' est un automorphisme de Mo induit par un element
de G qui respecte les fibres de J, c'est a dire un element de T ; le groupe .structural de la fibration analytique est donc un sous-groupe de T. La fibre-type Mo est une sous-variete complexe de M, fermee dans M donc compacte ; pour montrer qu'elle est connexe it suffit de reproduire le raisonnement de [7, p. 641: comme Mo est compacte, le nombre de ses composantes connexes est fini ; la variete obtenue a partir de M en assimilant les points
d'une meme composante connexe d'une fibre de J est, en vertu de ce qui a deja ete demontre, un recouvrement analytique fini de A(M), donc un tore complexe de meme dimension m ; la propriete universelle de J implique alors que ce revetement fini ne peut titre que l'identite. Ceci acheve la demonstration du theoreme de fibration analytique. Corollaire 1. Une variete complexe compacte homogene est fibree analyti-
VARIETES DE TYPE SURJECTIF
313
quement au-dessus de son tore d'Albanese par l'application de Jacobi. Corollaire 2. Une variete complexe compacte admettant une structure hermitienne a tenseur de Ricci non-negatif est fibree analytiquement au-dessus de son tore d'Albanese par l'application de Jacobi. Remarque 1. Dans le corollaire 1, nous retrouvons un theoreme connu,. figurant dans [2].
Pour les rapports du corollaire 2 avec [7], voir la remarque 1 du § 3 et 1'Introduction. Remarque 2.
Dans le cas oii M est une variete kahlerienne, a premiere classe de Chern non-negative, it est montre dans [7], que l'application de Jacobi est encore une fibration analytique et le groupe structural est reductible a un sous-groupe discret de G, admettant un nombre fini de generateurs. Une telle reduction semble impossible dans le cadre general des varietes de type
surjectif et meme dans celui des varietes hermitiennes a tenseur de Ricci hermitien non-negatif.
III.
LE THEOREME DE PARALLELISME PARTIEL 7.
Les varietes partiellement parallelisables
Nous rappelons la definition de la Remarque 2 du § 4. Definition. Soit Q un sous-espace complexe de H, l'espace des 1-formes d'une variete complexe, compacte, M; M sera dite partiellement parallelisee relativement a Q, si l'armulateur Qo de L dans Q est reduit a {0}. Si Qo = {0}, aucune forme non-nulle de Q n'a de zero sur M. Supposons, inversement que l'espace Q possede cette derniere propriete ; sa dimension complexe q est alors inferieure ou egale a n et it induit sur M un sous-fibre holomorphe (trivial) de T, Q avec la suite exacte
0,Q-T-(T/Q)-0 et la suite exacte duale (S)
-.,A-0
on -9Q = (T/Q)* est le sous-fibre holomorphe de dont la fibre en z est le sous-espace des vecteurs de .°l, annules par les elements de Qz ; le fibre quotient .,', anti-isomorphe a Q est, de ce fait, analytiquement trivial. Definition. Un sous-espace complex Q de H est dit distributif si aucun de ses elements en dehors de zero ne s'annule sur M. Proposition 7. Soit Q un sous-espace distributif de H, de dimension complexe q, sur une variete complexe, compacte M. La suite exacte associee de fibres holomorphes
314
PAUL GAUDUCHON
0se scinde analytiquement si et seulement si M est partiellement parallelisee relativement a Q. Demonstration. Supposons que Q soit tel que Qp = {0}. La proposition 3 vaut aussi bien pour n'importe quel sous-espace Q de H (cf. remarque 2 du § 4) ; soit done L1 un sous-espace complexe supplementaire de KQ dans L ; L1
induit un sous-fibre analytique 2 de .T, dont la fibre en z E M est 1'espace L1(z) des valeurs en z des champs de L1, de dimension complexe q, et qui possede q sections holomorphes globales, libres en tout point de M, qui sont les {XA} de la proposition 3; Y, est donc analytiquement trivial. Les deux sous-fibres -9Q et Y, sont supplementaires, car leur intersection se reduit a M et la somme de leurs dimensions fibrees est n ; la suite exacte (S) est donc 1
.analytiquement scindee.
Inversement, soit Q un sous-espace distributif de H tel que la suite exacte (S) soit scindee analytiquement ; it existe donc un sous-fibre holomorphe 2 1 de .T, supplementaire de -9Q et isomorphe a .,LL, done analytiquement trivial ,en vertu du lemme ; l'espace L1 des sections holomorphes de 2 1 est donc un sous-espace de L de dimension q, tel que
LInKQ={0}. De (7), nous concluons alors pour des raisons de dimension, que Q. = {0}. Remarque 1. Supposons que F lui-meme soit distributif, i.e., qu'aucune forme d'Albanese non nulle n' a de zero sur M ; it existe un systeme de m champs holomorphes locaux {XA}, dual d'une base donnee {bA} de F, sur un voisinage U de tout point z de M; it resulte alors de (3) que l'application de Jacobi est partout de rang maximum 2m. Le lemme 1 du § 6 est done vrai avec la seule hypothese : "F distributif" ; mais l'hypothese plus forte de surjectivite est necessaire pour les lemmes 2 et 3. Remarque 2. Si Q est distributif avec q = n, les fibres . et T sont analytiquement triviaux, Q coincide avec H et la suite (S) est trivialement scindee analytiquement. Si une telle situation se produit avec Q = F, M est alors de type surjectif, done fibree analytiquement, par J, au-dessus d'un tore complexe de meme dimension, a fibre connexe : l'applieation de Jacobi est donc un isomorphisme analytique de M sur son tore d'Albanese. Une telle conclusion ne vaut pas en general si nous supposons seulement que J (et T = -*) est analytiquement trivial, car m peut encore dans cc cas titre inferieur a n (cf. exemple de la remarque 3 du § 4) ; toutefois, si M est kdhlerienne et compacte, eompletement parallelisee analytiquement, F coincide avec H et J est alors un isomorphisme analytique de M sur son tore d'Albanese. Nous retrouvons ainsi, comme corollaire de la proposition 7 et du theoreme
de fibration analytique un resultat connu de H. C. Wang [8]. Cf. aussi le corollaire 3 du theoreme de parallelisme partiel du § 8.
315
VARIETES DE TYPE SURJECTIF
8.
Le theoreme de parallelisme partiel
Le theoreme que nous allons maintenant demontrer justifie a posteriori le terme "partiellement parallelisee" que nous avons adopte pour une variete a Qo = {0}.
Theoreme de parallelisme partiel. Sur une variete complexe compacte partiellement parallelisee relativement a un sous-espace Q de H, de dimension complexe q, it existe une famille de sous-espace L1 de L, de dimension com-
plexe q, dont aucun element, en dehors du champ nul, n'est annuli par l'ensemble des formes de Q. Pour tout L1 de cette famille, it existe une structure hermitienne g (non uniquement determinee) telle que les champs de L1 soient paralleles pour la connexion de chern associee a g. Demonstration. La premiere partie du theoreme est une redite de la pro-
position 3, compte-tenu de la remarque 2 du § 4: la famille des L1 est l'ensemble des supplementaires complexes de KQ dans L.
Un tel L1, quelconque par ailleurs, determine (cf. demonstration de la proposition 7), un sous-fibre holomorphe 21 de _OT qui scinde la suite exacte (S), i.e., tel que
9- =_qQ0+21. Soient p1 et p2 les projections de decomposition.
_OT
sur _qQ et 21 respectivement, liees a cette
A partir dune structure hermitienne (M, g) quelconque sur M, nous construisons une nouvelle structure hermitienne (M, g) definie comme suit, par la donnee de la metrique fibree hermitienne associee h sur le fibre 9-: (12)
h(X,, Y) = h[p1(Xz), p1(Yz)l + H[p2(Xz), pz(Y,)l
yXz , Y, E 9"z
,
Dans cette relation, h est la metrique hermitienne fibree sur G associee a g et H est un produit scalaire hermitien quelconque sur L1i le deuxieme terme du second membre de (12) doit s'entendre comme le produit scalaire-par H
-des deux champs de L1 uniquement determines par leurs valeurs en z respectives Xz et Y. Soient {Xi}, i = 1, , (n - q) un repere holomorphe local, au voisinage de z e M, du sous-fibre 2Q et {Xa}, a = 1, , q une base de L,; 1'ensemble {Xi}, {Xa}, constitue, au voisinage de z e M, un repere holomorphe local de T. Dans cc repere, nous avons, avec des notations evidentes,
hi; = hi; ,
hab = Hab
,
hia = haj = 0
d'oii it resulte pour les matrices w et w des connexions de Chem liees respectivement a h et h par (8) : _ Wi ,
as = 0 ,
V = CUa = 0
.
316
PAUL GAUDUCHON
Soit P l'operateur "derivee covariante" de la connexion de chern relative a g ; un champ X de L1 s'ecrit Ea=1 Aa Xa, oil All sont des constantes complexes.
Nous avons donc, dans le meme repere local que precedemment, q
4
q
PX = a=1 Z Aa.VXa =a,b=1 Z
n-q
a=1 ,:=1
Aa,ma.Xi
yXEL1,
cc qui acheve la demonstration du theoreme. Nous avons, avec les memes hypotheses, Corollaire 1. La restriction a 21 de la courbure de chern 0 We a g est nulle :
S(X:, F,)-Z, = 0
yZz E 21(z) , yXz, Yz E
,
:
Autrement dit la variete est "partiellement plate" (a l'ordre q). Corollaire 2. Soit T la torsion de connexion We a g, si L1 est une sousalgebre de Lie abelienne de L, on a
_'(X" Yz) = 0
,
yXz, Yz E 21(z) .
Autrement dit, la structure hermitienne g est, dans ce cas, "partiellement kahlerienne" (a l'ordre q). Le corollaire 1 resulte immediatement de la definition de la courbure de chern 6 = d"@; le corollaire 2 se deduit immediatement de
T(X,Y)=PXY-I P,[X, Y],
yX,YEL.
Remarque. Si Q est de dimension q = n, nous retrouvons dans le cas compact, comme corollaire du theoreme de parallelisme partiel, une propriete connue des varietes complexes analytiquement parallelisable : sur une telle variete it existe une metrique hermitienne naturelle telle que les champs holomorphes qui trivialisent soient a derivee covariante nulle pour la connexion de chern associee. (cf. par ex. [5, p. 217]). Si une telle situation se produit pour Q C B, L1 = L est une algebre de Lie abelienne puisque, dans ce cas, [L, L] C KQ = (01; it resulte alors du corollaire 2 que M est kahlerienne, et donc, en vertu de la Remarque 2 du § 7, Corollaire 3. Si le fibre cotangent holomorphe de M peut titre trivialise par des 1-formes fermees, l'application de Jacobi est un isomorphisme analytique de M sur son tore d'Albanese. En particulier, M est kahlerienne.
Ce corollaire 3, precise quelque peu le theoreme de H. C. Wang evoque plus haut (remarque 2 du § 7) ; inversement, it peut titre demontre aisement a 1' aide seulement de ce dernier theoreme et du resultat mentionne en debut de remarque, tous resultats que nous pouvons deduire eux-memes des theoremes
plus generaux (theoreme de fibration analytique et theoreme de parallelisme
VARIETES DE TYPE SURJECTIF
317
partiel) appliques au cas limite que constituent les varietes complexes compactes (completement) parallelisables dans le cadre plus large des varietes de type surjectif et des varietes partiellement parallelisables. Bibliographie [1]
A. Blanchard, Sur les varietes analytiques complexes, Ann. Sci. Ecole Normale Sup. 73 (1956) 157-202.
[2]
A. Borel & R. Remmert, Uber kompacte homogene Kdhlersche Mannigfaltigkeiten,
[3]
[4] [5] [6]
[7] [81
Math. Ann. 145 (1962) 429-439. P. Gauduchon, Tenseurs holomorphes et formes holomorphes sur une variete hermitienne compacte, C. R. Acad. Sci. Paris 279 (1974) 17-20. , Sur quelques problemes concernant les varietes complexes compactes et les fibres vectoriels holomorphes associes, These, Paris, 1975. S. I. Goldberg, Curvature and homology, Academic Press, New York, 1962. K. Kodaira & J. Morrow, Complex manifolds, Holt, Rinehart and Winston, New York, 1971. A. Lichnerowicz, Varietes kdhleriennes a premiere classe de Chern non negative et varietes riemaniennes a courbure de Ricci generalisee non negative, J. Differential Geometry 6 (1971) 47-94. H. C. Wang, Complex parallelisable manifolds, Proc. Amer. Math. Soc. 5 (1954) 771-776. 53, RUE DE LYON 75012 PARIS
J. DIFFERENTIAL GEOMETRY 12 (1977) 319-325
LE THEOREME DE FROBENIUS FORMEL JUNIA BORGES BOTELHO
Introduction
L'etude des algebres de Lie filtrees non transitives a fait apparaitre une certaine classe d'algebres de Lie que semble avoir une grande importance dans cette theorie. Geometriquement ces algebres de Lie correspondent aux pseudogroupes infinitesimaux associes aux distributions involutives, i.e., completement integrables. Il serait done souhaitable d'avoir pour ces algebres de Lie un theoreme correspondant au theoreme de Frobenius classique pour les distributions. Dans ce travail on montre que chaque telle algebre de Lie (appelee distribution involutive) est isomorphe a une distribution "canonique". D'une maniere plus precise, soit V un espace vectoriel de dimension finie et S(V*) 1' algebre de series formelles a coefficients dans le dual V* de V. Considerons 1'algebre de Lie D(V) des derivations de S(V*) munie de la structure naturelle de S(V*)-module. Une distribution involutive L sur V sera une sous-algebre de Lie de D(V) qui est an meme temps un sous-module libre de D(V) satisfaisant en plus a une condition de regularite. Le resultat principal de ce travail
est que L est egal, a un automorphisme de D(V) pres, a un sous-module de D(V) engendre par des derivations de la forme a/ax1, , a/axP, ou xi G V. La preuve de ce resultat s'ecarte de ce qui serait une traduction formelle de la demonstration du theoreme de Frobenius classique et on peut s'attendre a ce que notre methoque puisse servir de point de depart pour 1'etude algebrique des systemes differentielles avec des singularites. L'outil principal utilise est la
cohomologie de Spencer d'un sous-espace U de V a valeurs dans gr D(V). Dans le § 2 on montre que cette cohomologie est triviale. Dans le paragraphe final on reduit le probleme, en se servant de resultats cohomologiques, a un probleme de prolongement d'homomorphisme d'algebres de Lie transitives. On en deduit notre theoreme en utilisant des resultats de Rim [7] et Hayashi 15] .
Comme application du theoreme formel de Frobenius on peut dormer une demonstration immediate du Troisieme Theoreme Fondamental de Cartan [3], 181, dans le cas particulier oa l'algebre de Lie en consideration est une dis-
tribution involutive. D'ailleurs, nous croyons pouvoir l'utiliser dans 1'etude des algebres de Lie filtrees non transitives. Received June 5, 1975.
320
JUNIA BORGES BOTELHO
Ce travail, effectue a l'Universite de Sao Paulo, contient les principaux resultats de la these de doctorat de l'auteur, faite soul l'orientation du Professeur Alexandre A. M. Rodrigues. Je tiens a exprimer ma profonde reconnaissance a A. Petitjean pour des nombreuses conversations sur ce travail. 1.
Distributions
Dans tout ce travail K representera un corps commutatif de caracteristique nulle et V un K-espace vectoriel de dimension finie n. Nous munirons K de la topologie discrete. Soient S(V*) 1'algebre locale des series formelles a coefficients dans le dual V* de V et -9 son ideal maximal. On note .Ak, k > 0, la k' puissance de -9; on est une filtration decroissante posera -911 = S(V*) pour k < 0. Alors sur S(V*) et S(V*) munie de la topologie associee a cette filtration est une algebre topologique complete sur K. Soit D(V) 1'algebre de Lie des derivations de S(V*). Pour k E Z, on note
Dk(V) lensemble des X E D(V) tels que X(-&) C .ilk+' Alors {Dk(V)}kE2 est une filtration decroissante sur D(V) qui confert a D(V) une structure d'algebre de Lie filtree transitive et complete. Si L est une soul-algebre de Lie de D(V) on notera par {Lk}kEZ la filtration induite sur L par la filtration de D(V).
Par ailleurs, D(V) possede une structure naturelle de S(V*)-module libre de rang n dont la loi externe est definie par (fX)(g) = f X(g), pour f, g E S(V*) et X E D(V). 1.1. Definition. Une distribution de rang p sur V est un sous-S(V*)-module libre L de rang p de D(V) tel que dimK L/L° = p. Si en plus la distribution L est une sous-algebre de Lie de D(V) on dira qu'elle est involutive. Remarquons que si {Y1, , Y,} est une base d'une distribution L de rang p, la condition dimK L/L° = p s'exprime en disant que 7r(Y,), , 7r(Y,) sont
lineairment independants dans V of zr: D(V) - D(V)/D°(V)
V est la pro-
jection canonique. On note Aut S(V*) (resp. Aut D(V)) le groupe des automorphismes de l'algebre filtree S(V*) (resp. de l'algebre de Lie filtree D(V)). II est facile de voir que si H E Aut S(V*), alors H* : X E D(V) -+ H o X o H-1 E D(V) appartient a Aut D(V). Rappellons [6, th. 2.5, p. 456] le theoreme 1.2. Pour tout h E Ant D(V), it existe H E Aut S(V*) et un seul tel que
h = H*. Il en resulte 1.3. Lenune. Si h E Ant D(V) et L est une distribution involutive de rang p, alors h(L) est aussi une distribution involutive de rang p. Demonstration. C'est une consequence immediate de 1.2 et de la formule H*(fX) = H(f)H*(X) of H E Aut S(V*).
Si x E V, on note a/ax la derivation suivant le vecteur x, c'est a dire, la derivation de S(V*) definie par
THBOREME DE FROBENIUS FORMEL
0,...ek E Sk(V*)
321
0i(x)01. ..Bi.. .Bk E Sk-'(V*) i=1
L'application injective x e V-p a/ax e D(V) nous permet d'identifier V a une sous-algebre abelienne de D(V). Soit {x , xP} une famille libre de vecteurs de V. Il est immediat que le sous-S(V*)-module de D(V) engendre par a/ax,, , a/ax, est une distribution involutive de rang p. Le but de ce travail est de montrer que toute distribution involutive sur V est de cette forme, a un automorphisme de D(V) pres. La proposition suivante nous donne une caracterisation des distributions involutives. 1.4. Proposition. Une distribution L est involutive si et seulement si it , XP} de L telle que [Xi, X j] = 0 pour 1 < i, existe une S(V*)-base {X 1 < P. Demonstration. La condition est evidemment suffisante. Reciproquement, , YP} une base quelconque de la distribution involutive L. Par le soit {Y remarque au-dessus, it existe une base {a/ax, , a/axn} du S(V*)-module D(V) dans laquelle xi = ir(Yi) pour 1 < i < p, of ir : D(V) -p V est la projection canonique. On peut dons exprimer Yj = En=, a;a/axk ou a; e S(V*). est inversible. Soit (b;.)1sr,jsp son 11 est clair que la matrice carree
inverse et posons, pour 1 < r < p, Xr = E p=1 brY j ; on verifie que {X
, XP}
est une base de L et que l'on a Xr = a/axr + Ek-p+1 cra/axx of Or E S(V*). Il en resulte que [Xi, X j] est une combinaison 9(V*)-lineaire de {a/axP+1, , , XP} est une a/axn}. D'autre part, puisque L est supposee involutive et {X base de L, on peut ecrire [Xi, X J = EP=1 hijXr ou h2j e S(V*). Compte-tenu des expressions anterieures des Xr, [Xi, X II s'ecrit sous la forme [Xi, XI] _ 71, P-, hiia/axr + Ek=p+, l a/axx et, par suite on a, pour 1 < r < p, hzj = 0,
c'est a dire [Xi, X j] = 0 pour tout 1 G i, j < p. 2.
Cohomologie de Spencer d'un sous-espace U de V a valeurs dans gr D(V)
Soit gr D(V) = Oxez grx D(V) l'algebre de Lie graduee associee a l'algebre de Lie filtree D(V). On se propose de calculer la cohomologie de U a valeurs dans gr D(V) ou U est un sous-espace vectoriel de V. Rappelons [6, p. 454] qu'il existe un isomorphisme canonique 0 de l'espace vectoriel V ® S(V*) sur D(V) defini par O(x (3 f) = fa/ax. Au moyen de 0 on transporte sur V ® S(V*) la structure d'algebre de Lie filtree de D(V). Le crochet ainsi defini sur V (3 S(V*) est done donne par [x ® f, y ® g] = y ®
(fag/ax) - x ® (gaf lay) et la filtration par (V (3 S(V*))k = V ®., 'k1 On
identifiera par la suite les algebres de Lie filtrees D(V) et V ® S(V*) au moyen de 0. 11 s'ensuit que grx D(V) sera egale a V ® Sk+'(V*) Plus precisement,
322
JUNIA BORGES BOTELHO
un element Y E grk D(V) s'identifiera a 1'element de V O (Sk+I(V))* -- V O Sk+I(V*) defini par
Yl...Yk+I E Sk+I(V) - [Yk+I, [... [YI, Y]...]] E V V.
Soit maintenant U un sous-espace vectoriel de V et {XI, V telle que {x1,
, xP} soit une base de U. On note {x1,
, xn} une base de ,
x"a} la base duale
, x.n} et i l'inclusion de U dans V. Definissons ok+I : V O Sk+I(V*) - V O Sk(V*) O V* par ak+I(X)(x) = [x, X] _ NO i*) - ak+1 : V OSk+I(V*) of X E V (gSk+I(V*) et x E V. On posera de {x1,
dk+I,°
dk+1-m,m : V O - V O Sk(V*) O U* et Yon definira, pour 1 < m < p - 1, Sk+I-.(V*) O Am U* - V O Sk-m(V*) O Am` U* par dk+I-m,m(X O w) _
dk+I-m,'(X) A w of X E V O Sk+I-m(V*) et w E Am U*. On verifie que, pour t® P O w E V O dk+I-m,m(t O P (& w) = E1 t Sk+I-m(V*) O A m U*, on a dk+I-m,m = id® O aP/ax® O i*(x') A w et par suite on peut ecrire ak+I-m,m : Sk+I-m(V*) O U* Sk-m(V*) O OU Am A m+' U*, que l'on notera simplement a, s'exprime par a(P(& w) = Ei=1aP/axi®i*(xi) A w. Il est facile de voir que Oak+I-m,m
(2.1)
Sk(V*) O U*
Sk+I(V*)
Sk+I-P(V*) O
Sk-'(V*) © A2 U*
A P U*) 0
est un complexe, i.e., a c a = 0. En plus .dk+I-m, m, que l'on notera simplement
d, n'est autre que l'operateur cobord du complexe de Spencer d'ordre k de U a valeurs dans gr D(V) :
0> V O Sk+I(V*) (2.2)
V® Sk(V''`) O U*
)
V O SkI-P(V*) O AP U*
0
et dont on notera HIr(U, D(V)) la cohomologie en V O Sr(V*) O As U. On va montrer que ces groupes de cohomologie sont nuls en considerant le complexe dual de (2.1). 2.3. Lemme. Soit a*: Sr-I(V) O Am+I U- Sr(V) O Atm U l'application transposee de l'operateur a de (2.1). Si Q E Sr-'(V), on a
a*(Q O xi, A ... A xi.+,) m+1
=E (-1)'+I(xi,Q) O xi, A ... A xij A ... A j=1
xz,n+,
.
La demonstration de cc lemme est immediate. 2.4. Lemme. Pour tout r > 1 et tout m > 1, la suite suivante est exacte :
THEOREME DE FROBENIUS FORMEL
(2.5)
s'-'(V) ® Am+1 U
323
ax S'(V) o A U a* sr+l(V) ® Am-1 U U.
Demonstration. La preuve se fait par recurrence sur dims U. L'exactitude est immediate si dim U = 1. On suppose que (2.5) est exacte pour dim U = s - 1. Soit maintenant U de dimension s. On considere un sous-espace U' de U de codimension 1 et l'on prend xs E U - U'. Soit V un sous-espace de codimension 1 de V contenant U' et ne contenant pas xs. Si t E Sj(V) © A k U' et x e U, on definit x t et t A x de la facon evidente. En utilisant ces notations, posons Ar'm = Sr(V') © Am U', Br,m X .(Sr-1(V) O Am U') et Cr,m" = (Sr(V) 0 Am-1 U') A x8. Il est clair Sr(V) O Am U = Ar,m O+ Br,m a Crm et que a*Ar,,,, c Ar+1,m-11, B7'+l,m-1 O C7'+l,m-1 SOit t E Sr(V) O Am U, a*Cr,m C a*Br,m C Br+',m-1 et t = tA + tB + to Oil tA E Ar'm", tB E Br'm et to E C. En ecrivant tB = xs tB
et to = tc A xs, on obtient it = a*tA + xs a*tB + (-1)m+1x3. to + a*te A x8. Si it = 0, on a donc a*tA = 0, x, (a*t'B + (-1)m+1tc) = 0 et a*' A x, = 0 c'est a dire a*tA = 0, a*'+ (-1)m_1tC = 0 et a*t/ = 0. D'apres 1'hypothese de recurrence, tA est un cobord, i.e., tA = a*22A oU 22A E A7-',m+'. On
a alors tB + to = xs tB + (-1)ma*tB A xs = (-1)ma*(tB A xe), donc t = a(;IA + (-1)mtB A x,). On en deduit 2.6. Theoreme. Pour tout s > 1 et tout r > 0, Hs,r(U, D(V)) = 0. 3.
Theoreme de Frobenius formel
Soit L une distribution involutive de rang p sur V. Compte-tenu de la proposition 1.4, it existe une base {X1, , X,} de L telle que [Xi, X;] = 0 pour
1 < i, j < p. On se propose de construire une base du S(V*)-module D(V) en adjoignant a X1, , X. une suite X.+1, , X,, telle que [Xi, X,] = 0
pour 1 -1 (3.1)r
[X 1, Xr+1] + [X°, Xr] + ... + [Xz+l, X-1] = 0
Pour cela choisissons X-1 = xk oU {x1i
, x.} est une base de V telle que,
324
JUNIA BORGES BOTELHO
pour 1 < i on V such that
(4)
<X, Y>
=ao(x.Y)
By (2') and (4) we get
(5)
<x Y, z> + _ + <X, Y Z>
'The algebra V together with the linear function ao is said to be a clan corresponding to Q. If we define a bracket operation in V by
[x,Y] =xY-Yx,
'(6)
then V is a Lie algebra with respect to this bracket operation and q is a Lie .algebra isomorphism of g onto V. Therefore we may identify g with V by means of q. Following Nomizu [4], we shall express the Riemannian connection, the curvature tensor and the sectional curvature of Q in terms of its clan V ; those expressions were originally obtained by Y. Matsushima (unpublished). Proposition 1. The Riemannian connection P for Dot is given by
i.e., P,, is the skew symmetric part of L. Proof. According to [4], we have V,,Y = - [x, y] + U(x, y)
where 2 = + <x, [z, y])'. By (5), (6), we get
2 = + <x, z Y - y . z> = + <x, z Y> - <x . z, Y> - <x, Y z>
= <x-z, Y> + - <x . z, Y> - fix, Y z> = - <x, y z> = . Hence it follows that
329'
HOMOGENEOUS CONVEX DOMAINS
U(x, y) = 2 (L.xy
- tLyx) = I (Lyx - tLxy)
so that
Pxy = I (LxY - Lyx) + 2(Lyx - tL. y) = 2(L.x - tL.,)Y Proposition2. Then we have
Let S,x be the symmetric part of L,x, i.e., let S, =2l(L,+ IL,,)-
(i)
S,xy = Syx,
and the curvature tensor R and the sectional curvature k are given by (ii) R(x, y) = - [Sx, Sy], (iii)
k(x, y) =
S-Y I Iz - <S.xx, SyY> Ilxllzllyll2-<x,Y>z
x>. where IIxII = Proof. (i) is equivalent to (5). In fact we have
2<S.xy, z> = = <x y, z> +
= + <x, y z> = = 2<Syx, z> Since R(x, y) = [Px, Py] - PC.x,y], by Proposition 1, (2) and (6) we get
R(x, y) = 4 [L. - 1L,7, Ly - tL y] - (L Gx,y] - ILCx,y7) = 4{[L,x, Ly] - [L.x, tLyl - [tL.x, Ly] + [tL,x, tLy] - 2[L,,, Ly] + 2t[L,x, L,]}
_ - 4 [L.x + tL.x, Ly + tLyl = - [S,x, Sy] From (i), (ii) we obtain _ _ _ <S.xY, Syx> - <Syy, S.xx> = I S-Y IIZ - <S.xx, SyY> ,
which together with k(x, y) _
Ix I I z
I
gives (iii).
I Y I I, - <x, Y>
A clan V is said to be elementary if V satisfies the following conditions: (E.1)
V = Jul + P
(direct sum of vector spaces)
u=0,
(E.2) (E.3) (E.4)
and p.u=O forpEP, p q = D(p, q)u
for p, q E P
,
where t is a positive definite symmetric bilinear form on P.
330
HIROHIKO SHIMA
The domain Q corresponding to an elementary clan is the interior of a paraboloid (cf. [5], [61):
-1 for a ER, pEPI To prove our theorem, therefore, it suffices to show Theorem. Let V be a clan. Then the following conditions are equivalent: (i) The sectional curvature k < 0. (ii) V is an elementary clan. Proof. We first prove that (i) implies (ii). Since V is a clan, there exists a nonzero element u E V such that (cf. [5]) (7)
V Jul c Jul ,
(8)
and moreover putting P = {p E V ; p u = 0} we have :
(9) (10)
V = Jul + P
(orthogonal decomposition),
L,, leaves P invariant, and the eigenvalues of L,, on P = 0 or 2
Let p be an element in P such that L,,p = 0. By (7), (8) and (9) we obtain <Suu, q> = I = I + I = 0 for all q E P, so that Suu E Jul. Put S,,,u = Au (A E R). Then it follows from Proposition 2(i) that <Suu, SPp> = = A<SPU, P> _ A<S,,,p, p> = 2 = 0
Therefore by Proposition 2 (iii) we have
k(u, p)(
ul
2
I l p l lZ - Z) = Il Sup I
2
- <S.,u, Spp> = I
Sup 1 12 > 0 .
Since k < 0, we have p = 0. Hence it follows from (10) that the eigenvalues of L,, on P are equal to 2. By [5] this means that
forp,gEP,
(11)
where 0 is a positive definite symmetric bilinear form on P. Since <x, u> = ao(x) for all x E V, u is the principal idempotent of V and V = Jul + P is the principal decomposition of V, [6]. Therefore V is an elementary clan.
Conversely we shall prove that (i) follows from (ii). Let uo = Pi,
, p,n_I
1
ao(u)
be an orthonormal basis of V such that pi E P. Then we have
u,
HOMOGENEOUS CONVEX DOMAINS
u0 =
331
ao(u)
ao(u)
U0
(12) 1
Pi-U0 = 0 ,
ai j being Kronecker's delta. Let x = 20u0 + En-i iipi and y = pouo + be elements in V where 2, p j E R. By (12) we get 2Oi
X. Y =
(13)
n
Yn=1 2ipi n
2/() 2Opi
uo +
/mo(u)
En=i pipi
Pi
and therefore
<S'Y, u0> = C 2 (L.., + tLy)Y, u0) = 2
=
1
Aop0 + En=1 20i /ao(u) n-1
2\
<X. Y, U0> + 2
uo +
n-1
20pi
2 / a0(u) Pig u0>
N
+ 2 C/oUO
E Uipi, a u) u0 0
n-1
1
<SxY1 Pk> = C
2
1
2ipi)
22Op0 + E1
(L,,; + `Ly)Y, Pk) = 2 <x Yl Pk> + 2 R, where a1, i = 1, with respect to the torus T. From the above we obtain the following commuta-
tive diagram
L(T)
ai
(x1, ... , x'a) y
T (x,, ... , x'a.), mod Zm
>R
ai(x1, ... , x'a) = bi,x, + ... + 19i
y
S = R/T x'a), mod Z'a
= bi,x, + ... + bi'axm, mod Z
where m is the rank of H, and b1112 e Z, 1 G j, G 1, 1 G j, G M.
335
RIEMANNIAN S-MANIFOLDS
We assume that the symmetry so has infinite order, which means that at least one of the values 9i, 1 < i < 1, is an irrational number. From this we conclude that at least one of x;, 1 < j < m, is irrational. Therefore some, or all of the m-tuple numbers (x , xto which the symmetry so corresponds, are irrational. We substitute these irrational numbers by rational ones as close to them as we wish. Hence we obtain another symmetry s', which has finite order. Now we assume that so H°. Therefore there exists an integer 2 such that ' E H°. Since so has infinite order, equally so does so. Let T, be the maximal torus in H° passing through so.
The symmetry so can be considered as an orthogonal matrix. Therefore another orthogonal matrix a exists such that cos 27rz,
-sin 27rz,
sin 27rz, cos 27rz,
where at least one of the numbers z obtain cos 27r2z,
- sin 27r2z,
cos 27rzm,,
sin 27rz,,
- sin 2;r z-,,,
cos 27rz,,,,
is irrational. From the above we
sin 27r2z, cos 27r2z,
1
N's ON-1 = cos 27rA1z,,,
-sin 27r2r,
sin 27r2z,,, cos 27r iz,,,
Since ' E T there is another base such that s'' can be written cos 27r2zi
-sin 27rizi z
so
sin 27r2zi con 27r2zi
= cos 27r2z
sin 27r2zm
- sin 27r2zcos 27r2T where at least two of the numbers (1, 2zi,
are linearly independent , of the field of rational numbers. Therefore s' generates at least one-dimensional torus T, C T, and closure {so-, m > n°} = Ti and the elements of T,' commute with so. From the above we conclude that there exists an element a e T, which can be written
336
GR. TSAGAS & A. LEDGER
cos 27r(p' , - r;)
-sin 27r(p' , - r;)
sin 27r(p' , cos 27r(p' ,
a= cos 27r(p' - r;,,)
sin 21r(p' - r'm)
I
- sin 21r(p'. - r) cos 2ir(p'., - r'.) J
where pi,
,
p', are rational numbers close to zi,
, z-
, as we wish, respec-
tively, and p' = r', if ri is rational. The same element a with respect to the old base can be written cos 22r(p, - r,)
sin 22r(p, - r)
-sin 27r(p, - r) cos 22r(p, - r) Aap ' _ cos 21r(p,,,, - r.) - sin 22r(pm -
sin 22r(p,,, - r,n cos 27C(p,, - ,n)
Since a and so commute, we obtain cos 27rp1
sin 27rp1
-sin 27rp1
cos 27rp1
/3aso/-' = /3aJ-' soj-' = cos 27rpm,
- sin 27rp,
Il
sin 27rpm
cos
where p, i = 1,
, m, have the same meaning as p'. Therefore the symmetry aso belongs to the same component of H as the
given symmetry so, having finite order. Proposition 2.5. Let M = G/H be a compact Riemannian s-manifold. The symmetry so belongs to the identity component H° of H if and only if rank G
= rank H. We assume that the symmetry so belongs to H°. From so we obtain an automorphism A on G :
A:G-G
,
A : v - A(v) = sovso'
and an automorphism a on the Lie algebra g of G :
a:g=h+m- g=h+m, a:X-a(X)Eh,
VXEh.
Let T1, Tz be the maximal tori of H and G, respectively, through the element so. Since T1 C T, and all the elements of T, commute with so, so do the elements of T1. Since the vectors belonging to the tangent space of T2 at the identity element are invariant by a, we conclude that Tz C H and therefore
rank G = rank H.
337
RIEMANNIAN S-MANIFOLDS
The inverse is an immediate consequence of the assumption rank G = rank H ; then we have that So E H°. Corollary 2.6. Let M = G/H be a Riemannian homogeneous space such
that H is the largest isotropy subgroup of G at one point of M. If H is connected and dim H is odd, then M can never be a Riemannian s-manifold. If we assume that M is a Riemannian s-manifold, then so E H and there is always a maximal torus T in H through so. However since dim M is odd we obtain ad (so) having an eigenvalue 1. So we reach to a contradiction because ad (so) never has an eigenvalue 1. Therefore M can not be a Riemannian smanifold.
Remark 2.7. From the above we conclude that all Riemannian s-manifolds form a proper subset of all Riemannian homogeneous spaces. 3. Let M = G/H be a simply connected homogeneous space. It is known that M is isometric to the direct product M° X M1 X .. X Mr and that the identity component I°(M) of the group of isometries I(M) is naturally isomor-
phic to the group I°(M°) X I°(M1) X... x I°(M,.). We shall prove that each of the homogeneous spaces M0i M1,
, Mr is a
Riemannian s-manifold. To this aim we distinguish two cases. (i) If S E I°(M), then we have
s:M=MOXM1x ... XM. -M M0XM1X ... XMr, s : 0 = (0°, 01, ... , Or)
0 = (00/,''01, ... , Or) ,
S : x = (x0, x1, ... , xr)
S(x) = (y0, y1, ... , yr)
where yi = si(xi) = pi(s(x)), pi is the natural projection of M into Mi, and sz, is an isometry of Mi [4, p. 241]. By considering the de Rham decomposition theorem for the tangent space of M at 0, we have (3.1)
T0(M) = T0I°'(M) 6 To1'(M) O+ ... Q+ Tor'(M)
Since S E 7°(M), we have ad (s)(TOM (14)) = Toi'(14), where i = 0, 1,
,r
or ad (si)(Toi'(M)) = Toi'(M) = ad (s)(Ti(M)), [4, p. 240]. We also have si : Mi - Mi, si : 01 - Oi and hence si is symmetry at 0i for the manifold Mi. Therefore Mi, i = 0, 1, , r, is a Riemannian s-manifold. The order of s is the least common multiple of the integers {k°, k1i , kr} where ki, i = 0, 1, , r, is the order of si. (ii) If s 0 I°(M), then we obtain an orbit (Mz, M2'. , Mr) of the permutation group defined by s, and consider the product
M(i) = MI X M? X ... X Mz i If r1
1, then we can order Mi, Mz,
, Mi i such that s maps Mz isometrical-
338
GR. TSAGAS & A. LEDGER
ly onto M111, where 1 < 2 < ri - 1, and Mii isometrically onto Mi. This can always be done after some identifications. Therefore M be written
M = M0 x M> X ... xM(P), where Mo is the Euclidean part of M and Mci>, i = 1, , p, have the above meaning. With the same technique, as in case (i), we can prove that s can be written s = (*o, *1, , i,), where *i, i = 1, ., p, is a symmetry on the manifold M(i) having also the following properties
* i : My X MI X ... X M2 i - * M1 X MIX ... X W , Or) 1i (O1, 02, ... , Or) > (O10 O2, i : (Mz X 02 X ... X Or) -p (01 X Mz x ... X Or) ,
(3.2)
(3.3)
iy i
: (01 x 02 x ... X Ori-2 X
(3.4)
(3.5)
X Or)
- (01X02X ... XOr;-1xMii),
y'i:(0,X02x ... x0,.;-1XMP)-*(MiX02x ... XOri).
We can identify the manifold Mi with Mz, ing mappings
fv:Mi-*Mi 3
,
, Mii by virtue of the follow-
v =2,...,ri
.. , frg= where f2 = P'77 `2' ° 1G' i, f = f o p(3> `Y i, i °1 p22>,
Mri-1
2
fTi-1 o ... o f o picri) o i
and , Mii, respective2
pzri> are the natural projections of M(i) into Mz,
ly.
The mapping, defined by (3.5), can be considered as an isometry of Min onto M1 after the following identification 1
f' : Mi
Mli
,
f1 = f, ° f"_' °
° f2 ° pi
>
°
i
where pal) is the natural projection of Mci> into Mi. From the construction of f, we conclude that f 1 has 01 as a fixed point, Let T,,(M1ti>) be the tangent space of Mci> at the point 0' = (01, 02, , Ori). Then we have T,,(Mci>) = To'(Mci>) O To'(Mci)) O ... O To'i'(Mti>) ,
and ad (lp ) has the properties :
RIEMANNIAN S-MANIFOLDS
ad (*): To,(Mci)) -p To±1(M(1))
339
2
ad (*j): Tr,(Mci)) X Ari, where A,, j = 1,
from which we obtain ad (lr) = Al X A2 X ri, are defined as follows A,: To,(M(i)) -,, To,-1(M(i))
p = 1, ..., r2 - 1
,
,
,
Ari : Tri(M(i)) - To,(M(i))
We assume that the mapping f 1 is not a symmetry for the point Ol of M. Therefore there is a vector u1 E To.(Mc2)) = TO,(M'1)) which is invariant under d(f1)o, = ad (f ). From this vector we obtain the following sequence of vectors : u2 = ad (f)(u) E To.(Mci ), ... , uri-1 = ad (fri-1)uri-2) E To uri =
ad (fri)(uri-1) E Tr`(M(i)), ad (f)(uri) = u1 E T',(Mci)). Hence ad (ri), by the form of a matrix, can be written 0 0
Al
0 ...
0 A2
B= 1 0 10 lAri
0 0
...
0 0
0 ... Ari-2
0 0 0
0 0
0 0
II
0
Ari_, 0
Let u be the vector of TO,(M(1)) with coordinates u11 u2, have
(3.6)
0 0
A,
0
0
Ari
0
A2
Bu =
0 ... A,
0 0
[ u11 u2
I
-
[Ariul i Alu2
, uri,. Then we [ ul u2
0 ... Ari-1
0 ...
0
j
Uri
Ari-iUri
=u.
urt
From (3.6) we conclude that ad (*i) leaves the vector u fixed, and therefore *j is not a symmetry. But this is not true because *j is a symmetry. Therefore f 1 is a symmetry.
The order of the k-symmetric Riemannian space M is the least common , MI,,1, , k,, of the manifolds Mo, M(1), multiple of the orders ko, k1, respectively. Each order ki, i = 0, 1, ., ,u, has the form riq, where q is the least common multiple of (rank (A), , rank (Ari)). Hence we have Theorem 3.1. Let M be a simply connected Riemannian s-manifold. This X Mr each of which manifold splits into the product manifolds Mo X M1 x is a simply connected, irreducible Riemannian s-manifold. 4. Let M = G/H be a k-symmetric Riemannian space, and so the sym-
340
GR. TSAGAS & A. LEDGER
metry of M at its origin 0. From this symmetry so we obtain an automorphism A on G defined by
A: G -* G
A : v -* A(v) = sovsol
,
Proposition 4.1. Let M = G/H be a k-symmetric Riemannian space. Then the automorphism A on G has order k and preserves the isotropy subgroup H. From the definition of A we have
A:G-G,
A : v - A(v) = sovsol
A : sovso' -* A(sovsol) = sosovso'so, = sov(so')2 . A(sk-1v(s-1)k-1) - 0 0 =v A: 0 0 0 0 Skv(S-1)k
sk-1v(s-1)k-1
Thus we conclude that Ak = id., that is, A has order k. If p e H, then we obtain A(p) = sopsol. It is known that so : M - M, p : M -* M, sot : M -* M, so : 0 -* so(O) = 0, /1 : 0 -* p(0) = 0, so 1 : O - so 1(O) = 0, from which we obtain sopsol e H, that is, A preserves H. Definition 4.2. The triplet (G, H, A) is called a k-symmetric Lie group, where G is a Lie group, H is a closed subgroup of G, and A is an automorphism on G of order k with the property A(H) C H. Let M = G/H be a k-symmetric Riemannian space. We consider the Lie algebras g, h of G and H, respectively. Then we have
g=h+m, where m can be identified with the tangent space T0(M) of M at its origin 0. From so we can also obtain an automorphism a on g defined as follows :
a:g=h+m-*g=h+m,
a:X-*a(X)=Ad(so)X,
where Ad (so) = ad, (so). The following is also known :
exp:g-*G, (4.1)
exp:X-*expX,
exp {Ad (so)X} = so exp Xso'
Proposition 4.3. Let M = G/H be a k-symmetric Riemannian space, a the automorphism on g = h + m obtained by so. Then h is preserved by a, which has order k. If X e h, then exp X = 2 E H. Since 2 e H, we have so2so' e H, which implies so exp Xso1 e H. From this and (4.1) we obtain exp {Ad (so)(X)} = so exp Xso1 e H
341
RIEMANNIAN S-MANIFOLDS
which gives Ad (sa)(X) E h. Therefore h is preserved by a = Ad (so). From the definition of a and formula (4.1) we have a: g -* g ,
c: X - a(X) = a(X) = Ad (sa)(X) ,
exp {Ad (sa)(X)} = so exp Xso'
,
Ad (sa)(X) -* Ad (sa){Ad (sa)(X)} = Ad2 (sa)X , exp {Ad2 (sa)(X)} = sa{exp ((Ad (sa))(X)}so' = sa{sa exp Xso'}so' = so exp X(so')2 a:
which imply
exp {Ad' (sa)(X)} = so exp X(s-')x
showing that a = Ad (so) has order k. Definition 4.4. The triplet (g, h, a) is called a k-symmetric Lie algebra, where g is a Lie algebra, h is a Lie subalgebra of g, and a is an automorphism on g of order k with the property a(h) C h. Let M = G/H be a k-symmetric Riemannian space. If g and h are the Lie algebras of G and H, respectively, then we have
g=h+m,
a(h)Ch,
where a is the automorphism on g of order k, and m = g/h. It is known that the Riemannian metric g on M is G-invariant, which gives an Ad (H)-invariant nondegenerate symmetric bilinear form B on m = g/h defined by
B(X, Y) = 9(X, Y) , X, Y are the elements of g/h represented by X, Y, respectively. From the above we conclude that given a k-symmetric Riemannian space we then have a k-symmetric Lie group (G, H, A), a k-symmetric Lie algebra (g, h, a), and an Ad (H)-invariant nondegenerate symmetric bilinear form on
m = g/h. Definition 4.5. Let M = G/H be a k-symmetric Riemannian space. If the symmetry so commutes with all the elements of H, then M is called a regular k-symmetric Riemannian space or regular Riemannian s-manifold of order k. If a k-symmetric Riemannian manifold M = G/H is regular, then the automorphism A on G preserves the subgroup H as pointwise so that A (v) = v, Vv E H. The same is true of the automorphism a on the Lie algebra g of G which preserves the Lie algebra h of H pointwise so that a(X) = X, dX E h. The triplets (G, H, A) and (g, h, a), which are obtained by a regular k-symmetric Riemannian space, are called a regular k-symmetric Lie group and a regular k-symmetric Lie algebra, respectively.
342
GR. TSAGAS & A. LEDGER
Theorem 4.6. Let M = G/H be a regular Riemannian s-manifold. Then M is a reductive homogeneous space.
Let g and h be the Lie algebras of G and H respectively. Then we have g = h + m, where m can be identified with the tangent space of M at its origin.
If ad (H)m C m, then M is a reductive homogenous space. We assume that there exist X E m and R E H such that ad (p)(X) = Y E h. Since ad (p) o ad (so) =ad (so) o ad (9), we have ad (p) o ad (so)(X) = ad (so) o ad (p)(X), which im-
plies ad Y, where Z = ad (so)(X) E m. From ad" (so)(X) = X and the fact that ad is an automorphism, we conclude that Z = X and hence X = ad (so)X which is impossible because so is a symmetry. Hence we have reached a contradiction to our assumption. This implies ad (p)(m) C m. Theorem 4.7. Let (G, H, A) be a regular k-symmetric Lie group. Then there is a Riemannian metric on the homogeneous space M = G/H, which makes M a regular k-symmetric Riemannian space. First, we shall construct for each point P of M = G/H a diffeomorphism sp of order k on M, having P as an isolated fixed point. For the origin 0 of M we have the diffeomorphism so defined as follows :
so : M = G/H -> M = G/H,
so : vH -> so(vH) = A(v)H .
Let v(O) be a fixed point of so, where v s G. Then A(v) E vH. By putting p = v-'A(v) E H, since v E H we have p' = aA(a) = v-'A(v)A(v-')A2(v) and therefore p2 = v-'A2(v). But p2 E H implies A(p2) = p2. Thus p2 = A(v-1)A(v2). Similarly, for r < k we obtain pr = A(v-1)A'11(v) and finally pk = v-'A(v)A(v-')Ak(v) = id since Ak = id. Thus pk is the identity element of H. Now assume that v is sufficiently close to the identity element so that p is also near the identity element. Then p itself must be the identity element and therefore A(v) = v. Being invariant by A and near the identity element, v lies in the identity component of GA, where GA is the setwise of G by
A and hence in H. Thus v(O) = 0 proving our assertion that 0 is an isolated fixed point of so.
For the point P = v(0) we obtain as a diffeomorphism sp = v o so o v-' Then sp has P as an isolated fixed point, and its order is k. This is independent of the choice of v such that P = v(0). The Lie algebra g of G can be written in the known decomposition
g=h±m. We consider a special ad (H)-invariant nondegenerate symmetric bilinear form B on m. From B we obtain a G-invariant Riemannian metric g on M = G/H, which is given by the formula B(X, Y) = g0(X, Y) for X, Y E m. It can be easily obtained that s, is a Riemannian symmetry of order k on M at P. Hence M = G/H is a regular k-symmetric Riemannian space.
RIEMANNIAN S-MANIFOLDS
343
References
C. Chevalley, Theory of Lie groups, Princeton University Press, Princeton, 1946. P. Graham & J. Ledger, s-regular manifolds, Differential Geometry, in honor of K. Yano, Kinokuniya, Tokyo, 1972, 133-144. [ 3 ] S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. [ 4 ] S. Kobayashi & K. Nomizu, Foundations of differential geometry, Vol. I, Interscience, New York, 1963. [ 5 ] A. Ledger, Espace de Riemann symetriques generalises, C. R. Acad. Sci. Paris 264 (1967) 947-948. [1] [2]
[6] A. Ledger & M. Obata, Affine and Riemannian s-manifolds,
J. Differential
Geometry 2 (1968) 451-459. [71 K. Nomizu, Invariant affine connections in homogeneous spaces, Amer. J. Math. 76 (1954) 33-65. [81 J. Wolf, Spaces of constant curvature, McGraw-Hill, New York, 1967. [9] J. Wolf & A. Gray, Homogeneous spaces defined by Lie group automorphisms. I, II, J. Differential Geometry 2 (1968) 77-114, 115-159. UNIVERSITY OF PATRAS, GREECE UNIVERSITY OF LIVERPOOL, ENGLAND
J. DIFFERENTIAL GEOMETRY 12 (1977) 345-376
QUELQUES PROBLEMES D'INTERSECTIONS EN GEOMETRIE RIEMANNIENNE PIERRE MARRY & JEAN-LOUIS VERDIER
Cet article constitue essentiellement un travail de mise au point. On y exploite systematiquement certaines idees de S. S. Chern (cf. [6]). Quelques progres ont ete faits dans la presentation des calculs, certaines formes differentielles
introduites par Chern apparaissant naturellement comme integrales sur les fibres d'un fibre en spheres de formes classiques. Ces calculs se rattachent a ceux de certains caracteres differentiels.
Les auteurs se sont beaucoup inspire du travail fondamental de R. Bott et S. S. Chern (cf. [3]) qui traite de problemes analogues en geometrie Canalytique. 1.
Element d'aire relatif
1.1. Courant de Dirac. Soient X une variete differentielle C- de dimension n, Y une sous-variete de X de dimension p < n, TX le faisceau d'orientation de X et i : Y -* X 1'injection canonique. Soit T un faisceau localement libre de rang 1 sur X, egal a son inverse, induisant sur Y le faisceau d'orientation de Y. A toute p-forme differentielle T-tordue a support compact a sur X, le courant de Dirac 6Y de Y associe le nombre reel =
J
i'`(a).
Si a est fermee, ce nombre ne depend que de la classe de cohomologie a support compact de a, donc 6Y determine un element de Hom (HP(X, T), R). La dualite de Poincare nous donne un isomorphisme Hom (HP(X, T), R) Extra-P (T, Tx) et, comme T est localement libre, on a un isomorphisme Extra-P (T, Tx)
Hn-P(X, Hom (T, Tx)) = Hn-P(X, T O Tx)
.
L'image de 6Y dans Hn-P(X, T (9 Tx) est la classe fondamentale [Y] de Y dans X (par rapport a T). Elle provient en fait d'un element de HY P(X, TOO Tx) par le morphisme canonique HY P(X, T O Tx) -f Hn-P(X, T (D Tx).
Si V est un voisinage ouvert quelconque de Y dans X, [Y] a donc des representants qui sont des formes differentielles fermees, a support dans V, et T O Tx -tordues. Received July 1. 1975.
346
PIERRE MARRY & JEAN-LOUIS VERDIER
Si co, est un tel representant, co, E T(X, QX p © (T © T1)) et (a) dmp = 0, (b)
supp w c V,
(c)
pour toute forme fermee a E f'(X,,QX P (3 T) on o f i*(a) = J a A W. % JY
Dans la suite de cet article, on montre que, lorsque X est riemannienne, on peut determiner de telles formes mV de maniere canonique. On en tire quelques consequences pour des calculs de nombres de Lefschetz. 1.2. Cas des fibres vectoriels. Soit S X un fibre vectoriel reel de rang r > 0 sur une variete differentielle X de dimension n, muni d'une metrique sur les fibres. Comme les fibres de 7r sont orientables, le faisceau sur S d'orientation relative de S au-dessus de X est l'image reciproque d'un faisceau t(8) sur X, et le fibre vectoriel de rang 1 sur X associe a t(c) n'est autre que AT Notons Tx le faisceau d'orientation de X. Soit S X le fibre sur X en spheres unites de E. Definition 1.2.1. On appelle element d'aire relatif sur S toute (r - 1)forme a sur S, C`° et 2r,*s (t(S ))-tordue, telle que
(i)
1,
Q
J s
it existe une r-forme x sur X, C°° et t(E)-tordue telle que da = -Ir sx. (Dans cette definition, le symbole f designe l'operation d'integration le long (ii)
rz
Js
des fibres de ors, cf. [4, § 10].) Comme on le verra au § 1.3, it existe toujours des elements d'aire relatifs. Remarquons que la condition (i) n'est qu'une condition de normalisation. En effet, soul l'hypothese (ii) on a d J s a = J ns
et par suite,
J
da
= J -S
- sx = 0
a est une fonction localement constante.
Proposition 1.2.2. Soient a, et a2 deux elements d'aire relatifs sur S. Il ex-
iste alors une (r - 1)-forme 9 sur X et une (r - 2)-forme r sur S telles que al - aZ = 2rs(9 + Jr. Reciproquement, si a, est un element d'aire relatif sur S, 9 une (r - 1)-forme sur X et r une (r - 2)-forme sur S, aZ = al + 7rSM + dr est aussi un element d'airer rrelatif sur S.
Posons r= aZ - al. On a
r = 0 et dr = 7rs*. (O) oti 0 est une r-forme sur
f X. Soit a un nombre reel strictement positif, soit 0.: R+ , R une fonction C-, S
a support dans [0, e[ et egale a 1 sur le segment [0, 2e1. Notons p la projection canonique de E - X sur S (on identifie X a la section nulle de E). Soit p: E --> R+ la norme sur les fibres de 37. La forme d(O,(p)p*r) est C°° sur S - X et se prolonge a c en une r-forme fermee 0., C°° et 2r*t(S)-tordue,
347
PROBLEMES D'INTERSECTIONS
={ES
a support dans le voisinage tubulaire
sure - Xona
E} de X. En effet,
d(OE(p)p*r) = Oe(p)dp A p*r + 0E(p)p*(dr)
et, comme 0e(p) est nulle au voisinage de X et dr = ;rs¢, chaque terme se prolonge a S tout entier. Sur S - X, d9, = 0, donc, comme dim X < dim S, d9, = 0 sur S. Notons K la famille des supports de E dont la projection sur est compacte. On a, pour toute forme a E F,(S, Q, ©7r*Tx) fermee, rXE
fE
En effet, soit Sh le voisinage tubulaire de rayon h > 0 de X dans S, 7rh la projection de Sh = ash sur X. On a
cT A0,=lim
I(I
Ja
h-.0
= lim
f--Eh JX
Jr
aA0,=1im h-+0
X -IE-Eh
aAB,
(-1)nd(a A 0E(p)p*r) I
Eh
Par la formule de Stokes relative, on a, car a(E - Sh) = -Sh, nIE-ah
d(a A 0,(7r)p*r) r
r
d f fi
IE-ah
f
df
fi
Ia-Eh
a A 0s(p)p*r - (- 1)n-' $n
a A 0,(p)p*r -(-1)n
Ia(s-Eh)
a A Os(p)p*r
ffi a A 0s(p)p*rI sh
Jh
Donc
(1)
$
iE-ah
a A 8, = -J X J nh
(-1)na A 0E(p)p*r 1
,3,,
D'autre part, on a le lemme suivant. Lemme 1.2.3. Soit a une forme CO sur S. Soit ih : S -* S, la projection naturelle du fibre en spheres de rayon 1 de S sur le fibre en spheres de rayon h. Soit ri une forme C°° sur S telle que ih(ri ,,) soit localement bornee sur S, uni f ormement par rapport a h, et telle que f .1
72 1 sh converge uni f ormement sur h
tout compact vers une limite ri, lorsque h tend vers 0. Alors f
(a A )?) I sh ad-
met une limite uniforme lorsque h tend vers 0, qui n'est autre que (aI x) A ri°.
Cela resulte trivialement du fait que a est C° et que a l sh - 7r* (a I x) converge uniformement vers 0 lorsque h tend vers 0. Appliquant a ce lemme 1'egalite (1), it vient
348
PIERRE MARRY & JEAN-LOUIS VERDIER
lim
f
h-0 X J rzIS-Sh
a A 0, = -J (a j y) A lim fsh (p*D js) = 0 X
h-»0
J
.
h
a A 0, = 0 pour toute forme a r= 1,(H, STI O i*Tx) fermee. La dualite de Poincare nous montre 1'existence d'une (r - 1)-forme C-f3', sur 91 telle que 8, = dp'. Sur X on a 8, = *¢ = dp'. Toutes ces formes etant C°° sur Donc
Js
cette egalite reste vraie sur 3E12 tout entier. En particulier, par restriction a X,
on voit que ¢ = d(p' j x). Notons R1 = p' I x. On a dr = 7cs (df3,) =
done z = it 1 + r, oii r' est une (r - 1)-forme sur S telle que dr' = 0 et
SIsr'=0. On a une suite exacte de cohomologie (cf. § 2.2 de cet article) 0
H7-1(X, t(C)) > Hr-1(S, 7c8***t(E))
H°(X, R) - .. .
Donc, it existe une (r - 1)-forme R2 sur X, fermee, C°° et t(8)-tordue telle que r' _ ir8R2 + dr, of dr est une (r - 2)-forme sur S. En posant R = R1 + R2 on trouve bien pour r 1'expression desiree. La reciproque est immediate. On deduit directement de cette proposition le corollaire suivant. Corollaire 1.2.4. Si a est un element d'aire relatif sur S, la r-forme x sur X telle que da = -7rsX est fermee, et sa classe de cohomologie ne depend pas de l'element d'aire relatif a choisi. On a la proposition suivante. Proposition 1.2.5. Soient p : 8 - X - S la surjection canonique, a un nombre reel strictement positif et 0, : R+ - R une fonction C`° a support dans [0, e[, egale a 1 sur [0, e/2]. Notons p la norme sur .9' et B, l'ensemble des elements de S de norme strictement inferieure a e. Soit a un element d'aire relatif sur S. Alors : (a) la forme T. = (-1)'10,(p)p*a sur S - X est une section localement L' de SZ 1Qx ,c*t(c), c'est-a-dire que pour toute (n + 1)-forme k sur 8, a support compact, C° et 7r*TX tordue, A T. est integrable sur c ; (b) la forme - dr, sur S - X se prolonge a en une r-forme w, fermee, C- et ir*t(S)-tordue, a support dans 8, ; (c) si co, et rs sont les courants sur S associes aux formes w, et T., 8x le courant de Dirac de la section nulle X de on a 8X = Cu_ + d?E. (a) Pour que rE soit localement L', it suffit que p* . le soit. Soit ,J, une (n + 1)-forme de classe C° sur S a support compact. On a fe-XkAp*a=fXf°
[fr,
Ikh
A (P*6Ish)]dh
ob k, est, au signe pres, le coefficient de dh dans 1'expression de ,/i en coor-
PROBLEMES D'INTERSECTIONS
349
donnees polaires. Comme f ih A (p*o I sn) tend uniformement vers 0 lorsque nh
h tend vers 0 (lemme 1.2.3), la fonction h --> fn 'kh A (p*o Ish) est integrable Jn sur [0, + cc [. (b) Resulte du fait que ME(p) = 0 au voisinage de X. (c)
Soit a une n-forme differentielle sur 397, a support compact C°°, et Jr*TX
tordue. On a _ (-1)n+1
= (-1)n+ f
sx
(da) A r,
,
done
=JsaAco,+(-1)n+1Jf,
fS aAw,=
JB_X
x
(da)ArE,
aAw_= - J.p_x aAdrE,
done on a
< , + df,,
d(a A rE) _ (- 1)n lim f
fs-X
J &-Sh
lim(-1)n f aAr,=lim f h---0 J sh
f
h-0 J x J nh
d(a A rE)
(aAp*a)Ish=
aIx
La derniere egalite etant justifiee par le lemme 1.2.3. En particulier, pour toute forme fermee a E I',(S , S2 OO jr*Tx) on a Jx
aIx= f 3aAcoe.
Dans les cas ou Y est une sous-variete compacte de dimension p < n de X,
variete riemannienne de dimension n, notons NY le fibre normal a Y. On peut trouver un nombre reel so > 0 tel que, pour touts > so, 1'exponentielle induise un C°°-diffeomorphisme du Y-fibre en boules ouvertes de rayon s de NY sur le voisinage tubulaire V. de rayon s de Y dans X. L'image d'une forme w., construite sur NY comme ci-dessus par 1'exponentielle, represente la classe fondamentale de Y dans X (par rapport a tout faisceau T localement libre de rang 1 egal a son inverse, et induisant sur Y le faisceau d'orientation de Y). 1.3. L'element d'aire relatif canonique. On reprend dans ce paragraphe les notations du paragraphe precedent. Notons 17 une connexion sur le fibre i qui preserve la metrique. On sait qu'il en existe une. Notons d'autre part E _ S O le S-fibre image reciproque de par 7rs, JrE la projection canonique de E sur S, v : S-> E la section normale a S, define par pour
PIERRE MARRY & JEAN-LOUIS VERDIER
350
tout ., E S,, ou S, designe la fibre de S au-dessus de x e X, et N le sous-fibre de rang 1 de E engendre par v. Le S-fibre vectoriel de rang r, E - S est muni d'une metrique sur les fibres et d'une connexion V1, images reciproques de la metrique sur les fibres de S et de la connexion V sur E. Notons N1 l'orthogonal de N dans E, VN et VNl les connexions sur N et N1 respectivement, induites par V1, et F, la connexion sur E somme de Whitney de Vn, et FNL. Pour tout element t de 1'intervalle I = [0, 1], on definit sur E la connexion F, = ti7, + (1 - t)V0. Notons pr la projection de S x I sur S. Il existe sur pr*(E) une connexion unique V telle que (a) pour toute section s de E, is/at(pr*s) = 0, (b) la restriction de F a pr*EI sx[t} est F,. Les tenseurs de courbure de P0i V, et r peuvent etre consideres comme des 2-formes sur S et S x I a valeurs dans E* © E et pr*(E* (& E) respectivement. Du fait de 1'existence d'une metrique sur les fibres, on peut les considerer
comme des 2-formes Q0i Q, et Q sur S et S X I, a valeurs dans A2 E et pr*(Az E) respectivement. De meme, le tenseur de courbure de V peut etre considers comme une 2-forme Q sur X a valeurs dans A2 . Notons aussi F0 et r, les connexions images reciproques sur pr*E de V0 et G Q0 et Sl, les images reciproques sur S x I de Q0 et Q1. On peut ecrire
r=ti,+(1-t)I70=V,+(1-t)R, of R = F0 - i, est C°°-lineaire. Si a est une p-forme sur S X I a valeurs dans pr*E, on a R(a) _ v + (-1)P+'Kv, a>F,v ,
ou v = pr*v. On a
pop = r, o V, + (1 - or o R + R o V,] + (1 - t)ZR o R + R A dt
.
Lemme 1.3.0. Sur les sections de pr*E, on a les egalites suivantes entre les operateurs
p1 o R + R o V, _ V,v - 2(V,v, - >V lv R o R = r,v .
Ce lemme resulte directement de 1'expression de R, donnee plus haut, et des egalites w, v> = 1, (V,v, v> = w, V,v> = 0 et = 0 (cette derniere egalite etant vraie parce que V,v est une forme de degre 1 a valeurs dans pr*E). En passant aux 2-formes correspondantes' sur S X I, a valeurs dans pr*(A 1E) 1 a tout endomorphisme antisymetrique f d'un espace vectoriel V euclidien, on associe alt(f) e A2 V tel que <x n y, alt(f)> = 2 pour tout x et tout y de V.
351
PROBLEMES D'INTERSECTIONS
on obtient 1'egaliite
2 = .21 + 2(1 - t)[Viv A
V1v A V10
+(1-t)2V1 AV1v+2V1vAvAdt, ou encore (* )
= Q, + 2(1 - t)FiL, A f,
-
(t2 - 1)F,J A Flf) {- 2F1L A , A dt .
Soit as la (r - 1)-forme C`° sur S, a valeurs dans nr E define par
as-
(1)
as=
(2)
-1
22m7.mm !
(-1)m 22m+17C-M
J pr
2°'
v/ 20
sir=2m, sir=2m+ 1
.
Comme AT E est le fibre vectoriel de rang 1 associe a 7r t(S), as est une (r - 1)-forme C`° sur S et 7c*t(S`)-tordue. Proposition 1.3.1. as est un element d'aire relatif sur S. Cette proposition resulte des deux lemmes suivants. Lemme 1.3.2. da,; = - rs (((-1)m / 22m r °m ! )S2m), lorsque r = 2m et dcs
=10 lorsque r = 2m + 1. Remarquons tout d'abord que les connexions naturellement induites sur Ar E et pr*(A r E) par 1o et V1i v1 et 17 respectivement preservent la metrique. Comme ces deux fibres sont de rang 1, Io est egale a v1 et i, est egale a V sur ces fibres. (a) cas ou r = 2m. Nous utilisons la connexion F1. D'apres la formule de Stokes relative, on a
=
-
11v16S
(-1)(- 1)m
2m
171J
22m7cmm ! 22m7rmm)
.
pr
Qm (Qm) + (-l) 2M-1 I [Jpr M-) prla(SXI)
Par l'identite de Bianchi, P(2m) = 0. D'autre part, comme a(S X I) = S X {1}
-SX{0}ona
JPrIa (S XI)
Qm I
a(sxI) _ QT - Do
De plus 20 = 0 car Fo = FN +O 17N1 done 20 = QN + QN± et en elevant a la puissance m tous les facteurs sont nuls. Done
va
(-1)m s = 22m7.mm)
m1
22mnmJn !
s
352
(b)
PIERRE MARRY & JEAN-LOUIS VERDIER
cas of r = 2m + 1. Nous utilisons la connexion V0. Vo(v A 1Q0 = V0v A Sao + v A V0(Q0-) = 0
car Vov = 0 et V0(20) = 0 par l'identite de Bianchi. Lemme 1.3.3. (a)
f a,, = 1. r
cas r = 2m.
L'expression de .fZ (cf. (*)) nous donne QM
m! DP A [2(1 - t)F1D A Dl4 = P+q+s=m-1 p!q!s! A[(t2-1)V1 AP,vlsA2V1 ADAdt+A,
ou A est la somme des termes qui ne contiennent pas dt, et dont l'integrale sur les fibres de X X I Pr> S est nulle. D'autre part, les termes oil q > 1 sont nuls car f) A D = 0. 11 reste
f 2m =
J pr
m! 2(- 1)v A (V1v)2 .11 ' A Q. f 1 (t2 - 1)Sdt
E
P+s=m-1 p! S!
,
Jo
ou encore
DM = pr
m!(-1)s+122s+1(S!)2
E
P+s=m-1
p!s!(2s + 1)!
A (V,v)21+1 A
j71
ce qui donne pour a, 1'expression m-1
(1 bis)
as _ 71,
P=0
(- 1)P(m - p - 1) ! (2m - 2p - 1)!p!2 2P + 17r-
v A (V 1v)2m-2P-1 A \ QP .
Le seul terme, dont 1'integrale sur les fibres de rrs n'est pas nulle, est celui ou
p = 0 et donc (m - 1) !
f Srsas = (2m - 1)!2-7-r. J "s v A (V l
m-
(2m -
I
(2m-1)I
)2m-1
2;r
1)!
=1,
car la restriction de v A (F V)2--l a chaque fibre de S "s X est (2m - 1) ! fois la forme volume de cette fibre, dont le volume est 27r-/(m - 1)!. (b) cas r = 2m -{- 1. D'apres (*) on a .Q- = (.Q1 + 2V,v A v - V1v A V,v)"` ce qui s'ecrit
PROBLEMES D'INTERSECTIONS
1°0 -
353
(- 1)Sm! SGP A \ (2V1v A \ v)q A\ (V1 )28 P+q+s=m p!q!s!
,
cc qui donne
/
-p!s!M! v A (vlv)2s A/ \SG 1s
/ \ S20 = p+s=rn.
2-1
et par consequent rn
(2 bis)
Q3 =
p7o
22
+1
(- 1)P v /\ (V1)" `P A SJ ir P!(m - p) ! 7n
.
Le seul terme dont l'integrale sur les fibres de 7rs n'est pas nulle est celui ou p = 0, ce qui donne Q ns
1 7r",
f
ns
v
A
1
v
227,+1 7am
m! (2m)!
(2m)!
=1
Definition 1.3.4. La forme a, definie sur S par 1'egalite (1) si le rang de E est pair, par 1'egalite (2) si le rang de 37 est impair s'appelle l'element d'aire relatif canonique de S associe a la connexion V.2 Remarque. La connexion V sur 37 donne naturellement un scindage f de la suite exacte de S-fibres 0 -- 7rS SZX -- S? S 4----
f
S/I -- 0
d'oii l'on deduit une decomposition de SZs 1 en somme directe Qs 1 =
(1)
2s (SAX) ®2S/X
,et une decomposition des (r - 1)-formes sur S, t(E)-tordues SGg
I(& 2rst(ug)
=O S(SGX) ® S?g,1 ®7rst(C) . a+p=r-1
En particulier o se decompose selon cette somme directe. On peut aisement verifier que cette decomposition coincide avec la decomposition (1 bis) ou (2 bis), selon la parite de r. Soit xa E P(X, S?r (3 t(-P)) la forme define ci-dessus telle que - da, = 7rs (x,s).
On a dxs = 0, done x8 definit une classe de cohomologie [x,] appartenant a Hr(X, t(S)). La proposition 1.2.2 montre que cette classe ne depend pas de 2 on retrouve en (1 bis) une expression donnee par Chern dans [6] dans le cas du fibre tangent a une variete riemannienne de dimension paire.
PIERRE MARRY & JEAN-LOUIS VERDIER
354
1'element d'aire relatif choisi, donc de la connexion utilisee pour le construire. Lorsque r est impair, XE = 0. Definition 1.3.5. La forme SCE define ci-dessus s'appelle la forme d'Euler du fibre associee a la connexion V, et sa classe [xEj s'appelle la classe d'Euler
de E. Lorsque rgS = 0, on pose par convention xE = 1. 2.
Proprietes de 1'element d'aire relatif canonique
X un fibre vec2.1. Fonctorialite par rapport a la base. Soient E toriel defini comme au § 1, et f : Y , X une application C'° d'une variete differentielle Y dans X. Notons f *(E) le Y-fibre vectoriel image reciproque de E par f, muni de la connexion image reciproque de V par f. Alors 6f*(E = f*(6E)
et
= f*(;CE)
2.2. Homomorphisme de Thom-Gysin. Dans ce numero, F design soit le faisceau d'orientation 9-, de X, soit le faisceau simple de fibre R sur X, et 9 designe le faisceau inversible -F Ox t(E).
Remarquons tout d' abord que pour tout faisceau inversible d sur X, et tout nombre entier k, nous avons des isomorphismes
p* : Hk(S, icssd) _ Hk(E - X, lr*d I E-x) 7c*
Hk(X, a) _ Hk(E, it*d)
car S (resp. X) est retract de E - X (resp. E). Les isomorphismes reciproques, sont les morphismes de restriction correspondants. De meme, si 9 (resp. '') designe la famille des supports de E propres-au. dessus de X (resp. de projection sur X relativement compacte) on a pour-tout nombre entier k un isomorphisme
et comme X est retract de E, un isomorphisme 7C : H' (X, d) , H' (37, x *a) pour tout nombre entier 1. On a donc, par dualite de Poincare un isomorphisme transpose Hn +r
1 (E, *a O T, O ir*t(E))
H"-'(X, a O
x)
Donc, quitte a changer les notations, on a pour tout faisceau inversible et tout nombre entier k un isomorphisme
PRO13LEMES D'INTERSECTIONS
355
H;(S, it*d) =4 H"-"(X, d OO t(S))
,
qui n'est autre que l'integration sur les fibres de it et dont l'isomorphisme reciproque est l'application u qui a la classe [a] de Hk-''(X, .sad (9 t(S)) de representant ce associe la classe de 7r*ce A w., ou wE est define comme dans la proposition 1.2.5. Proposition 2.2.1. L'unique application de Hk 7(X, sad dans Hk(X,.sad) rendant commutatif le diagramme u
Hk(S, z*d)
I
H'-7(X, ,d (9 t(S )) -> Hk(X,.sad)
ou u est l'application naturelle, est l'application U [X,] qui a un element de H"-"(X, d (D t(,S)) fait correspondre son cup-produit a droite par [Xs]. Cette proposition se deduit immediatement de ce qui precede. Soit s un nombre reel compris strictement entre 0 et 1 et t la fonction de-
fine comme en 1.2.5. Posons * = 1 - 0E. On note toujours p : S - R, la norme sur les fibres de 8, p : S - X - S la surjection canonique. Notons 3: Hk-I(S, ir,*sd) - Hk(S, ;r*d) l'application qui, a la classe [a] de H"-'(S, 7rsd) de representant ce associe la classe de On a la proposition suivante. Proposition 2.2.2. On a un diagramme commutatif, dont les lignes sont exactes et les colonises des isomorphismes : Is g
Hk I(u,
Hk I(u - X, *&
Id
y Hk-I(,J
*d)
Is
w
IsI Hk-I(S, 7rsd)
Hk9(8, *d)
Hk(C, *.s) ) Hk(c - X, 7r*& Is v
Hk(,F , 7r*d)
Hk(S, zc*s s
...
) .. .
ou la ligne superieure est la suite exacte de cohomologie associee au ferme X
de E. La deuxieme ligne est la suite exacte de cohomologie a supports propres sur
X de 37, en interpretant S comme l'ensemble de points a l'infini de 37. Par construction 3 n'est autre que le cobord de cette suite exacte. La commutativite du diagramme resulte alors des compatibilites entre cohomologies a support. Corollaire 2.2.3. On a une suite exacte (suite exacte de Thom-Gysin)
358
PIERRE MARRY & JEAN-LOUIS VERDIER
S>
Hk(A')
Idl
Hk '( sl
Xl
l
H(u)1
(X
Hk(Shsy)
> H(s) -1--, Hk+1//lQX) " ' jna -
Id
I
( S2'sl iS2'
S' > Hk+y(fix) .
.
dont les lignes sont exactes. Le diagramme obtenu en ajoutant la fleche en pointille est toujours com-
mutatif. En effet, si R a Qs est une forme fermee, posons r = (irf p) A s
Qa
Alors dr = 7rS [(_1)r(f ) A x] et f r = 0. On en deduit, comme dans la proposition 1.2.2, qu'il existe des formes a et b, sur X et S respectivement, telles que r = icsa + db. Par consequent R et j(f R) sont rs dans la meme classe de Hk(QsIiQX(g, v)). \ / En revenant a la definition de la differentielle sur A', on voit que 8 n'est autre que le cup-produit a droite par [x], ce qui fait apparaitre la suite exacte de Gysin Hk-r(S2.)
Hk(p x)
4
Hk(S2s) f
Hk-r+1(Q. )
Comme la ligne inferieure est exacte aussi, on en deduit que la fleche verticale non-nommee induit un isomorphisme sur la cohomologie. D'apres le lemme des cinq, H(u) en est aussi un. Corollaire 2.4.3. Si [x] = 0 (ce qui est le cas par exemple lorsque r est impair) on a H*(S)
H*(X) O+ H*(X, t(S))a
,
oiu a est tel que a' = 0 si rest pair, et a' = 'p,,,, sir = 2m+ 1 et ou d°a = r - 1. D'autres consequences de la proposition 2.4.2 sont donnees dans [10]. 2.5. Restriction a un sous-fibre. Soient V une variete differentielle C- et n
F r V un fibre vectoriel de rang r sur V, muni d'une metrique sur les fibres et d'une connexion F compatible avec cette metrique. Etant donne un drapeau D d'ordre n de F, c'est-a-dire une suite strictement decroissante de n sous-fibres vectoriels de F telle que F = F, D FZ D Fn, si nous notons I = [0, 11 et pk la projection V X I" -> V pour 1 < k < n, nous definissons par recurrence une connexion Vk sur le fibre p *F de la maniere suivante : (a)
pour k = 1, F, = F,
pour k > 1, designons par q, la projection de V X I,'-' = (V X III-') X I sur V X I". Alors Ik est la connexion sur p S et p': S - X --> S' les projections canoniques et 6' un element d'aire relatif sur S'. Alors 6 = p'*6'Ig est un element d'aire relatif sur S, p*6 = p'*a' et it existe une r-forme X sur X telle que d6 = -7rsX et d6' _ 7r' *X. Donc si E(X, F, s; 3) ne depend pas de l'element d'aire relatif choisi, it ne depend pas non plus de la metrique. Soient 6o et 61 deux elements d' aire relatif s sur S. D' apres proposition 1.2.2,
it existe une (r - 1)-forme ri sur X et une (r - 2)-forme r sur S telles que 61 = 60 + 7r*,5 + dr. Si Xo et X1 sont les r-formes sur X telles que doo = - r
- d. Donc
et d61 = -7rs*X1, on a
p A s*p1 =
Sax
(3 A s*p*60 + fax $ A ri + fax B A s*p*(dr) .
D'apres la formule de Stokes, comme dd = 0, le dernier terme est nul et le d(p A ri), ou encore (- 1)n-r fx p A dri. Donc second est fx (-1)n f
x
p A X1 + (-I)' fax p
A s*p''`61
_(-1)n fX p A Xl +(-1)" If ax p A s*p*60 +(-1)n-T fx p A
_(-1)n fx p A
(
(X1 + dri)
diJ
f A s*p*6o + (- l)T fox p
d'ou (2) puisque X1 + da) = Xo
Supposons X compacte et prenons pour fibre S le fibre T. tangent a X, pour sections la section normale sortante v, et pour p la section canonique. Posons alors E(X) = E(X, Tx, v; p). Proposition3.1.2. On a E(X) = EP(X), ou EP(X) = E ,(-1)i dimHP(X,R) (cohomologie singuliere) est la caracteristique d'Euler-Poincare de X. Supposons que la dimension de X soit paire. Soit v un champ de vecteurs sur un voisinage de aX qui prolonge la section normale sortante v sur aX. On peut munir Tx d'une metrique et d'une connexion compatible V telle que Vv = 0. La forme A de courbure totale relative de v dans Tx est alors nulle.
D'apres remarqe 2.6.4, on a donc (*)
A=0
2) *P*6T, =
fax fax Notons X la variete sans bord obtenue en recollant deux copies X1 et X2 de
X le long de leur bord On a, d'apres (*),
E(X) = fX1 XT%= + f XT= = 2E(X) . 2
365
PROBLEMES D'INTERSECTIONS
Par la suite exacte de Mayer-Vietoris, on a
E(X) = EP(X) = 2EP(X) - EP(aX) et, comme dim (aX) est impaire, EP(aX) = 0, d'oi la proposition dans ce cas.
Supposons que la dimension de X soit impaire. On a alors, d'apres remarque 2.6.4, comme le sous-fibre de Tx normal a v au-dessus de aX n'est autre que le fibre tangent a aX, et comme XTx = 0, E(X) = 2E(aX) = 2EP(aX)
.
On a la suite exacte H*(X, R)
H* (,Y, R)
H*(aX, R)
d'ou EP(aX) = EP(X) -
(-1)i dim H'(X, R). Par la dualite de Poincare de X, on a
-
i
(-1)i dim Hi(X, R) _
i
(-1)i dim Hi(X, Tx)
Lorsque X est orientable on a donc EP(aX) = 2EP(X). Dans le cas contraire soit X le revetement canonique a deux feuillets orientable de X. On a Hh(X, R) = Hi(X, R) O Hi(X, Tx) ,
d'oti
EP(X) +
(-1)i dim Hh(X, Tx) = EP(X) = 2EP(X)
puisque X est un revetement d'ordre 2. Donc EP(aX) = 2EP(X) ce qui demontre la proposition. Corollaire 3.1.3. Soit Z C X un ferme dont X soit un voisinage cotubulaire. Pour tout i, H'(Z, R) est un R-espace vectoriel de dimension finie isomorphe a Hi(X, R). On a
EP(Z) = EP(X) = (-1)"- x f
x XTx +
v*oTx
(-1)aim x f
ax
En effet, comme Z admet un systeme fondamental de voisinages retracts de X, on a Hi(Z, R) Hi(X, R). 3.2. Termes locaux. Dans ce numero, on conserve les notations du nu-
368
PIERRE MARRY & JEAN-LOUIS VERDIER
Proposition 3.4.1. Pour toute forme C° fermee a sur X, de degre n - r et rx OO t(8)-tordue, a support compact, on a 0(W, s,
( A s*p*u
1)T lim f
En effet, par la definition 3.2.1 on a, pour tout k,
`y(W,s,p)=F'(Vk,3IPk,slaPk;M =(-1)n f PAz+(-1)rfavk pAs*P*o, Ti
Vk
et la premiere integrale tend vers zero avec la mesure de Vk lorsque k tend vers l'infini. Considerons le cas oii 8 est muni d'une connexion compatible avec la metrique. Supposons qu'il existe un voisinage V de W dans X qui soit une sousvariete a bord de X, fermee dans X, tel que s ne s'annule pas sur V - W et qu'il existe un diffeomorphisme F : V X ]0, 1] -f- V - W. Un tel voisinage et un tel diffeomorphisme existent par exemple des que la variete X, le fibre 8 et la section s sont analytiques d'apres les resultats de Lojasiewicz. Notons s': V - W - S 1'unique section telle que (a) s'lav = (s/IIsI)Iav = P°Slav, (b) 1 ou bien si codim W = 1 et rgE impair, on a O(W' S' R) = (-1)T fw (3 (9 s(4s) A x, , oh a est une section a valeurs entieres de t(3) Qx t(V) Q t(-Y). Lorsque codim W
= 1 et rg8 est pair it Taut remplacer dans la formule precedente xE par la forme de courbure totale relative A de I dans E. Resulte des corollaires 3.4.3 et 2.6.3 dans le cas codim W > 1 et de la remarque 2.6.4 dans le cas codim W = 1. Corollaire 3.4.5. Si rgE = codim W on a 0(W, s,
(-1)T
w
R © £(4s) .
Dans certains cas on peut determiner la classe d'homotopie de 4s par le calcul differentiel d'ordre 1. On a, sur B, la suite exacte de fibres
0 , 7r*3 , T, , 7r*T, , 0 oii T1 et T,, sont les fibres tangents, d'oiI une suite exacte sur X en prenant l'image inverse par s (*)
0,-: _*s*T&_*TX_0.
L'application tangente Ts: TX -* s*Ts est un scindage de (*). On a sur W,
370
PIERRE MARRY & JEAN-LOUIS VERDIER
suppose titre une sous-variete de X, la relation s = i ob i est la section nulle, d'ou s*T,31w = i*T,lw, et deux scindages Ts et Ti de la suite exacte
0 - Elw- s*T3Itiv-
T1Iw--> 0 .
La difference Ts - Ti est un morphisme de fibres vectoriels T. Jw --- l w = F w qui s'annule sur le fibre tangent Tw, d'ou par passage au quotient un morphisme Ds: Nw ---> Sw oil Nw = TxlwlTw
Corollaire 3.4.6. Si Ds : Nw --> Ew est un isomorphisme de Nw sur un sous-fibre de Cw,3 on a, lorsque codim W > 1, ou bien lorsque codim W = 1 et rg impair
0(W, s, p) =
(-1)T
w
p OO s A xDscnw:l
ou s est une section de module 1 de t(8) OO t(aV) OO t(7). Lorsque codim W = 1 et rg est pair, it Taut remplacer dans la formule ci-dessus la forme XDS(Nw)1 par la forme 2A oh A est la forme de courbure totale relative de Ds(Nw) dans
w et la section s est de module 0 ou 1. Donnons-nous une metrique sur NW. Notons SNw ---> W le fibre en spheres unites correspondant. C'est le fibre en spheres d'un voisinage tubulaire de W dans X. L'application Ds induit un homomorphisme de fibres Js: SNw ---> S,3w.
On constate que J's est daps la classe de W-homotopie de l'application Js introduite en proposition 3.4.2. Le corollaire resulte alors de 3.4.4 compte tenu de cc que le degre local s est de module 1 et de module 0 on 1 lorsque codim W = 1 et rg-F, pair. 3.5. Degre local. Precisons maintenant comment on utilise et comment
on determine la sections de t(E) Ox t(aV) OO t(7) du corollaire 3.4.4. En utilisant les isomorphismes canoniques t(2) © t(') -_ R et rx OO t(aV) -_ zw, 41e produit tensoriel par s induit un morphisme de faisceaux localement libres de rang 1 sur W, z® O t(S) ---> zw OO t(7). Ceci permet d'interpreter daps le corol-
laire 2.2.3 p (x s comme une forme zw OO t(7)-tordue sur W et par suite p Ox s A Xz (resp. p Ox s A A) est bien une mesure sur W. Lorsque rg = codim W, on a rg-y = 0 et it faut prendre par convention t(') = R. La forme p ©s est alors rw-tordue sur W et est donc une mesure sur W. Lorsque de plus, W est un point, zw = R et p est une section de zx OO t(8). Supposons p # 0 ; it exi ste alors un unique hombre reel p > 0 tel que p / l p I soit une section entiere qui engendre zx OO t(8). Donc (p/Jpp OO s est un nombre entier apple le degre local de s relativement a p. II existe parfois un choix canonique
de p, lorsque par exemple - = Tx le fibre tangent a X, et on omet de preciser 3 ce qui revient a dire que pour tout x e W, Ds(x) est injectif. 4 Si e est une base de t(E), a e C e correspond 1. Si Or (8V), Or (X) sont des orientations, a Or (X) ®x Or (a V) correspond Or (W) tel que Or (W) ®x Or (a V) = Or X dans l'isomorphisme naturel zw ® t(8V) -_ rx.
371
PROBLEMES D'INTERSECTIONS
dans ce cas que le degre local considers est pris relativement a (3. Pour determiner la section s, on proeede comme suit. Placons-nous sous les
hypotheses de 3.4.4 et supposons ae plus que codim W > 1. Soient x e W, dsx: aVx > S2,x l'application induite stir les fibres en x par Js, I, le sousespace orthogonal a -',x et Js,, : V -> Srs une application qui, composee avec l'injection SEs -> S,; donne une application homotope a dsx dans S8 - Sr. Donnons-nous au voisinage de x, une orientation Or (8) de 8, une orientation Or (DV) de aV et une orientation Or (s') de I. Lisomorphisme canonique tnax
t(-') __ t(8) O t (-Y)(deduit de l'isomorphisme canonique
7
77
permet alors d'orienter SEs a 1'aide de Or (8) et Or (E). Notons deg (s) le degre topologique de l'application dsx entre les spheres W, et S,-L,, munies des orientations Or (aVx) et Or (S,-L,,) respectivement. Proposition 3.5.1. On a e(ds)x = deg (s) -Or (8) Ox Or (aVx) Ox Or (1).
D'apres le corollaire 3.4.3 on a
O(W, s, ) = (- 1)r f
w
R 1 w A flav ds*6alw
Comme Js est homotope a une application ds : aV -> SE on a, vertu de la propo-
sition 2.5.1 et du theoreme 2.5.1,
0(W,S,R) = (- 1)r fw Rlw A f
av
As*(UEl A irBVxz)
Mais on a
6E1 A 7r YXE _ (-1)rmIr n C21
avec m = rg2' compte tenu de l'isomorphisme canonique de commutation i
t(-Y1-) Ox t(-Y) -=->. t(-Y) Ox t(-Y1-). Donc, par definition de l'integration par fibre,
O(W, S, p) = (-
1)r-r771 f
w
Rlw A xE
f
ds*(6E1) nav
Par definition on a x-,
av
's*(a,1) = deg (s) Or (-'1) © Or (aV)
par suite ds*(6,1) = ex-,
x nav
372
PIERRE MARRY & JEAN-LOUIS VERDIER
compte tenu de 1'isomorphisme i decrit plus haut et des isomorphismes de commutation de t(aV) avec t(f) et t(X--). On a donc O(W, s, p) _
(-1)rm. (-1)T
fw R w s A XE
avec la sections decrite par la proposition 3.5.1. Mais si m est impair et m G r - 1, on a XE = 0 et on peut toujours prendre le sign indique dans la proposition 3.5.1. La description des dans le cas codim W = 1 demande une etude particuliere laissee au lecteur. Notons seulement que le module de s est 0 ou 1 et que la formule 3.5.1, convenablement interpretee est encore valable lorsque rgS est impair.
3.6. Intersections de sous-varietes. Soient Z une variete de dimension n connexe sans bord, X et Y deux sous-varetes connexes de dimensions respectives p et q, Tx et TY des R-faisceaux inversibles d'ordre 2 sur Z qui induisent sur X et Y respectivement les faisceaux d'orientation. On a donc des classes fondamentales [X] e H4-P(Z, Tx) et [Y] e H11 -4(Z, vy). Pour toute forme fermee a sur Z, a support compact, zZ Q zx
O v -tordue et de degre p + q - n, on a [a] U [X] U [Y] e
TZ) ^_ R.
On se propose de dormer une autre expression de 1'application a H [a] U [X] U [Y]. Soient Nx le fibre normal a X, V un voisinage tubulaire de X, p : V -p Nx
un X-diffeomorphisme de V sur le fibre en boule unite de Nx pour une metrique sur Nx, N le fibre vectoriel p*q*Nx sur V of q : Nx -p X est la projection canonique. Soit s : V -p N la section deduite par changement de base de la section diagonale de q*Nx sur Nx. Notons N le fibre NlYr,V et s : Y n V N la sections restreinte Ay (1 V. Notons que l'ensemble des zeros de s est
XnY.
Proposition 3.6.1. Si X n Y est de mesure di ff erentielle nulle dans Y, on a
[a] U [X] U [Y] = o(X n Y, s, any)
.
Soient wx et wY des formes fermees sur Z dans les classes [X] et [Y] respectivement. On a
[a] U [X] U [Y]=JZ aAwxAwy=JY alyAwxly on peut prendre pour forme wx la forme p*w1,2 (proposition 1.2.5), et comme le support de aAwx est alors un compact contenu dans V, on a
PROBLEMES D'INTERSECTIONS
[a] U [ X] U [Y] = f
373
a Y A p* Win Y n V.
Soit Uk une suite de sous-varietes a bord de y fl V, fermees dans Y fl v telle que Uk+, C Uk et n Uk = X fl Y. Comme x fl Y est de mesure nulle, on a k
[a] U [X] U [Y] = lim fyn k-. o
-IIk
alyA p*Winz
Mais sur Y fl V - Uk, p*Q)1,2 = p*(-dr,,2) (proposition 1.2.5). Par suite a l y A p*r,iz [a] U [X] U [Y] =(-1)rgN lim k-- f auk
On a done
[a] U [X] U [Y] = lim (-1)'g7' fauk aly A s*p*o'N k-d'oii la proposition d'apres la formule (1) du § 2.1. Si W est un ouvert ferme de X fl Y, on a [a] U [X] U [y]_ 1(X fl Y, s, a iy) o(W, s, a1Y) + o(X fl Y - W, s, aly) L'application a 0(W, s, aly) est appelee le terme local de l'intersection de X avec Y relatif a W et note (X, Y ; W, a). Soient W un ouvert ferme de x fl Y qui soit une variete connexe de dimension strictement inferieure a celle de Y. On a, sur W, un diagramme de sousfibres de Tz I w :
Txlw
Tzlw
Tw
TyIw
d'oii une suite de fibres sur W et de morphismes de fibres : (1)
Tw -- Txjw
TYjw -I ) Tzw
avec i = (1) et j = (i3, -i4). On a j o i = 0 et i est injectif. Corollaire 3.6.2. Si l'image de j dans (1) est un sous-fibre et si (1) est-une suite exacte de fibres, la section s du corollaire 3.4.4 est de module 1. Resulte do corollaire 3.4.6.
374
PIERRE MARRY & JEAN-LOUIS VERDIER
3.7. Formule de Lefschetz. Soient X une variete compacte connexe riemannienne de dimension n et f : X -- > X une application differentiable. Soient 4 C X X X la diagonale, Ff c X X X le graphe de f, p1, p2 les premiere et deuxieme projections de X X X sur X. Comme p*TX induit sur F f le faisceau d'orientation de F f on a une classe fondamentale [F f] E Hn(X X X, p*rx). On a de meme une classe fondamentale [d] E H"(X X X, pa rX). Posons
(1)
Lef (X, f) = [4] U [F f]
.
A 1'aide de la dualite de Poincare on verifie immediatement que n
Lef (X, f) = E (-1)i Tr (f, H1(X)) i=o
.
Pour f = IdX on obtient
Lef (X, Id,) _
(-1)i rg (Hi(X))
.
i=O
On a donc Lef (X, Id,) = EP(X) caractteristique d'Euler-Poincare de X. Proposition 3.7.1. On a EP(X) = f, XTX = [XTX] En effet si wa est une forme differentille fermee sur X X X dans la classe [3] (proposition 1.2.5), on a
EP(X) = [d] U [4] = fXXX 0)'A w4 = Jd (011, = (_ 1)n fX XT-r' Remarque 3.7.2. Pour des raisons de parite on a EP(X) = 0 lorsque n est impair. Ceci resulte de la proposition 3.7.1 car dans ce cas XT., = 0. Ceci resulte aussi de la dualite de Poincare lorsque X est orientable. Supposons dorenavant que d f1 Tf soit de mesure differentielle nulle dans J. Notons a la section rxxx OO p*rx ®p2 rx qui au voisinage de tout point (x, y) s'ecrit Or,,, (X X X) ® Ors (X) ® Or, (X) avec Or.,,,, (X X X) = Or, (X) Or, (X) .
L'ensemble d n F f s'identifie par les projections a 1'ensemble des points fixes de f. Pour toute partie ouverte et fermee W de J n r f posons (cf. § 3.6) (2)
Lef (X, f ; W) = YQ, F ; W, a)
.
Si A n r f est une reunion finie de parties fermees disjointes Wi, on a donc
PROBLEMES D'INTERSECTIONS
Lef(X,f) _
375
Lef(X,f;Wi)
Le nombre Lef (X, f ; Wi) est appele le nombre de Lefschetz local de f relatif a Wi.
Soit W une partie ouverte et fermee de 4 fl F f qui soit une sous-variete connexe de J. Notons Tx le fibre tangent a X, Tw le fibre tangent a W, Nw
le sous-fibre de T1Jw orthogonal a Tw, p,: Tx - X et pw : Nw - W les projections canoniques, exp : Tx - X 1'application exponentielle. L'application e H (p(e), exp (a)) de Tx dans X X X induit un diffeomorphisme d'un voisinage de la section nulle sur un voisinage V de la diagonale. NW l'application inverse. L'application Notons (x, y) ti xy E
X induct un diffeomorphisme d'un voisinage de la section nulle sur un voisinage U de W dans X. Pour x e U, posons 2r(x) = pw o exp-1 (x) E W. L'inverse an voisinage de W de expINW est donc x H 7r(x)x. Comme W est fixe par f, it existe un voisinage U' de W tel que U' C U, f (U') C U, et tel que pour tout x E U', (x, f (X)) E V.
De plus le chemin t - exp t2r(x)x, t e [0, 1] est un chemin geodesique note g(2r(x), x) joignant 7c(x) a x.
Notons Bg : Tx,,, - Tx,.(,x) le transport parallele le long de g(rc(x), x). L'application x - Bg(xf (x)) est une W-application car on a px o Bg(xf (x)) = 7r(x)
et pour x e U' - W, Bg(xf(x)) # 0. Notons SxJw le fibre en spheres unites de TxJw-
Definition 3.7.3. On appelle indicatrice de f an voisinage de W, et on note Iw(f): U' - W - SxJw l'application x H Bg(xf(x))1IIdg(xf(x))IJ. Proposition 3.7.4. Soit V C U' un voisinage tubulaire de W pour l'appli-
cation rr. On a Lef (X, f ; W) _ (- 1)" J Iw(f)*(CTx). V Par la definition (2), on a Lef (X, f ; W) = '(4, F f ; W, a) et d'apres 2.4, on a ?'(4, Ff ; W, a) = O(W, s, c ). xy, En utilisant le voisinage tubulaire de 4 donne par l'application (x, y) on voit que le fibre N du § 3.4 n'est autre que pi Tx. Il induit donc sur Ff, identifie a X par la premiere projection, le fibre Tx. La sections du § 3.6 est S, J w West autre que l' applicaalors la section x H x f (x) et I(f): U' - W tion Js du § 3.4. La proposition resulte alors du corollaire 3.4.3.
Notons Sw, le fibre en spheres unites de Tw. Definition 3.7.5. Nous dirons que f est sans glissement an voisinage de W, s'il existe un voisinage tubulaire V C U' de W pour l'application rr, tel que Sw fl [Iw(f)(aV)] = 0. Supposons f sans glissement. Soient V C U' un voisinage tubulaire de W tel que Sw fl [Iw(f)(aV)] = 0 et x e W.
376
PIERRE MARRY & JEAN-LOUIS VERDIER
L'application Iw(f) av : aV. S,I w,., - Sw,, est homotope a une application Iw,.,(f) : aVy - SN,,,x oil SN,,,.., est la sphere unite de l'espace normal a W en x. Une orientation de X et de W au voisinage de x permet d'orienter V,, et SN,,,,, et pour ces orientations, Iw,.,(f) possede un degre note deg (W, f). Ce degre ne depend pas des orientations choisies ni du point x choisi. ,
Theoreme 3.7.6. Supposons f sans glissement au voisinage de W. Si codim (W, X) > 1, ou bien si X est de dimension impaire, on a Lef (X, f ; W) = (-1)n deg (W, f) EP(W)
.
Resulte de la proposition 3.7.4, du corollaire 3.4.4 et de la proposition 3.5.1. Notons Tw(f) : T1I w - Txl w l'endomorphisme de Txl w induit par la differentielle de f. Dans la decomposition: Txjw = Tw Q+ Nw, Tw(f) a une matrice
L de la forme (Id 0 Id + M)* Proposition 3.7.7. Pour que f soit sans glissement au voisinage de W, it suffit que det M z 0. On a alors deg (W, f) = signe (det M). Immediate. Bibliographie R. Bott, Vector fields and characteristic numbers, Michigan Math. J. 14 (1967)
231-244. A residue formula for holomorphic vector fields, J. Differential Geometry
1 (1967) 311-330. R. Bott & S. S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math. 114 (1965) 71-112. N. Bourbaki, Variete differentielles et analytiques, Actualites Sci. Indust., No. 1347, Hermann, Paris, 1971, §§ 10-11. J. Cheeger & J. Simons, Differential characters and geometric invariants, preprint, 1973.
S. S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. 45 (1944) 747-752. -, On the curvatura integra in a Riemannian manifold, Ann. of Math. 46 (1945) 647-684.
H. Flanders, Development of an extended exterior differential calculus, Trans. Amer. Math. Soc. 75 (1953) 311-326. S. Kobayashi & K. Nomizu, Foundations of differential geometry, Vol. II, Interscience, New York, 1969. P. Marry, Type d'homotopie rationnelle relative des fibres en varietes de Stiefel, Compositio Math. 34 (1977) 91-98. F. Takens, On the differential forms representing the Chern and Eider classes, Rev. Roumaine Math. Pures Appl. 14 (1969) 693-702. ECOLE NORMALE SUPERIEURE, PARIS
J. DIFFERENTIAL GEOMETRY 12 (1977) 377-401
HOLOMORPHIC AND DIFFERENTIABLE TANGENT SPACES TO A COMPLEX ANALYTIC VARIETY JOSEPH BECKER
An important invariant in the study of analytic varieties is the local embedding dimension. To measure this precisely one defines T(V, OP), the tangent space to V at p with respect to the analytic functions. Similarly one can define tangent spaces with respect to the infinitely differentiable functions Cm, and the k times continuous differentiable functions Ck, whose dimension is the local Ck or C° embedding dimension. It is known [6], [18] that T(V, CP) = T(V, OP). In this paper we strengthen that result as follows : there is a locally bounded function k : V - * Z+ such that T(V, Ck(P') = T(V, OP). An outline of the paper is the following. First show that for curves, k can be picked G N, where N is the exponent of the conductor. Then find a curve C in V such that T(C, OP) = T(V, Op). The local boundedness of k follows by showing there is an upper bound for the conductor number of all nearby linear one-dimensional sections of V. One finds this upper bound by stratifying V into finitely many "equisingular" varieties so that the conductor number is constant on each one. For curves, we derive some precise estimates for k, and in § 3 we give examples to show these estimates are in general the best possible. Also for each k we show there exists a variety V so that T(V, C11-1) # T(V, O), but T(V, Ck) = T(V, O), that is, k is the precise critical degree of differentiability. This enables us to construct a Stein complex space X with no C°° embedding in any C"°, but for every k there is a Ck embedding into some Cn. The author would like to thank K. Spallek and the referee for pointing out that Theorem 1 of this paper can be obtained directly via 1.1.5 and the last remark of [16]. The methods employed in [16] are somewhat different and do not seem to yield a proof of the curve selection lemma (Theorem 2) or the slicing lemma (Lemma 2) of this paper. 1.
Definitions and preliminaries
From [18] we have all of the following. Let V be a complex analytic variety in C", p e V, Ck the ring of germs at p of k times continuously differentiable Received July 3, 1975, and, in revised form, March 31, 1976. Research supported by a grant from the Purdue Research Foundation.
378
JOSEPH BECKER
complex valued functions on C11, k = 1, 2, , oo, and I(V, Cp) the ideal of functions in Cp vanishing identically on V. Then T(V, Cp) = {a e C71 = R'11:
(p) + aj
a.;
= 0 for all f e I(V, Cp)} J
_ (r ... , r2n) E Rn :
zn
az ri ax = 0 for all f e I(V, Cp)} J
where we identify C71 = R271 by ak = r2k_1 + ir21 . This is clearly a vector space
over the field of real numbers but not necessarily over the complexes : Write C71 = R71 O+ iR7L, C = R O+ iR. Then
a=ax+iay,
f=f.x+ify,
is = -ay + iax
if = -fy + if. ,
aeT
df=df.x+idfy, d(if) = -dfy + idfx = idf
0 = ax(dj., + idxfy) + ay(dyfx + idyfy) = axdxfx + aydyfz + i(axdxfy + adyfy) axdxfx + aydyfz = 0 = axdxfy + aydyfy
Hence it is sufficient to consider only the real valued f x and f y in computing the tangent space. By T(V, Op), we will mean the usual Zariski tangent space, sixth tangent cone of Whitney C6(V, p) _ {a e C1: adpF = 0 for all F e I(V, (9p)}. Other useful tangent cones are the third, fourth, and fifth of Whitney : C3(V, p) = {a e C": 3 sequences qj e V, 2i e C, qi -* p, 2 (p - qi) -* a} C(V, p) = {a e C71: 3 sequences qi e Reg (V), qi -* p, vi e T(V, (9q)
with vi-*a} C5(V, p) = {a e C": 3 sequences qi, pi e V, 2i e C, qi, pi -* p
,
2i(pi-qi)-*a} We have the following sequence of strong inclusions : C3(V, p) C C4(V, p) C C5(V, p) C T(V, Cp) C ...
T(V, Cp+') c
c T(V, C;) c T(V, (9p)
T(V, CP)
.
In addition, Bloom has shown [5] that if p is an isolated singular point of V, then T(V, ' ) is the complex linear span of C(V, p). 2.
One dimensional case
Throughout this section V will be a one-dimensional complex analytic subvariety of C71 with the origin as a singular point. If V is irreducible, 0: C -* V
379
HOLOMORPHIC TANGENT SPACES
will denote its normalization. Unless otherwise stated, V will be assumed to be holomorphically imbedded in its minimal possible dimension, that is, T(V, Oo) = Cn. We begin with some rather technical results, the first similar to paragraph 2.2 of [17]. Lemma 1. If f e I(V, Co), there is a holomorphic polynomial Pk(z) _ iQisk aaza, with Daf(o) = a!aa such that Pk(z) = o(jzIk) on V. Proof. By appropriate choice of coordinates, the normalization ¢ can be qn and , t°-un(t)), where q = q1 G q2 G the ui,'s are units ; hence o(z j) = o(1 I zz j) = o(z, J). There exists a polynomial Ak(z, 2)= kth order Taylor expansion off about the origin such that f - Ak = JPI=kz'gd(z) = o(zlk), where the gp are continuous functions such that g,(o) = 0. Let Ak = Pk + Qk be the sum of polynomials with Pk holomorphic and Qk having no holomorphic terms. Now composing with the normalization and writing holomorphic polynomial P(t) = Pk(cp(t)), polynomial Q(t, t) = Qk(cp(t)) with no holomorphic terms, and l = qk, we have written as ¢(t) _ (t°l, tQyu2(t),
P(t) + Q(t, t) = t`g(t) + tth(t) = o(I tl`)
,
where g and h are continuous functions such that g(o) = h(o) = 0. Hence nei-
ther P nor Q can have any terms of degree l or less, and we conclude that P(t) = o(tt). Thus Pk(z) = o(jzjk) on V. So far V has been assumed to be irreducible ; but if V is reducible the argument given is valid on each component, and the lemma as stated clearly holds if it holds for z in each component. Lemma 2. There is a biholomorphic change of coordinates in Cn so that the normalization has the form O(t) = (t4-UP), , t4-un(t)) where the uz are
units, q, < q2
(*k(t), g(t)) is not one-to-one either. Second Case. i]k is one-to-one. Then ikk is itself the normalization of a
curve, and letting R be the subring of C{t}, the ring of convergent power series in t, of convergent series in p1(t), , ck(t) it is well known that R contains all power series of high order. (The ideal J of universal denominators has locus just the origin, so by the Nullstellensatz rad J = m, the maximal ideal of C{t}. Hence there exists N > 0 such that for l > N, t'C{t} (-- R, so t' E R.) Now if the above process goes on for N(n - k) steps, ord 0k+1 > N and in one more step (subtracting off the corresponding convergent power series in R) we can make yak}1 - 0, which contradicts the fact that V is imbedded in minimal possible dimension. Proposition 1. Let V be irreducible, then there exists k > 0 such that T(V, Co) = T(V, Oo). Proof. Since V is imbedded in minimal dimension, coordinates on Cn can be chosen so that the conclusion of Lemma 2 holds. Then it is sufficient to
pick k = [qnlg1] + 1 where [r] for any real number r is the greatest integer less than or equal to r. Given f E I(V, Co), need to show dof = 0. Now f (z) - Pk(z) = o(Iz1k) on V. Write
Pk = Lk + Hk
Hk =
0 such that T(V, Co) = T(V, Oo). Proof. It is always the case that T(V, 0) D Complex Span {U71 T(V1, 0)}, but in general T(V, 0) might be larger. Similarly for all k and i, T(Vi, Ck) C T(V, Ck) so Real Span {Un 1 T(Vi, Ck)} C T(V, Ck) since T(V, Ck) is a real vector space. Now pick ki > 0 so that T(Vi, T(Vi, 0) and k = maxi {ki}. Then for each i, T(Vi, Ck) is a complex vector space so Real Span { U T(ViCk)} = Complex Span { U T(Vi, Ck)} and T(V, Ck) = T(V, 0). Remark. It is clear from the proof that k can be picked to be less than the maximal sheeting multiplicity of V. Proposition 3. Let V be any curve, then there exists k > 0 such that
HOLOMORPHIC TANGENT SPACES
381
T(V, Co) = T(V, (9a).
Proof. Let,6, be the germs at the point p of weakly holomorphic functions. An element u r= (9 is said to be a universal denominator if u©, C V . Let I be
the ideal of Op of all functions vanishing on Sing (V) and J be the ideal of universal denominators at p. Then locus (J) C Sing V, [10, p. 56], so by the Hilbert Nullstellensatz there is a positive integer N, called the conductor num-
ber, such that IiP C J. We shall show that k < N + 1. U V,, be the decomposition into irreducible components. Let V = V, U If Vi has normalization ¢i(t), the coordinate with minimal exponent is C3(Vi) = vi ; let w = r, aivi be a real linear combination of the vi with each ai # 0. Now take a new basis of Cn with w as the first element, w = z,; then o(j w I) = o(z I) on each component of V, hence on all of V. Also w r= Real Span (U C,(Vi)) C Real Span (U C5(Vi)) C Real Span (U T(Vi, C`)) (-- T(V, C11). If f E I(V, Ck), then of /aw = 0 since w E T(V, Ck). Now by Lemma 1, we have Pk(z) = Lk(z) + Hk(z) = o(jWIk) and Lk(z) has no w term. Hence Pk(z)/wk is a weakly holomorphic function. Furthermore since V imbedded in minimal possible dimension, w does not divide Pk(z) in 0. (Suppose Pk(z) = wg(z). Then ,(r(z) = Lk(z) + Hk(z) - wg(z) E I(V, 0) and do* = Lk - (g(o), 0, ... , 0), since Lk has no w term, do* # 0 (unless Lk = 0) and T(V, (9) # Cn, a contradiction.) Finally w`' is a universal denominator so w'N(Pk(z)/wk) is holomor-
phic. Hence k < N or Lk = 0. 3.
Examples
The estimates given for k in § 2 are, in general, the best possible (Example 1), but are not always the precise minimal values for k (Example 2). There exist space curves requiring an arbitrary large k (Example 3). Example 1. Let V be the irreducible space curve given by the image of ¢(t) = (t3, t4, t5). Then T(V, (9) = C3 because there is no first order f vanishing on V since any such f = I + H, I initial part, H higher order part, 0 = f (¢(t)) = I (P, t4, t5) + H(t3, t4, t5), order I = 3, 4, or 5, and order H > 6. Now the estimates given for k are [maximum multiplicity/ minimum multiplicity] +
1 = [5 / 3] + 1 = 2 and the conductor number + 1 = 2: Since the semigroup of Z generated by 3, 4, and 5 contains all integers > 3, the holomorphic functions considered as a subset of the weakly holomorphic functions ¢*(v&) C 0, which are generated by t3, t4, and t5 contain all tk, k > 3, and cC/¢*(v&) is generated by t and t2. Hence z, = t3, z3 = t4, z, = t5 are all universal de-
nominators, zic0/¢*(,6) = 0, so conductor number = 1. By either of the above estimates, T(V, Co) = C3. Now we show T(V, Co) _ Cz (first two coordinates)-to do this we use Bloom's result T(V, Co) = complex linear span of C5(V, 0). This is easily computed [5] to be C2. Example 2. Let q < p < r be three prime integers such that q > 5, 3q < r, r < 2p, and q divides none of 2p, 2r, r - p, and r + p, and r is not in the
382
JOSEPH BECKER
semigroup of Z generated by q and p ; for instance q = 7, p = 13, r = 23.
Let V be the image in C3 of ¢(t) = (tq, tP, tr); then T(V, 0) = C3, [max multi/min multi] + 1 > 4, conductor number > 4, T(V, Co) = complex linear span of C5(V, 0) = C2, but T(V, Co) = C3. only this last assertion will be verified here. Let f E I(V, Co) and show dot = (af/az af/az:, at/az2, of/az2i of/az3, of/az3) _ (0, 0, 0, 0, 0, 0). Now approximating by Taylor series : (Z - W)a
f(Z) -
Daf(W) = o(z - w 2)
.Z, WEE C3 .
,
Composing with the normalization, w = ¢(t), z = 0(s), writing fa(t) =
(t)),
and realizing the second derivative part of the Taylor series is bounded in comparison to Iz - wJ2: (sq
-
(Sq
-
(sP - tp)fz2(t)
+ (SP - tP)fz2(t) + (Sr - tr)fz9(t) + (Sr - tr)f3g(t) tr]2) O([ISq - tqI + I SP tPI + Sr
-
Now let w be a primitive qth root of unity, w = e2ii1q, and restrict the above equation to the lines s = wkt, k = 1, , q - 1 to yield (wkP -
1)tpfz2(t) + (a- 1)tpfi2(t)
+ (wkr - 1)trfzy(t) + (wkr - 1)trf23(t)
=
O(t2P(wkp - 1 12 + JwkP - 1 I
wkr
I wkr - 1 +
- 1 2)) = 0(t2P)
Now multiply this equation by rr, and let g,(t) = (tP/tr)fz2(t), g2(t) = (tP/tr)fi,(t), g3(t) = fz,(t), and g4(t) = (t/t)rfi,(t). It suffices to show each ai = limt-o gi(t) is zero. Now the gi satisfy the eq/uations : 0 = Jim (wkP - 1)g1(t) + (wkP - 1)g2(t) + t
0
/ 1)g3(t) + (wkr - 1)g4(t)
so it suffices to show the following matrix is nonsingular :
rwP-1 (0
1P
(0
1P
-1
w4P-1
wp-1
wr-1
Cw2p - 1
(t)3P - 1
w2r - 1 war - 1
w4P-1
W4r-1
wr-11 w2r - 1
t
( 03 r
W4r- 1
To compute the determinant, first factor out wP - 1 from the first column, woP - 1 from the second column, etc., and then perform row operations to bring it to the Vander Monde form :
383
HOLOMORPHIC TANGENT SPACES 11
1
(Op
6) p @2p
(0
3p
1
1
(0 r
@r
(0 (0
2p
2r
&)2r
3r
&j3r
1
Hence
determinant = (cep - 1)(up - 1)(wr - 1)(Cor - 1)(wp - _P)(cyr - (Lr) ((Lr - ,p)(&,r - ,p)(,r - 6,p)(@r - w-p) which is nonzero since oi _ w-1 and oil = 1 if and only if q divides 1. Example 3. Given any integer k > 0, there exists a curve in C3 such that T(V, (9) = C3 and T(V, Ck) = C2. Pick integers q G p G r as follows : q >
4k + 2, p = q + 1, r = (2k + 1)p - q(k + 1), and let V be the image of the map B(t) = (t4, tp, tr). By the Whitney extension theorem, one can show the existence of a Ck function / vanishing on V with a /8z3(0) # 0; we can also find another function in I(V, Ck) whose partial with respect to z3 is nonzero. We need to choose continuous functions '4 on V, *o = 0 on V so that ,Ga(x)
ISI 1 and either a5 > 1 or a0 > 0 , 0, f(al,a2,a3,a4)(t), as = 1 and a0 = 0 ,
where the f.'s are yet to be determined (except for f(0,o,0,0)8(t) = tr). In this no-
tation, the limiting condition becomes : letting m = k - 1, x = B(s), y = B(t),
fe(s) -
E
(B(s) - B(t))1fa+,0(t)/p! = o(B(s) -
B(t)lm-lal)
IEISm.- Ial
for all s, t e C, I a < m. Now the data will be chosen and different reasons given for the limit above to go to zero near the origin and away from the origin.
There are two notations of Ck on Reg V one given by Whitney's theorem and the other given by the differential structure of X as a complex manifold-
384
JOSEPH BECKER
we want to know that these are the same. (Generalizing Lemma 4.2 of [5].) Suppose f E Ck(Reg (V)) and we are given data fa which satisfy the chain rules : a at a
fa(B) =
fa(B) =
ael
ae
at
ae, f.ae al
at
1
at
+to,i,o,o>
I
at`
f.+(0,0,0,1)
where 0(t) = (0k, 02, B 04) = (tq, tq, tpu(t), tpu(t)). Then f I Reg (V) E Ck and satisfies
ai+j f(0) _ aitaja
Z
Cafa
II (Bca))fZ
a{{D(fo) I - 19 I + 1, and C, is an integer constant. Thus
lim) I
Z
Z
(t - s) DP(f(B))]/I t
- sIk-Iai} = 0,
and this limit can be seen to be the same as that in Whitney theorem by substituting (**). See [11, Chapter 1, § 6]. This can best be understood by considering the analogous computation with functions of one variable and k = 2. Suppose g = f (O) so g' = f'O' and g" = f"0'0' + f'O". We are given that lim [I g(t) - g(s) - (t - s)g'(s) - 2 (t - s)'-g"(s) I I t - s 1-2] = 0
,
and want to show
lim II AX) - f(y) - (x - Y)f'(Y) - 2(x - Y)2f"(Y) I Ix - Y I-9 = 0 . Since lim I (B(t) - B(s)) / (t - s) I exists and is nonzero, we can replace the later limit by
lim [I g(t) - g(s) - (B(t) - 0(s))f'(O(s)) - 2 (B(t) - 0(s))2f"(0(s))I It - s1-2] to see that this converges to zero, subtract it from out given limit, substitute for g' and g", and regroup terms to get lim U AS) IB(t) - 0(s) - 0(t - s)0'(s) - 2 (t - S)1011(s) I I t - s I - z]
+ lim
1A 2
(B(t) L
It
6(s))z
-
O'(s)B'(s)I
= 0.
Lest the choice of data appear altogether magical, we first show that for the
case q = 3, p > 7, the data are unique and lead naturally to the general choices. Now for f w2/z = t211 -3, m = [p/3] G 2, so f is supposed to be at least twice differentiable and
385
HOLOMORPHIC TANGENT SPACES
If(s) - f(t) - (sg - tq)aj(t) - (sg - tg)azf(t) - (SP - tp)awf(t) - (SP - tp)af l
Isq-t'2+ISp-tp12
is bounded as t, s 0. Letting s = wt or (o2t, where w is a primitive cube root of unity, we have t4 = sq and ((0r - 1)tr-2p - ((OP - l)t-Pawf(t) - ((OP - 1)(t/t)Pawf(t) and
(()2r - 1)tr-2P
-
(w2p
- 1)t-Pawf(t) - (0)2p - 1)(t/t2)Pawf(t) t2p-7 yields the matrix equation
are bounded as t --> 0. Multiplying by (Ur - 1
[ 27-1]
af(t)
P-1
(UP - 1
(02P
tI0
1
-
ll
111awf(t)
tp-r tP/t7
Since p and q are relatively prime, the above 2 X 2 matrix is nonsingular and amf(t)tP/tr = ((Ur - (0p)(1 - Dr)/(((Up - w)(1 - 6) P)) we can solve for limt.o = 1 and limto - @P)/ ((O)p - diP)(l - u)P)) = 0. Thus = (1 choosing awf = 0 and awf = t7 / i1, the chain rules awf(t)tP-o
rtr-1 =
()r)((,r
qtq-la2f + ptP-lawf
,
0 = qtq-lazf + pip-lawf
imply that a2f = (r/q)tr-q and azf = (-p/q)tr/iq More generally, we extend the above data by recalling that f. is supposed (a°1/az)(aQ2/az)(aa3/aw)(a°4/aw)F and defining higher deto represent D°F = rivatives inductively : any f. with a3 0 or a4 > 2 is identically zero ; terms f f cal,a2+1,o,o> with a4 = 1 satisfy the formula fa(t) = f(1,a2,(),0)(t)/tl; are determined by the chain rules. Hence we have
fa=
if a3>0or a4> 1,
(0, Catr-q°1/tga2+Pa4
otherwise ,
where C. = fl (r/q - i + 1) Ha21 (-p/q - 1 + 1). By the inequalities
r - mq > a - (m - 1)q - p > 0, note that fa is bounded on V if and only if al m. Let
gals, t) = f-(s) -
E
(B(s)
B(t))Y
1911n-IaI
We must show that ga(s, t) = 0(Isq - tqim-l°I + IsP - tP1m-H°l) uniformly in s .and t. Choose a real constant c > 0 so small that the set 12: 1212 - 11 < c} consists of q connected components about the qth roots of unity. We will treat
386
JOSEPH BECKER
the cases I sg - tq I < cI t l g and I sq - tq I > cI t I g separately. Case A. I Sq - tg l > cI t l q. We have
tql + Itqj < (1 + 1/c)Isq - tql
ISIq < ISq
M>0
ISP - tPI r} U {p E V I dim QV, p) > r + 1} where C4(V, p) and QV, p) are the fourth and fifth Whitney tangents cones to V at p, [24], [25]. Then C is an analytic subset of V of codimension at least two [22, Prop. 3.6] and every p E V - C has an open neighborhood so that after a local biholomorphic change of coordinates the following hold (and V is said to be equisingular at p) : (i) For each irreducible component Vi of V, v, fl Sing V = Sing Vi = Cr-1, [22, Props. 2.10, 2.12, and 4.5].
(ii)
Each component has a one-to-one nonsingular normalization [22, Prop.
4.2] 0: D -+ Vi given by ¢(t1, ... , tr) = (ti, ... , 4_1i tq, yr+1(t), ... ) ¢n(t)), where q is the sheeting order of it I Vi and lr(xl, , x,,,) = (x1, , xr). The branching set of this projection is just ¢({tr = 0}) = Cr-1 Now let Cond7(V) denote the conductor number of the variety at the point p. If Vi is a component of V, it is clear that any universal denominator for V is a universal denominator for Vi and since Sing Vi = Sing V, we have that Cond,(V) > Cond,(Vi). For any fixed s = (t1, , t,._1) consider the curve W3 in Vi given by tr --> ¢(s, tr). Since this curve WS lies in ES = s x C'z_r+', weakly holomorphic functions on WS extend to weakly holomorphic functions on Vi by ignoring the first r - 1 variables. Hence any universal denominator for Vi is a universal denominator for W, and Condp(Vi) > Cond7(WS). Note that for s in a neighborhood of p, Cond3(V) < Condp(V). (The ideal
sheaf of J is coherent [7, Theorem 22] because it is the kernel of (9
,
Home(©, 0/0), hence I (Sg(V)) /J is coherent ; the index of nilpotence of a coherent sheaf is an upper semi-continuous function.) We will show that for k > Cond,(V), T(V, C,) = T(V, (9P). Then defining k(p) to be the minimal such k, we have a function V -+ Z. Then k is bounded on compact sets. Now we need to prove Lemma 2. There is an analytic set A' C Cr-1 such that U,,,, T(V, (9,) is a complex vector bundle over Cr-1 - A' such that for s A', ES fl T(V, (9S) = T(WS, OS) and T(V, 0S) = T(WS, 0S) O+
Cr-1.
389
HOLOMORPHIC TANGENT SPACES
Then letting N = Cond,(V) + 1, for all s A', s near p, Cond,(V) < N so = T(WS, CS) = T(WS, OS) by Proposition 3. Now Cr-1, WS C V, so Cr-1
T(Cr-1, CS) C T(V, CS) via [18, Satz 1.2.11 and T(WS, CS) C T(V, QS). Hence T (WS, CS) O+ T (Cr-1, CS) = T (WS, OS) G+ Cr-1 = T (V, OS), so T (V, CS) T(V, CS') = T(V, 0S) But Lemma 2 just follows from a sequence of results of an earlier work [3, § 21. Let U be a polydisk in Ctm centered at 0 and (zi) coordinates in U. Given
, f. E I'(U, (91) we denote by R the sheaf of relations among (f,). For X Cq. For any integer q, 0 < q < n, we may write U = U,,_q X Uq C fl,
Cn_q
a E Un_q we set Uq = {b e U : b - a e Uq}. We denote by R I Uq the restriction of R to UQ and by R(Uq) sheaf of relations among (f; I Uo). Lemma a. For each integer q there is a negligible set Aq C Un_q such that each point p E (Un_q - Aq) X (0) has a neighborhood N, on which there are a,, , ak E I'(N,, R) with the property that (ai I N, (1 UQ) generate R(UQ) I N, (1 UQ. Hence R(UQ) and R I Uq agree off of Aq. Lemma,8. Let U be as in Lemma a and let X be a pure r-dimensional analytic subset of U. Assume for some fired q that given any a E Un_q, a is contained in every irreducible component of x (1 UQ. Denote by Ig the sheaf of germs of holomorphic functions vanishing on X. Then there is a negligible set A C Un_q such that given p E Un_q - A there are a neighborhood N, of p , h,,, E I'(NP, Ix) with the following properties : and hl, (a) (hi) generates Ix IN,,, (b) for any a E U, _ q - A, (hi JUQ (1 NP) generates I x, Un I UQ (1 N,. In these lemmas, "negligible" means the countable union of local analytic varieties. However it can be seen from the proofs that the set being removed is analytic in the event that the slices of the variety are one-dimensional. These proofs can be found at the end of the section. Now T(V, (S) = {a : a d,f = 0 for all f E I(V, OS)} but it is unnecessary to use , h,n, for I(V, (S) over 0S will all f E I(V, OS), any finite set of generators hl, , dh,,) and ? = the sheaf suffice f = (f 1, , fin.), fi E YO on V among (dh1, gm), fi e 0, gi E (9- on C rz among of relations (f, g) _ (f 1, ... , fm, g1, .. , ham,). Define 2r :", R by ir(f, g) = f. Clearly it is onto , (dhl, , dh,n, h1, and R I {s} = T(V, (9S). By Lemma a, there exists analytic Al C Cr_i so that for a A1i 9(UQ) = y J UQ and hence R(UQ) = R I U. Then for a A1, R(UQ) I {s} _ (R I UQ) I {s} and by definition (R I UQ) I {s} = ES (1 T(V, OS). For a Al U A, by definition R(UQ) I {s}= T(WS, OS). Thus letting m = maxpECr_1 {rank, (dh1,
dhJ} and A, = {p E Cr-1: rank, (dhl, , dh < m}, A' = A, U Al U A is the required set. Remark. It is actually unnecessary to remove the set A2, and the reason for this is extremely revealing for what is going on in the above discussion. Really Lemma 2 states that there exists a curve C in V, C = W, U coordinate axis in Cr-1, such that T(C, O) = T(V, (9); hence T(V, Ck) D T(C, Ck) = T(C, (9) = T(V, (9). At point s e A, however, to get such a curve it is not suf-
390
JOSEPH BECKER
ficient to just take intersections of V with linear subspaces. This is illustrated by the following example. Let V be the image in C4 of ¢(s, t) = (s, t3, t4, sty) ; none of 0i is a power series in {¢;};,i, so T(V, 0) = C'. Now if we restrict as above to the slice x1 = s = 0, we get (0, t3, t', 0) whose tangent space is C,2y,, not Cy'25,,, as needed. Taking a nonsingular slice back in the normalization, s = ctk, c constant, k > 0, yields (ctk, t3, t4, tk+5) ; since k + 5 is in the semigroup generated by 3 and 4, the tangent space is Cx2y, if k > 5, Cs1 if k = 1, C21x2 if k = 2, and a twodimensional subspace of C,1,2,3 if k = 4 or 3-in any case nothing in the x, direction. If instead one trys a linear section ax, + bx2 + cx3 + dx4 = 0 in the ambient space, one gets (- (b t3 + ct4) / (a + dt5), t3, t4, - (bte + ct9) / (a + dt5)) ; the tangent space is a two-dimensional subspace of Cxlx2x3. Hence we resort to nonsingular sections in the normalization sp = tq, q < p, p and q relatively prime integers, which itself has normalization 2 ___> (2q, ,ip). Composing gives 2 -> (2q, 23p, 24p, 25p+q) so it is possible for these to be all independent since 5p + q < 6p (the semigroup generated by 3p and 4p does not contain integers between 5p and 6p). In fact q = 7 and p = 11 works. Attempted proof of Lemma a (which does not quite work). Use induction on q-the relation of f , fm will be reduced to several relations of the type R(g , gm) in . for q each a E Un_q.
We may assume that at least one f i, say f,,, which is not independent of , zn (or else we get trivially an isomorphism of R Un_q and R(Un_q) and we are immediately reduced to the case q = 0). Choose coordinates in Uq so that f,, is not identically zero in the zn direction and writing z = (x, y) E Un-q X Uq, let A, = {x E Un_q : (akfm/aZk)(x, 0) = 0 for all k > 0} and A, = {x E Un_q : for each fi not independent of Zn, (akf i/aZk)(x, 0) = 0 for all k >
zn_q+i,
01; each is a proper analytic subset of Un_q and A' = (A, U A2) X Uq. If p A', each f i is regular in the zn direction so by the Weierstrass preparation theorem, there is a neighborhood NP of p, unit u and holomorphic polynomial gi E so that fi = uigi in Np. Now the lemma is a local result and permits multiplication by units so we can replace the f is by the gi's. A relation (a , am) E R is said to be a polynomial relation if each ai E n-10[Znl ; then R is generated over 0 by the polynomial relations. Let a E R
and for each i = 1, , m - 1 write ai = uigm + ri by the division theorem where ui E n0 and ri E n_,C[zn] has degree < deg gm. Let rm be defined by the equuations :
Tail a2
I:
0
gm
0
0
=u1
o
I
am J
I
II
igm +
+u21 I
-g1 J
I
0
I
L-g2j
0 gm
l -gm-1
391
HOLOMORPHIC TANGENT SPACES
, It remians only to show rm is a holomorphic polynomial. Clearly (r1, E2r
0,
i=1, ,r.
All of the entries in this equation are polynomials of bounded degree in zn. Thus we may view these last equations as a larger set of equations involving functions of n - 1 variables. These may be thought of as defining a system of relations equivalent to the restriction to NP of those we began with. Because of the lack of dependence on zn we may view these last equations as defining relations on NP n (C11-1 x (p 3). All of the above commutes with restriction to UQ, so we may assume inductively that our lemma holds on N. If we cover Un_q - A" with a locally finite set (NP=), then it is easy to see from [13] that A", together with the union of the negligible sets in each NP1, forms a negligible subset of Un,_q, and in its complement the conditions of the lemma are satisfied. This completes the proof. Remark 2. If q = 1 or 0, the set Aq is analytic-because we avoid having to use the divisor theorem in the complement of where we already used it. Our proof of Lemma 1 is modeled closely on Spallek's work in [13], [14], [15] in which he proved the following converse to our result : If F is a finitely generated subsheaf of 0, there is an analytic set A q of dimension at most n - q- 1 such that if g e 0(U) and for every a e Un,_q, g I UQ e F I UQ, then g
e
F(U - Aq). Because our applications do not permit the type of coordinate changes employed in [13], our result in Lemma a is weaker than the corresponding result in [131.
393
HOLOMORPHIC TANGENT SPACES
Proof of Lemma a. In order to apply Lemma a we need to express IX as a sheaf of relations in a manner which commutes with restriction. To do this we recall Cartan's proof of the coherence of I,[9]. Since Un_Q C X, we may change our last n - q coordinates so that projection on the first r coordinates induces a u-sheeted branched covering with branch locus B. (At this stage we may have to shrink U. We will only use the local form of this lemma.) Near each point of D - H(B) the map H has u local inverses of the form w;(x) = , xr, w,i,r+1(x), . , wj,n(x)). Using these we form ;=1 (zi - wj,i(z)), and this extends to a polynomial Pi(z) e C(D)[zi]. By a linear change of the last n - r coordinates we may insure that the discriminant a of Pr+1 is not , n we define polyidentically 0. Let C = {z e U : 5(z) = 0}. For i = r + 2, nomials Qi(z) as follows : if z e U - C, then near z (x1,
'/
u-1
Qi(z) = Z ak(x)zT+1 , k=0
ak(x) = d det [1, w j,,+1(x), ... , wi,r+1(X)1-1, a(x)wj,i(x), wj,r+1(x)k+1
.
. .
,
w,,1+1(x)u-1]
Here J1 = S and x = (z1, , Zr). Qi extends to an element of &(D)[zr+1] Un_Q be the canonical projection associated with our choice of Let p : U r - n + q is a coordinates. A' = {a e Un_Q : dimes C fl U4 = dim x fl UQ proper analytic subset of Un_Q since c fl UQ = (p I C)-1(a), {b e X : dimb f -1(f (b)) > l} is an analytic set for any holomorphic map f : X Y and any integer 1,
and if dim c fl U4 = dim x fl Uq then c fl UQ contains a component of X
fl Uq so a e c fl Uq by hypothesis and dimes C fl UQ =r-n+ q. Notice
D h U4 . Since B C C, if that for a e Un_Q restriction gives H : X fl Uq a e Un_q - A' then all of the above constructions commute with restriction to X fl UQ. From [9] we know that g e I,,a if and only if for all sufficiently large N, SNg is in the ideal generated by the germs of P,+1, , Pn, azr+z - Qr+z, , Sz, - Qn at a, and similarly for I Q a. Thus I,,,,, is the identified set of defining a relation begerms which appear as the first element in a e ()rn--1 X,a tween the germs of oN, Pr+1, azi - Qi at a. Now we can apply Lemma a to complete the proof. Now returning the proof of Lemma 2 (to show analyticity of the removed set), we must study the bad set (ala Lemma a) of the relations among P,+1, ,Pn, zr+zO - Qr+2, ,zno - Qn, which arises in reducing the relations from C11 to Cn-4. Since we are assuming each slice of the variety is one-di-
mensional, n - q = r - 1, q = n - r + 1. For the first n - r steps of the induction in Lemma a it is possible to use the method of the first proof and hence get no bad set (each Pi, ziO - Qi is a holomorphic polynomial in ,O[z,+1, . , zn], and after each reduction to less variables each s, has all entries holomorphic polynomials and by Remark 1, at least one entry a Weierstrass poly-
nomial-either the leading coefficient of P equal to 1, or all the Pi, i < j).
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JOSEPH BECKER
Then for the last two steps of the induction one can use the method of the second proof of Lemma a and by Remark 2, remove an analytic set. 5.
Homogeneous case
Now consider the case of a homogeneous algebraic variety, which is a set V = common locus in Cn of finitely many homogeneous polynomials. Here it
is easy to find an analytic curve (reducible) C in V such that T(V, (oo) = T(C, (oo). For analytic set V, let L(V) denote the complex linear span. First construct a curve C in V such that L(C) = L(V) as follows : pick finitely many points v,, - ,v , e V and let Ck = L(v,) U U L(vk) ; clearly Ck c V so L(Ck) c L(V). If L(Ck) # L(V), then V Z L(Ck) ; pick vk+, e V - L(Ck) and let Ck+, = L(vk+,) U Ck. Then dim L(Ck+1) > dim L(Ck) so eventually for
some m, L(C,) = L(V).
Now applying Lemma 3 below to both C and V, we have T(V, lr) = L(V)
= L(C) - T(C, (9).
Lemma 3. If V is homogeneous, then L(V) = T(V, (9). Proof. Any f e I(V, 0) is the sum of homogeneous polynomials which all vanish on V, so V is the common locus of the initial terms which are linear ; hence V C T(V, (9). Since T(V, (9) is linear, L(V) C T(V, 0). On the other hand, dim T(V, (9) is the minimal embedding dimension of V, so dim T(V, (9)
< dim P(V).
Remark. It is not at all surprising that the result is so easy for homogeneous varieties since the critical degree of differentiability is just k = 1 : By the methods [3, Lemma 3] of Lemma 3 one easily sees that L(C5(V)) = T(C5(V), (9)
D T(C3(V), (9) = T(V, (9), but C5(V) c T(V, C') so T(V, (9) c L(C5(V) c L(T, C')) = T(V, C') because T(V, C') is a complex vector space. Hence
T(V, (9) = T(V, C'). Alternately, one can see that the critical degree of differentiability is just one as follows : Suppose T(V, Co) # T(V, i9) = ambient space, then some differentiable function vanishing on V has a nonzero partial derivative at the origin, so considering the Taylor expansion of f restricted to V we have z i / I z l 0 on V as I z I , 0, for some zi 0 on V. But V is homogeneous and IAziI/IAzI = I zi l / I z 1 , so the values of I zi l / l z l do not change as I z J - 0. 6.
General case
Theorem 1. For any point p e V, a complex analytic variety, there exists an integer k > 0 such that T(V, Cp) = T(V, (&,). If k(p) is defined to be the smallest such integer, then the function k : V - Z is bounded on compact subsets of V and bounded for algebraic varieties. The first statement follows from Theorem 2, as pointed out in the remark at the end of the last section. The second statement follows from the proof of Theorem 2.
HOLOMORPHIC TANGENT SPACES
395
Theorem 2. For every p e V, there is a complex analytic curve C in V passing through p such that T(C, (9P) = T(V, (9P). Proof. This was inspired by [4, § 4] where it is shown that every differential operator on a variety is the finite sum of differential operators on curves in the variety. Unfortunately the proof given there does not seem to guarantee that first order operators are the sum of first order operators on curves. Proposition 4. Let V be an analytic variety with dim V > 1, p E V. Then there is an analytic variety W C V with dim W < dim V such that T(W, (9P) = T(V, Dr). It is clear that Theorem 2 follows from Proposition 4 by induction. Before starting on the proof we review some well known facts about completion of modules [26].
Let A be a local noetherian ring with maximal ideal m, and E a finitely generated A module. Then E is given the structure of a topological group with the fundamental system of neighborhoods mkE, called the natural topology. If F is a closed submodule of E, the natural topology of E induces on F the
natural topology of F, and the quotient E/F also has the natural topology. The completion (via Cauchy sequences) of E in this topology is E = lim E/mkE and also has the natural topology given by the fundamental system of neigh-
borhoods mkE. If lk mkE = {0}, the canonical map E > E is injective, E is
considered as a dense subset of t, and t is complete, that is, t = E. If 0 F - E > G -* 0 is an exact sequence of finitely generated A modules, then 0
0 is an exact sequence of finitely generated A modules,
consequently E/F = E/F, F fl E = F, and F is closed in E. Next E = AE, so if a, b are any two ideals of A, ab = AaAb = Aab = ab. If a is any ideal
of A, then (A fl a)^ = A A. (A fl a) C AA fl Id = la = a, in summary (A fl a)A C a. If a, b are ideals of A and a = b, then a=d fl A =b fl A = b. If {Fi} is a finite family of submodules of E, then (fl Fi)A = fl F. For an infinite family, we have (fl Fi)A C Pi since the latter is a closed set. For any submodule F of E, n ;k., (F + mkE) = F. , If A is the ring of convergent power series over the complexes, C{X1i the ring of formal power series over C, and then A = C[[X,, every ideal of either ring is closed. By an analytic ring we mean C{X1, where I is an ideal. If an analytic ring A is an integral domain, so is its
,
,
completion A, [10, Theorem 1], hence the completion of a prime ideal is again
prime. Conversely if A is an integral domain, then A is an integral domain since it is a subring of A ; if P is prime in A, then P fl A is prime in A. If A is a local noetherian ring, dim (A) is the largest integer k such that there exists a strictly increasing chain of prime ideals po C p1 C - C pk = m , Xn} and C[[X1, , Xn]] are both n. The of A. The dimensions of C{X1, C height of a prime p is the length It of the largest chain of primes p1 C p, C p. The depth of a prime p is the length d of the longest chain of primes C pd = m, so that Length, (p) + Depth, (p) = dim A. Depth and p C p1 C
396
JOSEPH BECKER
height of a prime and dimension of a ring are both preserved by completion. Now returning to the proof of Proposition 4, assume V is imbedded in minimal possible dimension, that is, T(V, 0,) = Cn so I(V) C m2, where m is the maximal ideal of 0. We want to show that there exists an analytic set W C V, dim W < dim V so that I(W) C m2. The most naive idea would be to say £9 is a unique factorization domain, so let W be the union of two different subvarieties W1i W2 of codimension one in V, where W1 is the locus of f1 so I(W) is generated by f1. Then any f E I(W1 U W) can be written as f = f1g, g E
I(W) so ord f > 2. However this does not work : Let V = locus of z3 - xy in C3, W1 = locus (x) = y axis = {(0, a, 0)}, W2 = locus (x - z) = W1 U {(a, a2, a)}, and f = x - z. Hence the proposition will have to be proven by contradiction of assumption that all lower dimensional subvarieties have tangent space not equal to Cn. Let dim V = r, V= V' U V", dim V' = r, dim V" < r - 1. Let Vi', , V1' , V'h the irreducible compobe the irreducible components of V", and Vi,
nents of V. Let I = I(V, 0). Then qi = I(V, 0) and pi = I(Vz', 0) are all prime and I = fl (n2=, pi). Pick a countable set W1+1, WZ+2, of irreducible subvarieties of codimension one in V such that U Wi is dense in V. (Take local parameterization 7r: V -+ Cr and a countable dense set ai E CPT-1, such that each ai determines a hyperplane Hi normal to it. Then U Hi is dense in Cr so 7r-1(Hi) is dense in V since Tr is a closed map. Let Wi be the irreducible components of 7r-1(Hi).) Then PZ+i = I(Wl+i, 0) is prime and f tiZ' Pi = I since ano continuous function vanishing on a dense subset of V is identically zero. For all k let I,= P1 fl fl Pk. Clearly we have
I1DII D ...IkD ... D fIk=I, k=1
h
I2
...
Ik
...
(1 Ik k=1
A
fl Ik/ = I . k=1
Now I C m2, so I C m2 = m2, and the proposition clearly follows from the below lemmas which imply f m2. Lemma 4. If no Ik C m2, then fl Ik Z int. Lemma 5. fk=1lk = (fk=I,)A. Proof of Lemma 4. Suppose f,, E Ik, ord fk = 1 for all k. Let Hk be the complex vector space given the image of the natural map Ik ---> m/m2. Then Hi D H? D is a decreasing sequence of finite dimensional vector spaces and hence is stable for large j, say H; H1 for all j. By assumption Hi zf- 0, choose 0 zf- h1 E R. Now define homogeneous polynomials hk of degree k inductively as follows : Suppose h1, , hk_1 are defined, 7o,:_1 = h1 + + hk_11 so that for all j, 3g, E 0, ord g, > k, and cpk_1 + g; E I. Let H; be the complex vector space spanned by the image S, of the natural map I, _ m/mk+1 restricted to those elements in I, whose image in m/mk is Ok_1 Then Hk
397
HOLOMORPHIC TANGENT SPACES
is a decreasing sequence of finite dimensional vector spaces and is stable for large j, say H; D Hi for all j. Choose hk E Sk. Then hk E S; for all j-apriori hk is only in H, but there exist finitely many ci E C, hji, gji E l9, hji homogeneous polynomial of degree k, ord g > k + 1, cDk-1 +hji + gji E I j, 1111 D
so that cok = (tOk-1 + hk = E Ci(rPk-1 + hji + gji) mod mk+1 Comparing terms
of orders k - 1 and k, we have E ci = 1 and E cih ji = hk. I j is a vector space so E ci( Cd+1 by 7, (a, b) = (p1(a), p,(b)). Then (r 1 W0)-'(0, 0) _
pi 1(0) x p2'(0) = (0, 0), so r gives a local parameterization of W. such that
r(Sg V) C Cd where Cd is identified with Cd x 0 in Ca+1 Let B, be the branching set of p1. Let B2 be the branching set of r-each irreducible component of which either contains Sg V or intersects Sg V in a set of dimension less than Sg V-and let B2 be the union of intersection of those irreducible components of B which do not contain Sg V. Let Z be the union of the components of Sg V of non-maximal dimension, and Z' the intersection with Sg V of all irreducible components of Sg W. which do not lie in Sg V. Let A = SgSg V U B, U B2 U Z U Z. Then for each p e Sg V - A, there is a neighborhood U of p in CZ's such that r 1 U fl Wo gives a local parameterization with branching set B a manifold contained in Cn X 0, r(B) C C d . Now r : Wo - B > C' - C d is a covering projection and induces a map on the first homotopy groups r, : r,(W, - B) r1(Cd+1 - Cd) Z. Since Z is a principal ideal domain, image (7r) qZ for some q. Let Dd+1 be a unit polydisc in C,1 + 1Dd = D11+1 n Cd, and +(t,, td, td+,) = (t,, ... , td, td+1). Then i!c*r,(Da+1 - Dd)) -- qZ. By a standard result in algebraic topology, there exists a map 0: D11+1 - Dd > Wo - B such
that rcp = i. (Given map + : Z > X and covering map r : X > X, then there
exists map 0: Z > I so r¢ = V if and only if i*r,(Z) C r*r,(X).) Then ¢ is holomorphic because locally it is r 1
.
Since r is a proper map (invers(
image of compact sets are compact), ¢ is bounded near Dd, so by the Riemam. removable singularities theorem it extends to a holomorphic map on Dd+1,
¢(t) = (t1,
, td, td+,, ¢d+2,
' '
, 02.). Then ¢ is one-to-one because r and
are both q to one off Dd. (Another standard result in algebraic topology is that the number of points in the fiber of a covering map r : X > X is the index of subgroup r*r1(X) in r,(X).) In summary, each irreducible component of W. has a normalization of the above form.
Let N, = Condo (W0), for p near 0, Condp (W0) < N1. Now we want to show that for p e Sg V - A, Condp (Wo) > Cond (Wo fl Ep =v n (Lo + p)), (*)
e.g., I(Sg W0)kop(W0) C C9p(Wo) implies
I(Sg (W0 n Ez,))'`ez,(W0 n Ez,) C (z,(W0 n Ep) Since for fixed s e Cd, ¢(s, td+1) is the normalization of Wo fl Ep, the restriction map 5p(Wo) > O(W0 fl Ep) is onto : let h¢ e Op(W0 fl Ep), h¢ e 9p(DI) C &,(Dd+1) extending the function by ignoring the other d variables so ho E 0p(W0). Also any element of I(Sg (W0 n Er))k is the sum of elements either identically zeo on Wo fl EP or in I(Sg (W0))k, in either case a universal denominator of Wo fl Ep ; the set of universal denominators is an ideal so line * is valid.
400
JOSEPH BECKER
Now we repeat the construction. Let Wo = UPEA P X v fl (Lo + p) and N2 = Condo (J'), take a local parameterization of Wo, and remove an analytic
set A' of strictly lower dimension to make Wo equisingular along A - A'; hence CondP (V fl (Lo + p)) < N2 for all p e A - A'. This finally gives a stratification of Sg V and an integer Ni associated to each strata so that for each point in that strata CondP (V fl (Lo + p)) < Ni. Just take the largest of this finite set of integers. References J. Becker, C', weakly holomorphic functions on analytic sets, Proc. Amer. Soc. 39 (1973) 89-93. [2] J. Becker & J. Polking, C", weakly holofnorphic functions on an analytic curve, Proc. Conf. Complex Analysis 1972, Rice University Studies, Vol. 59, No 2, [1]
1-12. [3]
[4]
J. Becker & J. Stutz, The C' embedding dimension of certain analytic sets, Duke Math. J. 40 (1973) 221-231. T. Bloom, Operateurs differentiels sur un espace analytique complexe, Seminaire Pierre Lelong 1967-1968, Lecture Notes in Math. Vol. 71, Springer, Berlin, 1968.
[ 5 ] -, C' functions on a complex analytic variety, Duke Math. J. 36 (1969) 283-296.
[61 R. Ephraim, C°° and analytic equivalence of singularities, Proc. Conf. Complex Analysis 1972, Rice University Studies Vol. 59, No. 1, 11-32. [71 R. C. Gunning, Lectures on complex analytic varieties, Math. Notes, Princeton University Press, Princeton, 1970. [8]
M. Jaffee, Differential operators on the curve X° = Yb, Dissertation, Brandeis University, 1972.
[9] S. Lojasiewicz, Triangulariation of semi-analytic sets, Ann. Scuola Norm. Sup.
Pisa 18 (1964) 449-474. R. Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Math. Vol. 25, Springer, Berlin, 1966. [11] B. Malgrange, Sur les fonctions differentiables et les ensembles analytique, Bull. Soc. Math. France 91 (1963) 113-127. [10]
[12] -, Ideals of differentiable functions, Tata Institute of Fundamental Research [13] [14]
Studies in Math., Oxford University Press, 1966. R. Remmert, Holomorphe and meromorphe Abbildungen komplexer Raume, Math. Ann. 133 (1957) 328-370.
K. Spallek, Verallgemeinerung eines Satzes von Osgood-Hartogs auf komplexe Raume, Math. Ann. 151 (1963) 200-218.
[15] -, Zum Satz von Osgood and Hartogs fur analytische Moduln. I, Math. Ann.
178 (1968) 83-118. [16] -, Zum Satz von Osgood and Hartogs fur analytische Moduln. II, Math. Ann. 182 (1969) 77-94. [17] -, Differierbare and holomorphe Functionen auf analytischen Mengen, Math. Ann. 161 (1965) 143-162. , Uber Singularitaten analytischen Mengen, Math. Ann. 172 (1967) 249-268. E. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. Y. T. Siu, On approximable and holomorphic functions on a complex space, Duke Math. J. 36 (1969) 451-454. [21] J. Stutz, The representation problem for differential operators on analytic sets, Math. Ann. 189 (1970) 121-133. [18] [19] [20]
[22] -, Analytic sets as branched coverings, Trans. Amer. Math. Soc. 166 (1972) 241-259.
[23]
H. Whitney, Extensions of differentiable functions, Trans. Amer. Math. Soc. 36 (1934) 63-89.
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[24] -, Local properties of analytic varieties, Difftrential and combinatorial topology, Princeton University Press, Princeton, 1965.
[25] -, Tangents to an analytic variety, Ann. of Math. 81 (1965) 496-549. [26] 0. Zariski & P. Samuel, Commutative algebra, Vols. I, II, D. Van Nostrand, Princeton, 1960.
PURDUE UNIVERSITY
J. DIFFERENTIAL GEOMETRY 12 (1977) 403-424
THE LENGTH SPECTRA OF SOME COMPACT MANIFOLDS OF NEGATIVE CURVATURE RAMESH GANGOLLI
1.
Introduction
Let R be a compact Riemannian manifold. In each free homotopy class f of closed paths on R, there exists a geodesic whose length is minimal among the paths in f ; let 1(y) be its length. The distinct members of the set of lengths l(?) as varies over all such classes can be arranged in increasing order 0 < 11 G 12 < . The sequence {li}ti,l, finite or infinite, is by definition the length spectrum of R. It may happen that l(y) = l(?') for two distinct classes. Let, for each i > 1, mi be the number of free homotopy classes f such that l(?) = li. The sequence {(li, mi)}till may be called the length spectrum with multiplicity. Let d be the Laplace-Beltrami operator of R. Then the space LZ(R) (with respect to the Riemannian measure) decomposes as the Hilbert space direct sum of finite dimensional eigenspaces for J. Let {2i}ti,l be the distinct eigenvalues, and ni the multiplicity of 2j. The sequence {(2i, ni)}till is the spectrum of J. We may assume the 2i to be arranged so that 0 > 21 > 22 > . In this paper, we shall study the length spectrum and its relation to the spectrum of d for a very special type of compact manifold of negative sectional curvature. Specifically, we shall consider a compact manifold R whose simply connected Riemannian covering manifold H is a symmetric space of noncompact type and of rank 1. As is well-known, H can then be represented as G/K, where G is a noncompact connected simple Lie group of R-rank one, with finite center, and K is a maximal compact subgroup of G. As a consequence R can be represented as I'\G/K, where F is a discrete subgroup of G, acting freely on G / K, such that F \ G is compact. F can be identified with the fundamental group of R. The metric on R is fixed to be the one obtained from the canonical G-invariant metric on G/K. Cf. [11], [27]. For such a manifold R, let {(li, mi)}ti,1 be the length spectrum with multiplicity, and for any 1 > 0, define QI(1) = Z (ti6ti,5t1 mi. Thus Q1(1) is the number of free homotopy classes 7 such that 1(r) G 1. It can be seen easily that QI(1) is finite for each finite 1. We shall show that the asymptotic behaviour of QI(1) as l --> co can be described precisely in terms of the covering space G/K. In fact, we find that Q,(l) - (21 p 1 1 ) -1 exp 2 l p h l as 1--> co, where p is the half Communicated by I. M. Singer, July 7, 1975. Research supported in part by the National Science Foundation.
404
RAMESH GANGOLLI
sum of the positive roots of the symmetric space G/K, and I is the usual Cartan-Killing norm. This is the main result of the present paper. In particuI
lar the asymptotic behaviour of Q1(1) depends only on the covering manifold, and is independent of the subgroup r, a somewhat unexpected result. In the course of proving this result, we shall also see that the length spectrum {1i}ti,l is determined by the spectrum of the Laplacian J. This has been known for certain kinds of manifolds, [1], [19], and the question has been raised whether it is true in general for an arbitrary compact manifold.' A result similar to our main result has been announced by Margulis [13].
See also Sinai [20]. Margulis works in the context of an arbitrary compact manifold of negative curvature ; his result is that Q1(1) Cl-' exp dl where C, d are positive constants. Bounds for d can be obtained. In our special context, the precise value of d can be obtained in terms of the structure of G/K. Margulis' proof has not appeared as far as the author knows. In any case, his proof is based on ergodic theory and is totally different. Cf. [13]. The free homotopy classes of closed paths on R can be easily seen to be in a natural one-to-one correspondence with the set Cr of conjugacy classes of elements of r. Thus our main result gives us some information about the distribution of these conjugacy classes. Actually we get somewhat more. An element r c r, r # 1, is said to be primitive if it cannot be expressed as a positive power of any other element of r. Let Prr, be the subset of Cr consisting of conjugacy classes of primitive elements of r. The corresponding free homotopy classes will be said to be primitive. Let Qo(1) be the number of
primitive classes r such that 1(r) G 1. Then we shall see that Qo(1) has the same asymptotic behaviour as Q,(1) as l - oo. A particular case of our main results was proved by H. Huber [12], who considered the case of compact Riemann surfaces of genus > 2. Thus G = SL(2, R). HUber's method is slightly different ; it was followed by BerardBergery in [1], where the case G = SO0(d, 1) was considered. Our method is to apply the Selberg trace formula to the fundamental solution of the heat equation on M, and analyse the resulting theta relation closely.
That this is useful for other problems in the context of P\G is indicated by [4], Eaton [3] or Wallach [22]. In [14] McKean considered G = SL(2, R) and by applying the trace formula to the heat kernel, gave an independent proof of HUber's result. Our method in proving the main result is a generalization of McKean's method.
HUber utilizes methods involving the Green's function of the upper half After this work was completed, the author came to know that recently J. J. Duistermaat and V. W. Guillemin [The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975) 39-79] have proved the general result that the length spectrum of any generic compact Riemannian manifold is determined by the spectrum of the Laplacian. The author understands that their method uses the wave equation on M. The method of the present paper uses the heat equation, as will be apparent below.
LENGTH SPECTRA OF MANIFOLDS
405
plane to prove a remarkable formula [12, p. 26], cf. (4.32) below, which is his main tool. We shall indicate below how Hiiber's formula can be generalized to our setting by an application of Selberg's trace formula. By using this, one can get some more geometric information. Specifically, for each x, y r= G, and r > 0, let Q(x, y, r) be the number of elements r r= I' such that the Riemannian
distance between rxK and yK is less than r. Then the asymptotic behaviour of Q(x, y, r) can be determined. Cf. § 4 below. This may be regarded as a `local' version of the main result. 2. Preliminaries Let G be a connected noncompact simple Lie group with finite center, and K a maximal compact subgroup of G. Let g, f be the respective Lie algebras of G and K, and let g = f + p be the Cartan decomposition, with respect to the involution 0 determined by f. Denote by < , > the Cartan Killing form ; for any X e g, we put I X 12 = - <X, OX>. Then I is a norm on g. Let a, be a maximal abelian subspace of p. Throughout this paper, we assume that dim a, = 1. Extend a, to a maximal abelian 0-stable subalgebra a of g, so that I
a = at - a,, where a, = a f f, a, = a (l p. Then a is a Cartan subalgebra of g. Denote by gc, ac etc. the complexifications of g, a, etc, and let ftc, ac) be the set of roots of (gc, ac). Order the dual spaces of a, and a, + ia, compactibly as usual (Cf. [11]), and let 0+ be the set of positive roots under this order.
0 on a,}, and let Let P+ = {a E 0+ ; a * 0 on a,} and P_ P = 2 EaEP+ a. Let X. be a root vector belonging to a e 0, and let nc = E.EP+ CXa. Then, if n = nc f g, we have the Iwasawa decompositions g -
f - a, + n, G = KA,N where A, = exp a,, N = exp n.
n is equal to normalizer of A, in EQEP+ RXa. Let M be the centralizer of A. in K, M' the
K, and W = W(G, A,) the Weyl group M'/M. W operates naturally on A,, a, (a,)*, (a,c)*, etc. Let A be the real dual of a,, and Ac its complexification. For 2 E Ac, we put 2 = Re 2 + i Im 2 with Re 2, Im ), in A. We extend the form to ac, Ac, in the obvious way. W preserves < , >. We let dk be the normalized Haar measure on K. Let da, do be the Haar measures on A N given by the Euclidean structure on A, n furnished by the inner product - <X, OY)', and the exponential map. Then the Haar measure dx on G can be so normalized that for any f E C,(G), we have f f (x)dx = f f JG
JK
A,
f f (kan) exp 2p(log a)dk da do . +
These narmalizations will be fixed from now on. Denote by C, (K\G/K) the subspace of C°(G) consisting of those f E C- (G) such that f (k,xk2) = f (x), x E G, k k, E K. Such functions are said to be spheri-
cal. The spaces L,(K'\G/K), L2(K\G/K) etc. are defined analogously. For
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RAMESH GANGOLLI
any x E G, let H(x) E a, be the unique element of a, such that x E K exp H(x)N.
Then for any ,l E Ac, the function ¢,(x) = f exp (il - p)(H(xk))dk is the elementary spherical function corresponding to ,l. For f E L,(K\G/K) and ,l E Ac, define the spherical Fourier transform (2.1)
f(2) = Jf(x)c2(x)dx,
where dx is the Haar measure on G. Let f E L,(K\G/K), and define (2.2)
F f(a) = exp p(log a) f f (an)dn Jv
.
Then F f E L,(A,). Ff is the so-called Abel transform of f, and it is known that (2.3)
f (,l) = f F1(a) exp il(log a)da = F f (,l)
,
A,
where Ff (2) is the Euclidean Fourier transform of Ff.
For x E G, we have x = k exp X, k E K, X E p. Put a(X) _ JX J. a(x) is spherical, smooth and will play a role below. Let 8(x) be the elementary spherical function 0,(x) = f exp - p(H(xk))dk. The Harish-Chandra-Schwartz x
space '(G) is then defined as in [10]. For each left or right invariant differential operator D on G, and an integer r > 0, define TD,,(f) =
Sup,,,, c(x)-1(1 + a(x))' lDf(X) 1, for f E C°°(G). '(G) then consists of those f E C`°(G) for which TD,r(f) < co for all D, r. '(G) is a Frechet space under these seminorms. Similarly we define seminorms 1 D,,.(f) = Sup,,, -8(x)-2(1 + o(x))' lDf(x)l, and put le,(G) = {f E C`°(G) ; 1 D,,.(f) < co for all D, r}. Then W,(G) C W(G) C L2(G). W,(G) C L,(G). The space W,(G) was introduced and studied by Trombi-Varadarajan [21].
The spaces of spherical functions in le(G), W,(G) will be denoted by W(K\G/K), W,(K\G/K) respectively. Let _' be the set of restrictions to a, of elements of P+. Then one knows, since rank (G/K) = 1, that we can select (3 E _' such that 2(3 is the only other possible element of _'. Let p be the number of roots in P, whose restriction to a, equals (3, and let q be the number of remaining elements. Let H, be the element of a, such that (3(H0) = 1, and Hj3 the element such that = (3(H), H E a,. Then it is known that = 2p + 8q, p(H0) = 2(p + 2q)
and H, = (2p + 8q)-1Ho. It follows that = 4(p + 2q)2(2p + 8q)-1, which will be used below.
407
LENGTH SPECTRA OF MANIFOLDS
3.
The trace formula
Let T be a discrete subgroup of G such that T \G is compact. Fix a GifG nvariant measure dx on T \G by requiring that for each f E C,(G), we have f (x)dx =
J r\G
(Z',, f(rx)dx. Let T be an irreducible unitary representation
of T on a finite dimensional vector space V, and denote by U the representation of G induced by T. Thus U acts on the Hilber t space H consisting of functions f : G --> V which satisfy (i) f (rx) = T(r)f (x) and (ii)
J r\G
(f (x), f (x))dz
< co where (., ) is the inner product on V. The action of G on H is by right translation. Thus (U(x)f)(y) = f(yx), x, y E G, f e H. U is a unitary representation of G. Under our assumption of compactness for T\G, it is well known that U is a discrete direct sum of irreducible unitary representations of G, each occurring with finite multiplicity. Denoting by 61(G) the set of equivalence
classes of irreducible unitary representations of G, we let nr(w, T) be the number of summands of U which lie in the class w. Then we can write U )1-GB(G) nr(w, T)w,, and nr(w, T) < co for each w.
For f E L1(G) let U(f) = J f(x)U(x)dx. U(f) is a bounded operator on H. G
As in [18], [7], we say that f is admissible if (i) the series E, f(y-lrx)T(r) converges absolutely, uniformly on compacts of G X G, to a continuous End (V)-valued function F(x, y, T) and (ii) the operator U(f) is of trace class. When f is admissible, we have the trace formula (3.1)
E nr(w, T) Trace U,(f) =
.G.(G)
fr\G
Trace F(x, x, T)dt ,
where U. is a representation of class w E &(G). Of course, U.(f) has a trace because U(f) does. As in [18], one rewrites the right side of (3.1) to get the Selberg trace formula nr(w, T) Trace U,(f) =
(3.2) mEB(G)
E Trace T(r) Vol (T,\G,)I,(f) 7EGr
where Cr is a complete set of representatives in T of the conjugacy classes of
elements of F, and G, is the centralizer of r in G, F,= T fl G,. Since T \G is compact, every element of T is semisimple, and G, is reductive, and T,\G, is compact. We fix a Haar measure dx, on G, in a manner analogous to the manner in which the Haar measure on G was fixed, following the Iwasawa decomposition of G,, and put dx, for the invariant measure on T,\G,. The volume Vol (T,\G,) is computed with respect to this measure. Finally, I,(f)!=
f(x-'rx)dx*, where dx* is the G-invariant measure on G,\G normalized fG,\G
so that dx = dx,dx7 .
408
RAMESH GANGOLLI
The use of (3.2) depends on having a stock of admissible functions. The following proposition was proved in [7]. Proposition 3.1. Let f E '1(K\G/K). then f is admissible. A similar assertion holds if f E W1(G) and is left and right K-finite. We shall only need this special case.
Let f E '1(K\G/K). Then U,(f) = 0 unless is of class one with respect to K, i.e., unless the restriction of U., to K contains the trivial representation of K. When U., is of class one, there is associated with it a unique positive definite elementary spherical function ¢,. say, A., E A. Then Trace U,,(f) = where ? is as in (2.1). (Cf. [6]). Thus, when f E '1(K\G/K), we get (3.3)
E nr((o, T) f(j.,) _ E Trace T(r). Vol (rr\Gr)Ir(f)
.Er(G,1)
7E Up
where (9(G, 1) stands for these elements in (9(G) which are of class one. We shall now compute the integrals I7(f) for f E W1(K\G/K), in a form suitable for use in § 4. An element x e G is said to be elliptic, if it is conjugate to some element of K and is then automatically semisimple. x E G is said to be hyperbolic, if it is semisimple but not elliptic. In all other cases x is said to be parabolic. When
G/I' is compact, r does not contain parabolic elements. It is well-known that r e r is elliptic if and only if it is of finite order. Both these properties are equivalent to the property that r has a fixed point on G/K. We assume throughout that r contains no nontrivial elliptic elements. Thus each r E r, r # 1, is hyperbolic. The integrals I7(f) can be computed for hyperbolic r quite simply, and can be expressed in terms of the Abel transform Ff of (2.2) when f is spherical. Let J be a Cartan subgroup of G with Lie algebra j, 0+ a set of positive roots for 0 _ ftc, jc). For any a e 0+ let , be the corresponding character of J. Put p, = 2 a, and PT = exp p(log h). We may assume that Pj is a well-defined character of J. Put 4,(h) = P(h) 11 aEm+ (1 - ba(h)-'), h e J, and let 0f be the invariant integral of f relative to J (cf. [9]). Thus (3.4)
0f (h) =
(h) f
f(x lrx)dx* .
J\G
Here a',(h) = sign IIaEIR (1 - ba(h)-'), the product being over the set 0 of real roots in 0+, i.e., those which are real on j, the Lie algebra of J. The Haar measure dh on J is normalized as mentioned in § 2 above, and dx* is the G-invariant measure on J\G such that dx = dh dx*. 0 is defined and smooth on T= J n G' = the regular points in J. For r E r, let Gr be its centralizer with Lie algebra gr, and let jr be a 0stable Cartan subalgebra of gr which is fundamental. Then one knowns t hat I7(f) and 0'r are related to each other, thanks to a theorem of Harish-Chandra
409
LENGTH SPECTRA OF MANIFOLDS
[10, p. 33]. If we let 0; be the set of positive roots of (gG, jr), and put H, _ rf a E ; H,, then we know that (3.5)
J,(f) = C,r fr(r ; H) ;
C, # 0
,
where 0 fr(r ; H) is the result of applying the differential operator H, to the function 0fr, and evaluating the result at T. All this is well-known and can be found, e.g., in [23]. The value of C, will be useful for us. It can be computed by using [10, Lemma 23], and [23, II, Chap. 8]. One should bear in mind that our normalizations of Haar measure differ from those used in [23, II, Chap. 8]. The value of C, is found to be pKr>-1 .(2)- r.2,-ar
(-1)mr[Wx,]
Cr =
aE 0 y K
(3.6)
x
fZ
(1 -
a E o9/gr
Here m, = -(dim G, - rank G, - dim K, + rank K,), n, = -(dim (G,/K - rank (G,/K,)), Wx, is the Weyl group of K, and [W,,] is its cardinality, 0r K stands for the compact roots in 0r , px, is the half sum of these roots, and
i9r is the complement of 0. in t+. Recall that we have assumed that rank (G/K) = 1. In this case there can be at most two nonconjugate Cartan subgroups. One of these is always noncompact, namely A = A,A,, and dim A0 = 1. When another nonconjugate Cartan subgroup exists, it is compact, and we may call it B. Thus there are two invariant integrals 0f and 0' .
We shall compute 0f for f E '1(K\G/K) and relate it to F f. Let a be a regular element of A, and let a = a,a,, ar E A, a, E A, Then (3.7)
F f(a,) _ ,(a,) f f(an)dn = Y
_
f (an)dn
fZv
Since f(an) = f(a,an) = f(an).
For regular a, the map n->a-In-lan is a diffeomorphism of N onto N whose Jacobian is computable. (See e.g. [11, Chapter X]). Thus
Ff(a,) = ,(a,)
fZ aEP+
(1 - a(a)-') f x f(n-lan)dn
(3.8)
_ ,(a')
I
fZ
aEP+
(1 - a(a)-') f"K f N
f(k-'n-1ank)dn dk .
since f is spherical.
The last integral can be transformed as in [10]. It equals f
f(x-1ax)dx, , Ap\G
410
RAMESH GANGOLLI
where dx = da, dx1*. Since Al is compact and carries normalized Haar measure,
this last integral equals f A\G f (x-lax)dx*. Also, if a E P+, so does a. Hence the
product ]]aE p_ (1 - a(a)-1) is real and has the same sign as ]]aEP+ (1 -a(a) ') a real
which of course is precisely ER(a). Using all this, we get
Ff(ap) = ,(a,)ER(a) fl (1 - a(a)-1) 5A,G f(x-lax)dx* a
(3.9)
fl (1 - Sa(ar)-1)-1o f(a) _ bo(ar)-' aEP_ where we have used the fact that for a E P_, a(a) = 1 so that a(a) = Sa(ar). Thus finally, we have (3.10)
of (a) _ P(ar)
fl (1 -
aEP_
a E A'
Now suppose that r E F, r # 1, so that r is hyperbolic. Let h = h(r) be an element of A to which r is conjugate. Then I,(f) = I.W. Let h = hph, ; then h, # 1, since r is hyperbolic. Clearly, ac is a Cartan subalgebra of gc. If a E
0+(ge, ac), then ba(h) = 1, so a(hdga(h,) = 1. Since a is real on a,, and purely imaginary on a, it follows that ga(h,) = 1. Since dim a, = 1, and a is real on A., we conclude that a - 1 on A,, and so a vanishes on a,. Thus a E P_. Therefore +(g°, ac) C P_. It follows that Gh C MAC, and A, is in the
center of Gh. Hence A is fundamental in Gh. The operator Hh, equals II {aEP_7;sa(h)=1} H. In particular, each H. occurring here is in a, Thus, in ap7 plying 11 h to (3.10), we need only worry about the factor g,(ar) fl.,,_ (1 -
a(ar)-1), since H h will not act on F f(a1) at all. The result of applying H h to this function and evaluating the result at h is seen to be equal to [W1 ]
{aEP_; ea(h)=1}
a PKn X ,(hr) X {aEP_;flea(h)#1} (1 - ba(h)-1)
.
Cf. [10, Lemma 24] for a similar computation. Using (3.5), (3.6), we have the following proposition. Proposition 3.2. Let r be a hyperbolic element of G, and let h = h(r) be
an element of A to which it is conjugate. Let h = hrh,, hr E A, h, E A,. Then
I,(f) = Ih(f) =
(3.11) C(h)
=
flaEP+ (1 - ba(h) 1)) 1
One should note that C(h) is actually positive. For later use, we shall examine C(h) a little more carefully. Since ,(hd = exp p(log h,) _ exp 2 f aE + a(log h,), we see that C(h) equals ER(h)
fl (exp 2a(log h) - a(hr)-1 exp - 2a(log
aEP+
411
LENGTH SPECTRA OF MANIFOLDS
Since any a is purely imaginary on a, we must have ea(h,)-1 = a(hp) ; if a is a real root, then of course ga(hp) = 1. Now it is well-known in our case that there is at most one real root in P+. Denote this root by ao when it exists. Then the factor corresponding to it is exp ao(log h) / 2 - exp - a,(log h) / 2. The remaining roots in P+ will be denoted by P° . These are all complex, and occur in conjugate pairs a, a. Thus we can find a subset Q+ of P+ so that P+ Qo UQo Now let a e Q+, and consider the factors corresponding to a and a in the above product. We have a(h,)-1 = ga(h1) = ga(h,). Let Oa(h,) be the argument of ga(h,). Thus ga(h,) = exp iea(h,). Then these two factors have the product exp a(log hp) + exp - a(log h) - 2 cos Oa(h,). Now all the numbers a(log h) are of the same sign, depending on which Weyl chamber hp lies in. Using this remark one quickly finds that
(3.12)
C(h) = exp - I p(log hp) I x (1 - exp - I ao(log hp) I) X fl (1 - 2 cos Oa(hr) exp - I a(log h) I
aGQ+
+ exp - 2 I a(log h) I)-1;
when P+ contains no real root, the factor corresponding to ao is, of course, absent.
4.
The length spectrum
As we have said in § 1, our results follow from applying the trace formula to suitable admissible functions, mainly to the fundamental solution of the heat equation on G/K. Let Q be the Casimir operator of G, and for t > 0 let g,(x) be the fundamental solution of the heat equation Qu = au/at on G/K, with u assumed spherical. The properties of g, are discussed in [4]. Let us briefly recall them. As a function on G, g, is spherical, nonnegative real valued, and gc+8 = g*g8,
for t, s > 0. g, is the fundamental solution in the sense that for any f e C(K\G/K), for example, the function U(x, t) = (g*f)(x) is the unique spherical solution of 2u = 8u/at such that u(x, t) - f (x) --> 0 uniformly on compact sets as t, 0. The function g, is in L1(K\G/K) for each t > 0, and k, can be computed. Indeed, k,(2) = exp - ( + )t. Since g, is integrable, k, is defined for all 2 such that 4°x is bounded, thus in the tube A + iCe, and the above formula for k, holds there. It follows, for example by using [21], that
g, e'1(K\G/K). In particular, g, is admissible. Since k,(2) is known, it is possible to compute the Abel transform Fg, by using the Fourier inversion formula. We get, remembering dim Ap = 1, (4.1)
Fgt(ap) =
(4;rt)-1/2 exp
- (t + Ilog apl2/(4t))
412
RAMESH GANGOLLI
Of course, a similar formula would hold when the dimension of A, > 1, but we would not be using it. Now applying (3.3) to gt, using (3.5) and (3.6) we find o
(G,1)
nr(w, T) exp
(4.2)
- ( + )t
Z Trace T(r) Vol (I',\G,) I,(gt)
rECr
On the right side we get from the term corresponding to r = 1, the contribution gt(1) (degree T). Vol (r\G). The remaining elements Cr are all hyperbolic since r is assumed torsion-free. Call the sum of these remaining terms JH(t). It can be shown (cf. Eaton [3], or [4]) that limt-o JH(t) = 0. This is actually done in Eaton [3] under the additional hypothesis that T is the trivial representation. But the expression for JH(t) when T is nontrivial is clearly dominated in absolute value by a multiple of the correspondiug expression when T is trivial, since gt > 0. Hence JH(t) -> 0 in our case also. If L(t) denotes the left side of (4.2), it follows that lim tn'2L(t) =
tn2gt(1))(Vol. r\G) (degree T) o
/
Here n = dim (G/K). It is shown in [4] that limt-o tnl2gt(1) exists and equals C'G, a constant which
depends only on G. Thus limt.o tn'2L(t) = C'G Vol. (I'\G) degree (T). Now introduce, for r > 0, the function (4.3)
N(r, T) _
nr(w, T) , 19. I co, (4.4)
r-n'2N(r, T) - C'GP(2 + 11 Vol. (I'\G). degree (T) ,
which is analogous to a classical result of H. Weyl [24]. When T is trivial this result is implied by that of Minakshisundaram and Pleijel [15]. Of course (4.4)
LENGTH SPECTRA OF MANIFOLDS
413
is just a step away from Eaton's result.' When T is trivial, N(r, T) is just the Weyl function of the manifold T'\G/K. More precisely, if {(ai, ni)}i,1 is the spectrum of the Laplacian on T"\G/K, is easily seen that N(r, 1) = Z (i; 12d:srj ni. We shall write N(r) for N(r, 1). Clearly the knowledge of N(r) is equivalent to that of the spectrum of the Laplacian. In particular, the spectrum of the Laplacian on F \ G/ K determines Vol (F \ G). Cf. [4], [15]. This will be needed below.
We now turn to the consideration of the length spectrum of R = T'\G/K. For this purpose, we have to compute the terms in (4.2) explicitly, with T = 1 ; this will be done next, resulting in (4.7) below. Cearly G/K is the simply connected covering manifold of R, and we can identify F with the fundamental group 'r1(R). It is well-known that the free homotopy classes of closed paths on R are in a natural one-to-one correspondence with the set of conjugacy classes of F, and hence with the set Cr. For any r r= Cr, the corresponding free homotopy class always contains a periodic geodesic g, say, which has minimum length among all the paths in that class [2]. Let l(r) be the length of g,. Any closed path in this homotopy class can be lifted to a path of equal length on G/K which joins some point m r= G/K to the point rm. It follows that the length l(r) of g, is the minimum of the lengths of paths joining some point m e G/K to its image rm under r. In fact, l(r) = Inf,, G/K d(m, rm) where d(., ) is the Riemannian distance on G/K. Now, if m = xK with x r= G, we have d(m, rm) = d(xK, rxK) = d(K, x-'rxK) = a(x-'rx), where a is the function introduced in § 1. It follows that l(r) = infXEG a(x-'rx). Notice that l(r) depends only on the conjugacy class of r, as it should. Moreover, for the computation of l(r), we can replace r by any element h of G conjugate to r, even if h does not lie in F at all. This remark enables one to compute l(r) more explicitly. Recall that r is conjugate to an element h = h(r) E A. Let h = h,ht ; h acts as an isometry on G/K, with no fixed points. Since G/K is of negative curvature, it follows from [2], [16] that there is exactly one geodesic of G/K which is stabilized by h. This geodesic is characterized by the property that a point p e G/K is on the geodesic if and 2 Actually, one does not need to assume that P is torsion free. In that case the right side of (4.2) splits into three terms, namely Jc(t), JE(t), JH(t), coming respectively from
central, elliptic and hyperbolic elements in Cr. Cf. [4]. One sees that Jc(t) = gt(l) Vol. (P\G) E,Ezncr Trace T(1), where Z = center (G), so that Jim t' /2JG(t) = C'' Vol. (P\G) E Trace T(1) . t-o EZncr One can show as in [3] that limt-0 t'/2JE(t) = 0, so that one gets r n/2N(r,T) -
C'GF(n/2 + 1)-' Vol. (I'\G) E,Ezncr Trace T(-,), which implies that E,Ezncr Trace T(7) must be nonnegative. If now T is irreducible, then T(7) is a scalar for ; E Z n r by Schur's lemma, and T(r) = x('). Identity, where x is a character of the finite abelian
group z n r. If x is nontrivial character, it follows that E7EZncr T(1) = 0, so that r-1a2/N(r, T) - 0 if T jzncr is a nontrivial irreducible representation. When T is not irreducible, Framer x(7) = E deg Ti, where Ti runs over those irreducible summands of
T which restrict to the trivial character of z n P.
414
RAMESH GANGOLLI
only if d(p, hp) = inf EGix d(m, hm). Now it is easy to see that the geodesic Exp A, (where Exp is the exponential map of G/K from p to G/K) is stabilized by h, (recall here that dim AP = 1). Moreover, if p c Exp a,, then d(p, hp) =
6(h). This shows that inf,EGI, d(m, hm) = 6(h), so that l(r) = u(h(r)). Of course, u(h(r)) = I log hp(r) 1.
Note that l(r) = l(7-1), (indeed the geodesics in the homotopy class r-1 are just reverse to those in r), and l(ri) = fl(r) for any integer j > 1. Lemma 4.1. Let r E I', r # 1. Then I', is isomorphic to Z. Proof.
r is hyperbolic, and by conjugation, we may assume r E A, rp # 1.
Let r', r" E f', and suppose r' = r'rt, r" = rvrrt Since G, C MAp as we have seen above, and ri commutes with r, we have 7'E MAC. Thus It follows that the set of elements {r, r' E f',} is a subgroup of rr(rv)-1rp(rz) Ap. Clearly this is a discrete subgroup, hence it is isomorphic to Z. Let o, be a generator for it, and let a E I', be such that 3 = 3p3,. We claim that 3 generates f', freely. In fact let r' E F_ Then rv = 8i for some j E Z. We claim that r' = V. Indeed, r(O ' Thus r'O-i E r fl K, so that r'3-j = 1 since F contains no elliptic elements # 1. Hence r' = 3i and our 1
assertion follows. Remark. Using the negative curvature of G/K, this result could also have been deduced from the theorem of Preismann [17], which is more general. In our special case, the above proof is more direct. Definition 4.2. An element r E f', r 1, will be said to be primitive if r
is a generator of f',. Clearly every r E f', r # 1, can be written as o3 with j > 1 integral, and a primitive. The integer j is unique and will be denoted by j(r). We will next compute Vol. (f',\G,). We may again assume r E A. Then G, C MA,. In fact G, = M,Ap, where M, = M fl G,. Let r = rvr,. Each element of M, commutes with both r and rv, hence with rz. If follows that rz commutes with G,, so rz acts trivially on G,/K,. Thus the action of r on G,IK, is the same as the action of rp. Now it is clear that K, = K fl Gr = M,, and since G, = M,Ap we conclude that the action of I', on G, /K, is the same as the action of jai, j E Z} on AP, acting by left translation. Here we identify AP ~ G,/K,. We thus get (recalling that the measures have been so normalized that K, carries normalized Haar measure), (4.5)
Vol. (I',\G,) = Vol. (f',\G,/K,) = Vol. (A,/Jai, j E Z})
.
The last term is clearly equal to Ilog 8,1 = 1(8). Moreover, since r = OJcr), we have l(r) = j(7)l(3). Thus (4.6)
Vol. (I'r\Gr) = l(r)j(r)
1
.
Using all this in the trace formula, (3.3) with T = 1
415
LENGTH SPECTRA OF MANIFOLDS
L(t) =
(4.7)
mEa(G,1)
nr(w,
1) exp -
= g,(1) Vol. (z'\G) +
( + )t
E
7E Cr-[1}
1(r)j(r)-'I7(g )
Moreover, if r is conjugate to h = h(r) E A, we also know that I7(gc) = I h,(gt)
(4.8)
= (4'rt)-1"2C(h(r)) exp - (t + 4 log h,(r) j2/t) = (47rt)-112C(h(r)) exp - (t + 4l(r)2/t)
because, as we have seen above, l(r) = log hp(r) 1.
It follows that for each t > 0, the series E7ECr-[1} l(r)j(r) exp is convergent; one sees from this that the numbers {l(r), r E C, - {1}} have no finite point of accumulation. In particular, one may indeed order them 0 < 11 < 12 , and the multiplicity mi of each li is finite. (This can also be inferred on general grounds of course.) One immediate consequence of (4.7) is that the length spectrum {li}i,1 of R is determined by the spectrum of the Laplacian, or what is the same, by the function L(t). For, as we saw before, L(t) determines the volume Vol. (T\G), 4l(r)2lt
and hence the first term on the right side of (4.7). Then the smallest of the numbers {l(r); r E C, - {1}}, which is of course 11, is seen to be equal to the supremum of the set > 0; lim ((47ct)1I2 exp (t + 4s2/t)(L(t) - g8(1) Vol. (F\G))) = 0} fill
c-.o
This means that 11 is determined by L(t). Moreover, it is seen that lim (47rt)112 exp (t + 4li/t)(L(t) - g,(1) Vol. (I'\G)) c-o
=
E
1(r)j(r) 1C(h(r)) = 11
{Y; c (7) = t11
E
j(r)
[7; a (7) = L1}
which is positive. Call this number e1. One can now subtract off the contribution to L(t) from IT; l(r) = l1}, and putting
L2(t) = L(t) - g,(1) Vol. (F \G) - {(47ct)-1I2e1 exp - (t + 4li/t)} we find 12 to be the supremum of > 0 ; lim ((47Ct)112 exp (t + 4e2/t) L2(t)) = 0}
and that 1im, (47rt)1"2 exp (t + 4lz/t)L2(t) is positive and equals e2 = I2 7117;1(7)=1211(r)-1C(h(r))
416
RAMESH GANGOLLI
Proceeding in this way, we see that L(t) determines both the numbers {Ii}i, and {si}i,,, where si = li E[rECp;t(r)=zi j(r)-1C(h(r)). Conversely, a knowledge
of these numbers and of Vol. (r\G) clearly determines L(t), and hence the spectrum of the Laplacian ; indeed L(t) = g,(1) Vol. (r\G) + E (47rt)-'"Isi exp - (t + 4E,/ t) i>1
When G = SL(2, R), C(h(r)) depends on r only via 1(r). In fact C(h(r)) _
2 cosh (l(r)/2f), and so si = 2li cosh
j(r)-1 Thus
in this case, knowledge of the sequence {(li, si)} is equivalent to the knowledge of the sequence {(li, )ii)}, where )7, ={rECr,acr)=t j(r)_1 Since {(li, si)} characterizes L(t), we see that in this special case {(li, 7)j)} characterizes L(t). This }
result was originally observed by Hiiber [12]. As we have seen in § 3, the expression for C(h(r)) is more complicated in the general case, and does not depend merely on l(r). Returning to the general case, we let Prr, be the set of primitive elements
in Cr - {1}. Then we can write (4.9)
L(t) = g,(1) Vol. (P\G) +
BEPrr, 1>1
l(3)Iar(g,)
where (4.10)
Ia3(g,) = (4ret)-1/2C(h(3!)) exp - (t + II2l(o)2/t)
The set {l(o) ; 3 E Pr,} can be ordered in a sequence 0 < r1 < r2 < ; let pi be the cardinality of the set {8 e Prr, ; 1(5) = ri}. We call the sequence {ri} the primitive length spectrum, and the sequence {(ri, pi)} the primitive length spectrum with multiplicity. One can ask to what extent these are determined by L(t). Obviously, the set {r1} is contained in the set {li}, which is determined by L(t). So one must try and decide from a knowledge of L(t) whether a given number l; is in the set {ri} or not, i.e., if it is a primitive length or not. Obviously, if l; is not a multiple of some smaller lk, it must be a primitive length. However, if 1, is a multiple of some smaller lk, it could happen that l; is also the length of some other primitive geodesic as well. The author has not been able to decide this question in general by using the above formula. However, when G = SL(2, R), one can answer this question. Indeed in this case, L(t) is characterized by {(li, r)i)} which we can assume known. Now l1 is obviously equal to r1, and ri, equals p1i since j(r) = 1 for all r such that l(r) = 11. Now consider 2r1. It must be one of the numbers {li}i>,. Suppose 2r1 = li,. Then
the numbers {ls ; s < i, - 1} must all be primitive lengths. Thus r, = is and 7)s = ps for all s < i1 - 1. We can now decide whether li, is a primitive length or not. For if lit = ri1, then we should have rii, = 2p1 + pi,, and pi, > 0. Thus, if riil > 2P, = I , we can conclude that li, is a primitive length, li, = ri, and pi, = rii, - 27)i On the other hand if 7)j1 = 2pi, then li, is not a primitive
LENGTH SPECTRA OF MANIFOLDS
417
length. Next, let li2 be the smallest member of the set {li}i>ii, which is an integral multiple of some number l; smaller than it. By the definition of lie, it is clear that the numbers {13 ; it < s < iz} are primitive lengths, and so , = p, for these. As to lie itself, we can decide whether it is a primitive length by comparing i2 with the sum Z ((k,j),jrk=ail j>1) 1 /j. If 12 is strictly larger, then lie is a primitive length, and the difference between i2 and this sum gives its
multiplicity. Proceeding in this way, we see that L(t) determines both the primitive length spectrum and its multiplicity. Finally, let Si = {k > 1, jr, = li for some j > 1}. Then we have mi = Z,kGSi Pk Hence the length spectrum
with multiplicity is also determined by L(t) in this case. When G is not SL(2, R), these questions are not settled by the present method, and a close look at the computations seems to indicate that in general L(t) probably would not determine the primitive length spectrum or the multiplicities. To return to our main topic, define for any l > 0, (4.11)
Q,(l) _ [{6 e Prr, ; 1(6) < l}]
,
Q,(1) = [{r E Cr - {1},1(r) < l}]
[S] stands for the cardinality of S. We shall now determine the asymptotic behaviour of the functions Q0(l), Q1(l) as l -> co. For h e A, with h, # 1 put (4.12)
C, (h) = exp - p(log h,) I jj+ (1 + exp - j a(log h,) I) -'
(4.13)
C_(h) = exp - j p(log h,) I jj (1 - exp - I a(log h,) p-'
(4.14)
C0(h) = exp - I p(log h,) I
IX
,
.GP+
and define (4.15) F(t) _ (42rt)-1/2(exp - t)
Z l(r)j(r)-'C(h(r)) exp - 4l(7)2/t rEer-{1)
and let F, F_, Fo be defined analogously by replacing C(h) by C+(h), C_(h), C0(h) in (4.15). Lemma 4.3. Let H(t) be any of the four functions F(t), F+(t), F_(t), F0(t),
and let, for r > 0, ft(r) = f e-r'H(t)dt. Then H(t) -> 0 as t --> 0, H(t) --> 1 as t -p co, and rH(r) --> 1 as r --> 0. Proof. We know that for r E Cr - {1}, l(r) = log h,(r) I is bounded away from zero. Hence, if p = sup.ep,,rEcr-{I} exp - ka(logh,(1))j, we conclude that p < 1. Let D = ((1 + p)/(1 Then for each r E Cr - {1}, (4.16)
C+(h(r)) < C(h(r)) < C_(h(r)) < D C+(h(r))
where we used the expression (3.12) for C(h). Therefore
418
RAMESH GANGOLLI
F+(t) < F(t) < F_(t) < DF+(t)
(4.17)
and similarly
Fo(t) < F_(t)
(4.18)
Now we know, by the remarks immediately following (4.2), that F(t) (called JH(t) there) approaches zero as t - 0. From (4.17), (4.18) it follows that F+(t), F_(t) and F0(t) all do the same. We next claim that F(t) - 1 as t - oo. In fact by
F(t) = 1 + (4.19)
nr(w, 1) . exp - ( + )t v Ee(G,1)
- gt(1) Vol. (1'\ G)
.
As t - 0o , each term in the sum approaches monotonely to zero, because + > 0 ; so the whole sum approaches zero. Next, we know [4] that
gc(x) = [W(G, A,)]-' f A exp - ( + )t ¢2(x) Ic(2) I-z d2 , where c(2) is the Harish-Chandra c-function. It follows that oat(1) = [WI-1 f A exp - ( + )t Ic(2)L-z d2
again by monotone convergence, we conclude that gt(1) - 0 as t
Now
(4.19) shows F(t) - 1 as t - oo. We will now show that F+(t) - 1 as t - 00 . The other functions F_, Fo can be treated similarly. Using (3.12) it is easy to see that C+(h(r))/C(h(r)) - 1 as l(r) = I log h,(r) I - oo . Lets > 0 be given, and choose and fix N so large that for l(r) > N, we have (4.20)
(1 - s)C(h(r)) < C+(h(r)) < (1 + s)C(h(r)) .
Let FN(t), Fl(t) be the tails of the series defining F(t), F+(t) beyond l(r) > N. Then one sees (4.21)
(1 - s)FN(t)
F`v(t) < (1 + s)FN(t)
For each fixed N, the sum (4irt)-111 exp
- t t(r)s:v l(r)i(7)-1C(h(7)) exp - 4l(r)z/t
is a finite sum and approaches zero as t -f c . Since F(t) -f 1 as t -f 00 , it follows that FN(t) -f 1 as t -f oo. Thus from (4.21) we deduce
LENGTH SPECTRA OF MANIFOLDS
419
(1 - s) < Jimt-- FN(t) < liil FN(t) < 1 +
t--
Now by examining the sum F+(t) - FN(t) we can similarly conclude that limt__ (F+(t) - FN (t)) = 0. This together with the above shows that (4.23)
(1 - e) < lim F+(t) < lim F+(t) < 1 +
s
Since s is arbitrary, we conclude F+(t) -* 1 as t - o o. The first assertion of the lemma is proved by proceeding similarly for F_, Fo. Since F(t) is nonnegative and F(t) -+ 1 as t -p 0, Karamata's theorem [25] shows that
rP(r) -p 1 as r -p 0, where P(r) = f e-rtdF(t)
.
Also, the functions F+(t) - F(t), F_(t) - F(t) do not change sign, and approach 0 as t -+ o o. So by the same theorem, we must have r(P+(r) - P(r))
- 0, r(F'-(r) - P(r)) -p 0 as r -p 0. Finally, F0(t) - F(t) does not change
sign, and approaches 0 as t --p o o. So we get r(P0(r) - P_(r)) --p 0 as r --p 0. Since rP(r) - 1 as r --p 0, the proof is finished. Theorem 4.4. Let Q0(l), Q1(1) be the functions defined in (4.11). Then we
have
(4.25)
2 p l exp - (2 p l)Q0(l) -* 1
as l-p 0
2Ipl l exp - (2IpI l)Q1(l) - 1
as l - 0
where 2 IpI = 2112 = (p + 2q)(2p + 8q)-1'2. Proof. We deal first with Q0(l). The result for Q1(l) will be deduced from
it. Recall first the notations of § 1. Let h(r) be in A, and h(r) conjugate to r e C, - {1}. log h,(r) is a multiple of Ho ; say it equals u,H0. Then 1(r) = I log h,(r) I = I ur I. I Ho 1. Also I p(log h,(r)) I = I ur I I p(H0) I. Then
I p(log hv(1)) I = 1(r)
-
I p(Ho) I / I Ho I
It can be computed easily that Ip(Ho)I IHoI = 2(p + 2q)(2p + 8q)-"2 = IpIHence 2IpI = (p + 2q)(2p + 8q) -1/2 and I p(log h,(r)) I = I pIl(r). Since each r equals 81(r) with 8 primitive, and l(r) = j(r)l(8), we have (4.26) Fo(t) _ (42rt)-1'2 exp - IPI2 t Z Z exp - (j I p I 1(8) + 6EPrp j>1 Thus
f
e-rtFo(t)dt 0
4j2l(8)2/t)
.
420
RAMESH GANGOLLI
_Z Z l(o) exp - j I P 110) aEPrr j>1
(4.27)
f (47t)exp - (( p 2 + r)t + 'j21(8)2/t)dt 0
f(4zrt)-1'2 exp (-x2t - Iy2/t)dt = (2x)-1 exp - xy to get
Use the formula Fo (r)
= 1 (r + I P
I2)-1/2
12 (r +
2)-112
Z Z 1(8) eXp - (jl(3)(P 1 + -%/r -+I P I2))
aEPrr j>1
2
(4.28) P
aEPrr
/r +
eXp -
1(8)
P
1 - exp - l(3)(pI + r -+p I2)
Let (4.29)
Pie)-1/2
Go(r) = 2(r +
aEPrr
1(8) exp - l(8)(p + /r + pI2)
which converges by comparison with Fo(r). The ratio of the corresponding terms in G0(r) and Fo(r) approaches 1 as 1(8) - co. So an argument similar to that of Lemma 4.3 shows that rG0(r) and rF0(r) have the same limit as r - 0. Since we know rF0(r) - 1 as r - 0, we conclude rG0(r) - 1 as r - 0. Now rG0(r) = 2 r(r + I p
r(r +
I2)-1/2
aEPrr IPI2)-1/2fo
(4.30)
l(8) exp - l(8) (I P I + Ir + I P
lexp
-
r IPI2 - IP1)
IPI2
Wr+
IpI
X(IJP r+ r-IPI)
p2-
0
Writing z = r + -I p I2 - I p I , we see that z - 0 as r - 0. Letting r - 0 in the above expression we conclude lim z fo exp - zl l exp - 2 I p I l dQ0(l)
Now Karamata's theorem gives us the first conclusion of the theorem. (See the note added in proof.) As to Q1(1), we have
(4.31)
Q0(l) = [{3; 8 e Pry, 1(8) < l}] G Q1(l) _ [{r E Cr - {1} ; 1(r) G l}. _ [{(8, 8 e Pry, j > 1, jl(8) < l}] l [SEPrr;b(a)Sll 18
=f
a
l
0Y
a
2 lQo(Y)dY
vow = Qo(l) + f0 Y
421
LENGTH SPECTRA OF MANIFOLDS
Since we know the asymptotic estimate for Q0(l), the estimate for Q1(l) follows easily from this expression. This finishes the proof of the main result. One notes that the asymptotic behaviour of Qo and Q1 depends only on the
metric structure of the covering manifold G/K and not on the particular manifold R (or what is the same, on r). This theorem generalizes a result of H. Huber [12] who treated the case G = SL(2, R). Huber's method is slightly different; it was followed by Berard-Bergery [1] to G = SO0(d, 1), d > 2; Our method generalizes the method of McKean [14] who works with G = SL(2, R). These authors use a
metric on G / K which gives it curvature - 1 in their cases. Our metric is somewhat different. This introduces an inessential discrepancy between the values of I p I which they get there and we get here. Huber also proved the remarkable formula [12, p. 26], 2,v/7cr(s)
(S -
1)r(s - y) 2s--1
(4.32)
+ rlS 1
-
+
2)
r(s)
r(s - 1)
nr((, 1)r( 2 (S - S-(2m)))F(2 (S -
Vol. (r/G)
2
Z
recr-{11
l(r)i(r)-1(cosh l(r) - l)-"'(cosh l(r))
1i2
where s±(2,.,) are the roots of S2 - S - J. = 0, and G = SL(2, R). J. is the eigenvalue of the Laplacian. One must bear in mind that Huber used the metric which gives curvature - 1 to G/K. HUber's proof of (4.32) utilizes methods involving the Green's function of the upper half-plane. Huber used the above formula together with the theorem of Ikehara to get the analogue of Theorem 4.4 for G = SL(2, R). A generalization of (4.32) for G = SO0(d, 1) is presented by Berard-Bergery in [1, p. 118], and is used there similarly to obtain Theorem 4.4 for G = SO0(d, 1). Both (4.32) and its generalization to SO0(d, 1) in [1] result from the traceformula by the choice of a suitable admissible function f,. One must, of course, compute f s and Ff.* In fact, let x e G, and x = ka,k', k, k' e K, a, e 4 be its polar decomposition. Put J Ho j = c (recall that this equals ,/2p _+8q). Let
Q e I be as in § 2, and put t = t(a,) = p(log a¢). Then t can be regarded as a coordinate on 4 Consider, for a complex S, the function fs(x) = (cosh t)-s where t = t(a,) and x = ka,k'. f s is clearly spherical. If Re s > p + 2q, one can show that f, E '1(K\G/K), so that f, is admissible. (4.32) and its generalization result from applying the trace formula to this f s. It is possible to compute the analogue of (4.32) for all the groups of rank (G/K) = 1 by computing j,, F f. directly. Since the main application of these formulas was to get
422
RAMESH GANGOLLI
Theorem 4.4 which we have obtained by other means, it does not seem worthwhile to give details of the derivation. We will content ourselves with quoting the result, which may amuse the reader : 72r(U), 1)7r(P+q+1)/2
(4.33)
rQ'(s - s-lam)))1 (2(s - s+lam)))
r(2s)I'('(s - q + 1)) = Vol. (r1G) + 7r(p+q+1)/2 21-s+(p+2q)/2 r(s - s(p + 2q)) r(ss)r(I(s - q + 1))
xE
, E Cr- (i}
1(7)1(7)-'C(h(7))(cosh l(r))-s+(P+2q)/2
where st(2) are the roots of the equation
S2 - s(p + 2q) + (p + 2q)2 + 2 (Ho)2 = 0 . Thus
st (2.) = (p + 2q) ± 2
-.1(Ho)2 = p(Ho) ± ia.(Ho)
The reader will easily check that when p = d - 1, q = 0 (which is appropriate for G = SO0(d, 1)), one gets from this the formula of [1, p. 1181. (4.32) results from p = 1, q = 0. The difference of metrics must be borne in mind. For the other groups G, the values of p, q are as follows : When G =
SU(d, 1), p = 2(d - 1) and q = 1 ; When G = Sp(d, 1), p = 4(d - 1) and
q= 3.When G=F,(-2o),p=8andq=7.
A final application of these methods which may be worth mentioning is the
following. Let x, y e G, and let for any r > 0, Q(x, y, r) be the number of elements 7 e I', such that a(y-'rx) < r. Q(x, y, r) is the number of points k on G/K which lie in a ball of radius r around the point yK. The computation of 1, alluded to above enables us to find the asymptotic behaviour of Q(x, y, r) as r--->oo ; (cf. [1]). Briefly, the method is as follows : Since fs is admissible, ,Er fs(xry 1) converges nicely and can be expanded as a series Z-E6(G,1) Z$=1 fs(2) (x) (y), where +;m, 1 < i < nr((o, 1), are
eigenfunctions of P. in L2(r\G/K), corresponding to the eigenvalue Q.. Now Z, f s(y-'rx) _ Z, (cosh a(y-'rx)/c)-s, with c = 8q as before, which
can be viewed as a Dirichlet series, convergent if Re s > p + 2q. On the right side, the computation of 1, allows one to conclude that this Dirichlet series has a single simple pole at s = p + 2q whose residue can be computed. Applying the theorem of Wiener-Ikehara one gets (4.34)
Q(x,Y, r)
2.7r (p+q+1)/2
e21P1,
r(2 (p + q + 1)).21pI Vol. (r\G)
2P +2q
as r - oo
.
LENGTH SPECTRA OF MANIFOLDS
423
We leave the details to the reader. A result analogous to Theorem 4.4 has been announced by Margulis [13]. See also Sinai [20]. These authors use ergodic theory. Margulis' result is the stronger one. His context is that of an arbitrary compact manifold of negative curvature, and he shows that Q0(l) - Cl-1 exp dl, for some positive d. In our special situation, we have been able to relate this constant d to the structure
of the manifold. Margulis' proofs have not appeared, as far as the author knows.
Added in proof. After this paper went to press, D. Hejhal pointed out to me that the proof of Theorem 4.4, as well as of the analogous theorem in McKean's paper, is based on an incorrect application of Karamata's theorem. However, the conclusion of the theorem is correct. There are several ways of filling the gap. One is to use Huber's method as indicated above, exploiting (4.33). The other is to use the heat kernel in the trace formula, and to study the behaviour of that formula for complex t in a sector. The third, and the most satisfactory, method is to study the Dirichlet series Z 1(6) exp -sl(d), s e C. By using the analytic properties of the Selberg zeta function (See R. Gangolli, Ill. J. Math. 21 (1977) 1-41), one can show that this series is
meromorphic in Re (s) > 2 1 p - e for some e > 0, and has a single simple pole at s = 2 1 p I with residue 2 p 1. Now Wiener-Ikehara's theorem yields Theorem 4.4. (This method is described for noncompact G/I' in a forthcoming paper of G. Warner and the author.) For yet another method, and a better result, see D. DeGeorge, Ann. Sci. Ecole Norm. Sup. 10(1977) 133-153. Bibliography L. Berard-Bergery, Laplacien et geodesiques fermees sur les formes despace hyperbolique compactes, Seminaire Bourbaki, 24ieme Annee, 1971-72, Exp. 406.
R. L. Bishop & B. O'Neill, Manifolds of negative curvature, Trans. Amer. Soc. 145 (1969) 1-48. T. Eaton, Thesis, University of Washington, Seattle, 1972. R. Gangolli, Asymptotic behaviour of spectra of compact quotients of certain symmetric spaces, Acta Math. 121 (1968) 151-192. , Spherical functions on semisimple Lie groups, Geometry and analysis on symmetric spaces, Marcel Dekker, New York, 1972. , Spectra of discrete uniform subgroups, Geometry and analysis on symmetric spaces, Marcel Dekker, New York, 1972.
R. Gangolli & G. Warner, On Selberg's trace formula, J. Math. Soc. Japan 27 (1975) 328-343. Harish-Chandra, Spherical functions on semisimple Lie groups. I, II, Amer.. J. Math. 80 (1958) 241-310, 533-613. -, Invariant eigendistributions on a semisimple Lie group, Trans. Amer. Math. Soc. 119 (1965) 457-508. Discrete series for semisimple Lie groups. II, Acta Math. 116 (1966) 1-111.
S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. H. Huber, Zur analytischen theorie hyperbolisher raumformen and bewegungsgruppen. I, Math. Ann. 138 (1959) 1-26.
424 1131
:[141
[15] [161
[17] 1181
1191
[20]
,[21]
RAMESH GANGOLLI
G. A. Margulis, Certain applications of ergodic theory to the investigation of manifolds of negative curvature, J. Functional Anal.i Prilozen 3 (1969) 89-90, (Russian). H. P. McKean, Selberg's trace formula as applied to a compact Rienzann surface,
Comm. Pure Appl. Math. 25 (1972) 225-246. S. Minakshisundaram & A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Canad. J. Math. 1 (1949) 242-256. V. Ozols, On the critical points of the displacement function of an isometry, J. Differential Geometry 3 (1969) 411-432. A. Preismann, Quelques proprietes globales des espaces de Riemann, Comment. Math. Helv. 15 (1943) 175-216. A. Selberg, Harmonic analysis and discontinuous subgroups in weekly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc.
20 (1956) 47-87. Seminaire Berger, Varietes a courbures negative, Universite Paris VII, 1970/71.
Y. Sinai, The asymptotic behaviour of the number of closed geodesics on a
compact manifold of negative curvature, Izv. Akad. Nauk. SSSR Ser. Mat. 30 (1966) 1275-1296. P. Trombi & V. S. Varadarajan, Spherical transforms on semisimple Lie groups,
Ann. of Math. 94 (1971) 246-303.
[22]
[23]
N. Wallach, An asymptotic formula of Gelfand and Gangolli for the spectrum of I'\G, J. Differential Geometry 11 (1976) 91-101. G. Warner, Harmonic analysis on semisinzple Lie groups. I, II, Springer, Berlin, 1972.
[24]
H. Weyl, Das asymptotische Verteilungsgesetz der eigenschwingungen eines beliebig gestalteten elastischen korpers, Rend. Circ. Mat. Palermo 39 (1915)
[25]
D. Widder, The Laplace transform, Princeton University Press, Princeton, 1941. N. Wiener, Tauberian theorems, Ann. of Math. 33 (1932) 1-100. J. A. Wolf, Spaces of constant curvature, McGraw-Hill, New York, 1967.
1-50.
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UNIVERSITY OF WASHINGTON, SEATTLE
J. DIFFERENTIAL GEOMETRY 12 (1977) 425-434
GLOBAL PROPERTIES OF SPHERICAL CURVES JOEL L. WEINER
Let a be a closed curve regularly embedded in Euclidean three-space satisfying suitable differentiability conditions. In addition, suppose a is nonsingular, i.e., free of multiple points. In 1968, B. Segre [4] proved the following about such curves. Theorem. If a is nonsingular and lies on a sphere, and 0 denotes any point of the convex hull of a with the condition that 0 (if lying on a) is not a vertex of a, then there are always at least four points of a whose osculating plane at
each of those points passes through 0. If 0 is a vertex of a then there are at least three points of a whose osculating plane at each of those points passes through 0. All terms used in the statement of the theorem are defined later in this paper. To quote H. W. Guggenheimer [2] who reviewed [4], "The 12-page proof
is rather complicated." Here we present a shorter and hopefully more transparent proof of this theorem. In addition, we need only require that the spherical curve a be of class C2 whereas Segre's proof requires a be of class C3. Also, we obtain, with no extra effort, a similar theorem which holds if a's only singularity is one double point ; in this case, the above mentioned minimums must be reduced by two. In the last section of this paper we characterize spherical curves with the following property : for every point 0 of the convex hull of a, other than a. vertex of a, there exists the same (necessarily even) number of distinct points of a whose osculating plane at each of those points passes through 0. The proofs of many results in this paper ultimately depend on ideas contained in a paper by W. Fenchel [1]. Throughout this paper we use the following conventions. By a curve we mean a regular C2 function a : D , E3, where D is an interval (with or without. end points) or a circle, and E3 is Euclidean three-space. We let a denote both the function and its configuration a(D) in E3. When D is a circle we say a is closed. If D is a closed interval we may sometimes refer to a as an arc. We say a point P in E3 is a multiple point of a if it is the image of k > 1 points of D. If k = 2 then P is called a double point. At a multiple point P we will think of P as k distinct points each traversed once by a as we traverse D once. If a has no multiple points, then we say a is nonsingular. Received July 16, 1975. This research was supported by NSF Grant GP 43030.
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1.
Geodesic curvature
Let a be an oriented spherical curve; i.e., a lies on a sphere S in E3 and has a preferred direction of traversal. Let S be oriented, say, with respect to the outward pointing normal. We denote by k the geodesic curvature of a as a curve in S. It is defined by k = (d2a/ds2) n, where s is the arc length parameter of a consistent with its orientation, and n is da/ds rotated +90° in the tangent plane to S at its point of contact with S. Since a is C2, k is a continuous function on a. At each point P of a there is in S a circle tangent to a which best approximates a near P. This circle w(P) is the osculating circle to a at P ; it is easy to see that w(P) is the intersection of the sphere S and the osculating plane ir(P) to a at P, when a is viewed as a curve in P. We have the following obvious lemma. Lemma 1. Let a be a spherical curve and P E a. Then k(P) = 0 if and only if 7r(P) goes through the center of S. We will need some lemmas about spherical curves proved by Fenchel [1]. Actually we state mild generalizations of these lemmas; see [1], [5] for their proofs. In these lemmas we speak of a set on the sphere being to the left of a curve. By this we mean that when the tangent vector to the curve in the preferred direction is rotated x-90° it points into the set. Also when we say a point P is between points A and B we mean that either A and B are antipodes or if A and B are not antipodes then P lies on the shorter geodesic arc through A and B. Lemma 2. A nonsingular spherical curve a with k > 0 and not identically zero connects two points A and B of a great circle r without otherwise meeting it. Then A and B are not antipodes of one another. In addition the region bounded by the curve and the smaller great circular arc AB of r and lying in a hemisphere is to the curve's left. Lemma 3. Let a be a nonsingular spherical curve with k > 0, and let r be an arbitrary great circle which meets a in at least two points. Then there is a subarc a, of a with the following characteristics: 1. The end points A and B of a, lie on r. 2. a, has otherwise no points in common with r. 3. All other points of intersection of a with r lie between A and B. Remark. If a, contains a point P for which k(P) > 0, then A and B are not antipodal by Lemma 2. In particular, more than a half circle of r is free of points of intersection with a. 2.
Fenchel's theorem
The convex hull of a point set M in Euclidean space is the smallest convex set containing M. Let Q be the convex hull of a spherical curve a. The next lemma characterizes the points of Q ; for its proof see [1, Satz All.
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427
Lemma 4. For 0 to be an element of 2 it is necessary and sufficient that there exists a plane 2 through 0 such that 0 is in the convex hull of a fl ,. Throughout this section we take 0 to be the center of the sphere S on which a lies. With this choice for 0, Lemmas 3 and 4 lead immediately to a theorem due to Fenchel [1, Satz II']. This theorem is restated to include the possibility that 0 is an element of the boundary of 2 as well as the interior of D. Theorem 1 (Fenchel). Suppose a is closed and nonsingular except perhaps for one double point. If 0 E 2, and a does not contain a great semicircular arc, then the geodesic curvature of a changes sign at least twice. The same lemmas can be used to prove the following extension of Theorem 1. This will be shown here. Theorem 2. Suppose a is closed and nonsingular. If 0 E Sl, and a does not contain a great semicircular arc, then the geodesic curvature of a changes sign at least four times. Remark. It is easy to construct examples of closed nonsingular spherical curves whose geodesic curvature changes sign only twice and which necessarily contain a great semicircular arc. It is a consequence of Lemma 2 that these curves lie in a hemisphere determined by the great semicircular arc. The remainder of this section is devoted to a proof of Theorem 2. Before we proceed we introduce some notation. If a is a non-closed spherical curve, and P, Q are two points of a, then by PaQ we mean the oriented arc running along a from P to Q. If P, Q are two points of the sphere S which are not antipodal, then PQ denotes the smaller great circular arc through P and Q oriented from P towards Q. To denote the larger great circular arc connecting P and Q, we write PAQ where A is on the great circle through P and Q but A PQ. By a Jordan curve we mean a nonsingular continuous image of a circle. Proof of Theorem 2.
Let a be a closed nonsingular curve lying on a sphere
S with center 0, and suppose that a contains no great semicircular arc. In particular, a's geodesic curvature k is not identically zero. Also suppose 0 E 2, the convex hull of a. By Theorem 1 we already know that k changes sign at least twice. We will show that the supposition that k changes sign only twice leads to a contradiction. Therefore suppose k changes sign twice at the points
A and B of a. Let a' and a' be the two curves into which a is separated by A and B, both oriented so that their geodesic curvature is nonnegative (and, of course, not identically zero). Suppose a' and a2 begin at A and end at B. By Lemma 2 there is a plane 2 through 0 such that 0 is in the convex hull
of 2 fl a. Let r = 2 fl s; it is, of course, a great circle. There are two cases to consider. Either 1. a meets r in at least three points and these points do not lie in an open half circle of r, or 2. a meets r in two points, which are necessarily antipodal.
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Case 1. Let C, D, E be distinct points at which a = a' U a2 meets r and which do not lie in an open half circle of r. We may suppose that C and D are points of a'; in fact, suppose C precedes D in a'. Since a' meets r in at least two points, Lemma 3 implies that there exists a subarc a; with the characteristics 1, 2, and 3 of that lemma. Also a7 is not a great semicircular arc. The remark following Lemma 3 implies that E must be a point of a2. We may assume that C and D are the end points of a'; if the new C, D, E lie in an open half circle of r so do the old C, D, E. Let H be the closed hemisphere determined by r and not containing a; ex-
cept for the end r pints C and D. Let L be the region to the left of the oriented Jordan curve' Ca'D U DC together with its boundary. Lemma 3 implies that a' C H U L. In particular A, B E H U L ; hence a2 must begin and end in H U L. The boundary of H U L is the Jordan curve a7 U DEC. Now if a2 is not contained in H U L, it must cross the boundary along DEC (excluding
the end points D and Q. Remember that a' and a2 meet only at A and B. We assume without loss of generality that a2 crosses DEC. If a2 did not cross
DEC, then it would be tangent to r at E. We could then rotate 2 a bit about the diameter of S through C or D so that a crosses r at points which we still call C, D, E and which still do not lie in an open half circle of r. Since a2 meets r at least twice, Lemma 3 implies the existence of a subarc c:1. Let a' begin at F and terminate at G. Characteristic 3 of a; implies that at least one of the points F and G is not between C and D. At this stage of the argument we suppose that F does not lie between C and D. The argument is similar if we suppose that G does not lie between C and D. Consider the oriented Jordan curve Aa1D U DF U Fa2A. If D and F are antipodal, then here DF is the half great circle not containing G.; see Fig. 1.
Fig.
1
Note that Da'B and Fa2B cannot cross the Jordan curve. That F012B does not cross DF is the only part of the preceding statement which may not be im-
PROPERTIES OF SPHERICAL CURVES
429
mediately clear. However Fa2B may only cross r along FG which is less than a half circle ; also DF is at most a half circle. Thus DF meets FG only at F. Thus Fa2B meets DF only at F. Now Da1B and Fa2B are on opposite sides of the Jordan curve near D and F, respectively. This is clear since a' is entering
H at D and al is leaving H at F. Thus B is both to the right and the left of the Jordan curve, which is a contradiction. Case 2. Let C and D be the two points in which a meets T. As already noted C and D are necessarily antipodal. This case can be reduced to Case 1 since there must be a great circle through C and D which intersects a at a third point E. Clearly C, D, E do not lie in an open half circle.
Remark. We do not use the fact that a' and a' join at A and B in a CZ fashion, but only that they begin and end at A and B, respectively. 3.
Segre's theorem
Generally, if P is a point of a curve a then at P a passes through the osculating plane to a at P. However if this does not happen we call P a vertex of a. Thus by a vertex of a curve a we mean a point P of a with the property that near P a lies on one side of the osculating plane to a at P. Theorem 3. Let a be a closed curve on the sphere S and let 0 E SQ, a's convex hull. Then (i) if a is nonsingular and 0 is not a vertex of a, there exist at least four points of a whose osculating plane at each of those points passes through 0,
(ii) if a is nonsingular and 0 is a vertex of a, there exist at least three points of a whose osculating plane at each of those points passes through 0, (iii) if a's only singularity is one double point and 0 is not a vertex of a,
there exist at least two points of a whose osculating plane at each of those points passes through 0. The idea behind the proof lies in the observation that Theorem 3 follows trivially from Theorems 1 and 2 by means of Lemma 1 if 0 is the center of S. So if 0 is not the center of S we let a* be the projection of a into a sphere X centered at 0 and apply Theorems 1 and 2 to a* to get the required num-
ber of points of a* whose osculating plane at each of those points passes through 0. If 0 E a, then a* is not a closed curve but one can still show that a* has the required number of points whose osculating plane at each of those points passes through 0. Finally we observe by Lemma 5 that an osculating plane at a point of a* passes through 0 if and only if the osculating plane at the corresponding point of a does so. We now introduce the notation which will be used in the proofs of Lemma 5 and Theorem 3. Let a be a closed curve on S, and SQ the convex hull of a. Suppose that 0 is any element of D and X is a sphere centered at 0. Let p : S --> X be the projection of S into X through 0. When 0 E a, p is understood to be defined only on S - {0}. Denote the image of P E S under p : S - > .Z by P*.
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If 0 is in interior of S, we let a* denote the image of a under p. If 0 e a, note first that p(a) is contained in a hemisphere H with boundary r*, where r* is the intersection of the tangent plane to S at 0 with 1. Assume 0 is not a multiple point of a ; then the limits of P* as P approaches 0 along a first from one side and then the other are two antipodal points on r*. We adjoin these points to p(a) and denote the resulting arc by a*. When 0 is a multiple point of a, we adjoin points of r* to p(a) as above to get a collection of arcs denoted by a*. Then let Q* be the convex hull of a*. Let ir(P) and 7r*(P*) denote the osculating planes to a at P and a* at P*, respectively. Lemma 5. Suppose P zt- 0. Then ir(P) passes through 0 if and only if ft*(P*) goes through 0. Moreover, if rr(P) passes through 0, then P is a vertex of a if and only if P*is a vertex of a*. Proof. The projection p : S - is a C°° difeomorphism of S onto its image. Thus the order of contact between two curves on S and their images under p on _Y is preserved (except if the contact is at 0 E a). Let w(P) and w*(P*) denote the osculating circles to a at P and a* at P*, respectively. Suppose 7r(P) passes through 0. Since w(P) lies in ir(P) which passes through 0, its image under p is a (great) circle on _Y if 0 a and is a half (great) circle on _Y if 0 E a. Let w(P)* denote the circle in which p((O(P)) lies on J. Since the order of contact is preserved, w(P)* _ w*(P*). Thus both ir(P) and it*(P*) contain w(P)*. Hence 7r(P) = 2r*(P*) passes through 0. The converse is proved in an identifical fashion. Now suppose ir(P) passes through 0. Then, by the above, ir(P) = 7r*(P*). If a lies on one side of ir(P) near P, clearly a* lies on one side of it*(P*) near P* and conversely. That is, P is a vertex of a if and only if P* is a vertex of a*. Proof of Theorem 3. We separate the proof into two cases according as 0 e a or not. Suppose 0 a. Then it is clear that 0 E Q* since 0 e Q. Thus we may apply Theorems 1 and 2 to a* lying on 1. If a is nonsingular, so is a* ; thus a* has at least four points where its geodesic curvature is zero. If a has just one double point, so does a* ; thus a* has at least two points where its geodesic curvature is zero. By Lemma 1, at each of these points of a* the osculating plane passes through 0. Hence by Lemma 5 the osculating planes at the corresponding points of a pass through 0. Thus we have proved (i) and (iii) for the case 0 a. Suppose 0 e a and 0 is not a multiple point of a. Assume now a is oriented. By means of p we orient a*. Denote the beginning of a* by A and the end by B. Let w be the osculating circle to a at 0. Its image under p including end points, denoted by w*, is a half great circular arc of 1. It is easy to see that uI* also begins at A and ends at B. Also w* and a* are tangent at A and B. If 0 is not a vertex of a, then a* is on opposite sides of m* in H near A and B ; see Fig. 2. If 0 is a vertex of a, then a* is on the same side of m* in H
PROPERTIES OF SPHERICAL CURVES
Fig.
431
2
near A and B. Let k* be the geodesic curvature of a*. Then using Lemma 2 and the idea of parity, one can show the following hold : 1. k* changes sign at least twice if 0 is not a vertex of a and a is nonsingular, 2. k* changes sign at least twice if 0 is a vertex of a and a is nonsingular,
k* changes sign at least once if 0 is not a vertex of a and a's only singularity is one double point. Again apply Lemmas 1 and 5, in that order, to prove (i), (ii), and (iii) for the case where 0 e a and 0 not a multiple point of a. If 0 is the double point of a the proof of (iii) is immediate. Corollary. Let a be a C3 closed nonplanar curve in E3 with no pair.of directly parallel tangents. Then a has at least four vertices. For the proof of this corollary see Segre [4, p. 263] where the same result is proven for C4 curves. Our results allow his proof to go through for C3 curves. Actually the corollary follows immediately from Theorem 2 and the remark following Theoerm 2 since the tangent indicatrix of a nonplanar curve cannot lie in a hemisphere. 3.
4. A characterization In this section we find a characterization for a (possibly singular) closed curve a lying on the sphere S and having the property that for each point 0 in its convex hull Q except for vertices of a there exists the same (necessarily even) number of distinct points of a whose osculating plane at each of those points passes through 0. The next lemma is especially important in this section. It follows by means of stereographic projection from a similar fact for plane curves due to Kneser ; see [3, p. 48] for Kneser's theorem and its proof. When we say that the circle cu lies between the (disjoint) circles m' and cue on the sphere S we mean that w is in the connected component of S - ((u' U cue) whose boundary is a' U cue. Lemma 6. Let a be spherical arc with monotone geodesic curvature k. Let
P, Q, and R be three points of a with Q between P and R. Then w(Q) is be-
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JOEL L. WEINER
tween co(P) and co(R) if it is not equal to co(P) or c,(R). Moreover, w(Q) = c,(P) (respectively, co(R)) only if k(Q) = k(P) (respectively, k(R)). At this point we make some additional assumptions about the closed spher-
ical curve a which will hold throughout the remainder of this section. First, we require that there exists at most a finite number of points of a at which the geodesic curvature k takes on an extreme value. This is equivalent to requiring that a has at most a finite number of vertices since the vertices of a occur at the extremes of k. Secondly, we assume k is strictly monotone between the vertices of a. This second condition rules out the possibility of a having an arc of points with the same osculating plane. Let B denote the closed ball whose boundary S contains the closed curve a. Clearly Q C B. Theorem 4. Suppose a has n vertices. If 0 E B, then there exist at most n points of a whose osculating plane at each of those points passes through 0. Proof. Let V1, V2, , V, denote the vertices of a as they occur in making one circuit of a. Using the notation of § 2, we set ai = ViaVi+1 for i = 1, 2, , n, where V,t+1 = V1. We will show for each integer i, where 1 < i < n, there exists at most one point P E ai such that 0 E 7r(P). This immediately implies the theorem. Suppose, to the contrary, that ai contains two points P and Q such that 0 E 7r(P) fl 7r(Q). In particular, 7r(P) fl 7r(Q) # 0; hence co(P) fl w(Q) * 0. This is impossible by Lemma 6 since k is strictly monotone on ai. Remark. Note that vi E ai-1 fl ai for i = 1, 2, , n, where a° = an. Hence if 0 E B and, in addition, 0 E 7r(Vi), then there exist strictly less than n points of a whose osculating plane at each of those points passes through 0. Corollary. Suppose a has n vertices. If 0 E Q, then there exist at most n points of a whose osculating plane at each of those points passes through 0. Let V1, V21 , V, be the vertices of a. Note that n is necessarily even since it is the number of extreme points of the geodesic curvature of a. Theorem 5. Suppose (o(Vi) fl a = {Vi} for i = 1, 2, , n. Then for every 0 E Q - {V1, V2i , Vn} there exist exactly n points P11 P21 ., Pn of a such that 0 E 7r(Pi) for i = 1, 2, , n, and conversely. Proof. Let B' = B - U2=17r(Vi). Also let B'm be the set of points 0 in B' with the property that there exist exactly m points P11 P21 . . , P,n of a such that 0 E 7r(Pi) for i = 1, 2, . , m. Let Q' = Q - {V11 V21 ... , Vn}. For i = 1, 2, , n, the assumption w(Vi) fl a = {Vi} implies Q fl ;r(Vi) = {Vi}. Thus Q' is a connected subset of B'. The theorem is proved by showing that for any nonnegative integer m, B' is an open and closed subset of B'. This implies Q' c B' for some nonnegative integer m. Then we show m = n. The fact that B'is both open and closed in B' follows in three steps Step 1. B' c interior U,m,Sj B. Let 0 E B' and suppose there exist m points P1, P2, , P,n, of a such that 0 E 7r(Pi) and Pi is not a vertex of a for
433
PROPERTIES OF SPHERICAL CURVES
i = 1, 2,
, m. We will show for each integer i, where 1 < i < m, there
exists a neighborhood Ni of Pi in a with the property that Ui = UPEN, 7r(P) fl B' is an open set of B' containing 0. Moreover, we may assume N1, N2, .. , N are mutually disjoint. It is then clear that u = n Ui is a neighborhood 1
of 0 in U,n,S; B';.
Consider the point Pi. Since Pi is not a vertex there exists an open neighborhood Ni of Pi in a on which k is strictly monotone. By Lemma 6, Ni does not contain Pj, where j i. Let P' and P2' be the boundary points of Ni. It follows from Lemma 6 that UP E w(P) is an open set of S ; it is the component of S - [w(Pi) U w(Pi')] containing Pi. Then Ui = UPE,, 7r(P) f1 B' is an open set of B'. In fact Ui is the component of B' - [7r(P') U 7r(Pz')] containing Pi. Clearly 0 E Ui since Pi E Ni. Step 2. B'm is closed in B'. Let Oi, i = 1, 2, , be a sequence of points in Bm approaching 0 E B'. Thus for each i = 1, 2, , there exist exactly m , m. By taking points Pil, Pie, , Pig, of a such that Oi E 7r(Pi;) for j = 1, 2, subsequences if necessary, we may assume that Pi; approaches a point P5 as i
approaches infinity for j = 1, 2, , m. By continuity 0 E 7r(P;) for j = 1, 2, , m. Thus there are at least m points of a whose osculating plane at each of those points passes through 0 unless P; = Pk for some j k. Suppose this ; then in any neighborhood of P; = Pk there exist the distinct points PiJ. Pik, for i sufficiently large. Since Oi e 7r(Pi,) f1 7r(Pik), w(Pi,) fl w(Pik) 0. By
Lemma 6, P; = Pk is a vertex of a. But this contradicts the assumption 0
Ui=17r(Vi). Thus P;
Pk for all j
k between 1 and m inclusive. By Step
1 there exist at most m points P P2, , P of a with 0 E 7r(P;). Step 3. Bm is open in B'. This step follows immediately from Step 1 and Step 2 since B'm = 0 for m > n by Theorem 4. We now know that Q' c B' where m < n. Suppose m < n. We will show this leads to a contradiction. Let o E a n d.'. Since 0 E Q', there exist m points Pl, P2, , m. In the notation of the proof , P with 0 E ir(Pi) for i = 1, 2, of Theorem 4, there exists an are ai for some integer between 1 and n inclusive with the following property : there exists no point Q E ai such that 0 E 7r(Q).
Thus w(Vi) and w(Vi+,) do not have 0 between them. Hence, say, w(Vi) and 0 are separated by w(V2}). In particular Vi and 0 are on opposite sides of w(Vi+). Thus a must meet w(Vi+) at points other than Vi,,. The converse follows from the remark following the proof of Theorem 4. q.e.d. It may still be that for every point 0 of Q' there exists the same number of points of a whose osculating plane at each of those points passes through 0
even though w(Vi) n a {Vi} for some integer i, 1 < i < n. For this to happen the following must be true : if, say, V1 is a vertex of a and w(V) intersects a in more than V1, then there must be another vertex Vi for some integer i, 2 < i < n, such that ir(Vi) = 7r(V1). Also, for points P near V, and Q near Vi, ir(P) and 7r(Q) must be on opposite sides of rr(V,) = 7r(Vi).
434
JOEL L. WEINER References
W. Fenchel, Uber Krummung and Windung geschlossener Raumkurven, Math. Ann. 101 (1929) 238-252. [2] H. W. Guggenheimer, Rev. #4787, Math. Rev. 39 (1970) 871. [ 3 ] -, Differential geometry, McGraw-Hill, New York, 1963. [ 4 ] B. Segre, Alcune proprietd differenziali in grande delle curve chiuse sghembe, [1]
Rend. Mat. (6) 1 (1968) 237-297.
[5 ] J. L. Weiner, A theorem on closed space curves, Rend. Mat. (3) 8 (1975) 789-804. UNIVERSITY OF HAWAII
J. DIFFERENTIAL GEOMETRY 12 (1977) 435-441
THE DIMENSION OF BASIC SETS JOHN M. FRANKS
Let f : M -p M be a C' diffeomorphism of a compact connected manifold M. A closed f-invariant set A C M is said to be hyperbolic if the tangent bundle of M restricted to A is the Whitney sum of two Df-invariant bundles, i.e., if TM = Eu(A) +O E3(A), and if there are constants C > 0 and 0 < 2 < 1 such that lD fn(V) l < CA" I v I I D f -n(V) I < CA" I V I
for v E Es, n > 0, for v E E", n > 0
The diffeomorphism f is said to satisfy Axiom A if (a) the non-wandering set
Q(f) = {x E M : U n U..>0 f "(U) # 0 for every neighborhood U of x} of f is a hyperbolic set, and (b) Q(f) equals the closure of the set of periodic points of f. If f satisfies Axiom A, one has the spectral decomposition theorem of U Al where Ai are pairwise disjoint, Smale [9] which says Q(f) = A, U f-invariant closed sets and f J,, is topologically transitive.
These Ai are called the basic sets of f, and it is the object of this article to investigate restrictions on their dimensions imposed by the homotopy type of f and the fiber dimensions of the bundles E8 and E. In [11] S. Smale showed that any diffeomorphism can be isotoped to a diffeomorphism satisfying Axiom A with all basic sets of dimension zero. This disproved earlier conjectures that some homotopy classes might contain only diffeomorphisms with a basic set of positive dimension. Theorem 1 below shows that if one restricts either the fiber dimensions of the bundles E" or the total number of basic sets for f, then there are indeed homotopy classes all of whose diffeomorphisms (subject to these restrictions) have basic sets of positive dimension. In Theorem 2 we investigate diffeomrphisms with a single infinite basic set, the others being isolated periodic orbits. It is a pleasure to acknowledge valuable conversations with R. F. Williams. We consider diffeomorphisms which in addition to Axiom A satisfy the no-
cycle property [10] which we now define. If Ai is a basic set of f then its stable and unstable manifolds ([5] or [9]) are defined by W3(Ai) = {x E M I d(f n(x), Ai) - 0 as n - oo} Communicated by R. Bott, July 11, 1975. This research was supported in part by NSF Grant GP42329X.
436
JOHN M. FRANKS
Wu(Ai) = {x E M d(f -'i(x), Ai) - 0 as
n,
co } .
One says Ai G A; if Wu(A;) fl WI(Ai) zf- 0. If this extends to a total ordering on the basic sets Ai, then f is said to satisfy the no-cycle property and we re-index so that Ai G A; when i < j. If Ai is a basic set of f : M , M then we define the index ui of Ai with respect to f to be the fiber dimension of Eu(Ai). All homology and cohomology will be singular with real coefficients unless otherwise stated. Theorem 1. If f : M , M satisfies Axiom A and the no-cycle property and Hk(M) zf- 0, then there is a basic set Ai satisfying dim Ai > Ik - u,,I where ui is the index of A. Hence, if f has fewer basic sets than nonzero cohomology groups, it must have a basic set of positive dimension, or equivalently : Corollary 1. If f has only basic sets of dimension zero, then there is a basic set Ai with index u1 = k for each k such that Hk(M) zf- 0. Theorem 2. Suppose f : M , M satisfies Axiom A and the no-cycle property and has one infinite basic set A, the others being isolated periodic orbits. If f*: Hk(M) --> Hk(M) has an eigenvalue which is not a root of unity, then dim A > I n - 2k I where n = dim M. It A is an attractor, then dim A > max {(n - k), k}. We note that M. Shub [8] has shown that whenever f * : H*(M) , H*(M) has an eigenvalue which is not a root of unity, then f must have at least one infinite basic set. In case M is the n-dimensional torus Tn we can strengthen Theorem 2 because either f * : HI(T11) --> HI(T11) has an eigenvalue which is not a root of unity or f* : H*(Tn) --> H* (T11) is quasi-unipotent (i.e., has only roots of unity as eigenvalues). Corollary 2. If f*: Tn , Tn satisfies Axiom A and the no-cycle property and has only one basic set A which is infinite, then either f*: H*(Tn) --> H*(Tn)
is quasi-unipotent or dim A > n - 2. It is not difficult to construct diffeomorphisms on Tn with a single infinite basic set of dimension n, n - 1, but the author does not know if there is a diffeomorphism of T3 which is not unipotent on homology and with a single infinite basic set of dimension one (dimensions 2 and 3 can be realized in this case). The hypothesis that f* not be quasi-unipotent on cohomology is necessary since it is easy to construct f : Tn , Tn homotopic to the identity with a single infinite basic set of dimension zero. We review briefly the filtrations of [10] associated with a diffeomorphism which satisfies Axiom A and the no-cycle property. It is possible to find submanifolds (with boundary and of the same dimension as M),
M=MZD ... DM,DMo=0, such that
437
DIMENSION OF BASIC SETS
Mi
Ai =
/U f(Mi) c int M1 , I
I
mEZ
f'(M1 - Mi_1)
Wu(Ai) U Mi_1 = Mi-1 U n fm(Mi) mZ0
Henceforth f : M -p M will be a diffeomorphism of a compact manifold D satisfying Axiom A and the no-cycle property and M = Mz D M1_, D M, = 0 will be a filtration for f. The proofs of Theorems 1 and 2 use the following proposition which may be of some independent interest. Proposition 1. Suppose f : M -p M satisfies Axiom A and the no-cyclic property and Ai C Mi - Mi_1 is a basic set of f. Let S = {k I fk Hk(Mi, Mi_1) Hk(Mi, Mi_1) has a nonzero eigenvalue}. Then dim Ai > max S - min S. We procede now with a sequence of lemmas leading to the proofs of the results above. We will use closed local stable and unstable manifolds of a point x E A, denoted Ws (x) and Wu(x) (see [5] or [9]). Since it is not in general true that dim (X X Y) = dim X + dim Y it is necessary to use the concept of cohomological dimension over R [3] defined as :
follows : If X is a compact Housdorff space, then dim, X = sup {k I Hk(X, A ; R) # 0} where A runs over all closed subspaces of X and ft' is Cech cohomology
with real coefficients. By a result of [7, p. 152] dim, X < dim X. Lemma 1. Suppose Ai C Mi - Mi_1 is a basic set for f and Mi, Mi_1 are
the elements of a filtration for f. If k > dim, Ws (Ai), then the map f*: Hk(Mi, Mi_1) - Hk(Mi, Mi_1) is nilpotent. Proof. This is essentially the same as [4, Lemma 6] which drew heavily on [1]. Let X = Wu(Ai) U Mi_1 and let Ilk denote Cech cohomology with real coefficients. We use the closed local unstable manifolds of [5]. The inclusion (Wu(Ai), aWu(Ai)) - (X, W) is a relative homeomorphism where W =
cl(X - Wu(Ai)). Hence by a standard result [12, p. 266], HN(WI. (A), aWu (Ai)) = Hk(X, W)
By definition of dim,, Hk(WE (Ai), aWu(Ai)) = 0 ,
when k > dim, Wu(Ai). Since W is compact and X C {f n,o f n(int Mi_1)} U Ai it follows that f -(W) C Mi_1 for some m > 0. The diagram
(X, Mi-0 - (X, W)
f'
If.
(X, Mi-1)
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JOHN M. FRANKS
commutes. Thus the map (f-)*: Hk(X, Mi_1) -* IIk(X, Mi_1) factors through Hk(X, W) so that (f'")* = (f*)- = 0 when k > dimR Wu(Ai). Now if f*: Hk(Mi, Mt_1) -* Hk(Mi, Mi_1) is not nilpotent, there is a subspace V # 0 with f *(V) = V. By [1, Lemma 1], the map h* is one-to-one on V where h*: Hk(Mi, Mi_1) = Hk(M1, Mi_1) -* Hk(X, Mi_1) is induced by the inclusion h : (X, Mt_1) -* (Mi, Mi_1). Thus we have a commutative diagram Hk(Mi, Mi-1) ('*) Hk(Mi, Mi-1) h*
H k(X,/Mi-1)
h* IHk(Xy.,
Mi-1)
But, (f*)mh*(V) = h*(f*)'nV = h*(V) 0, which is a contradiction if k > dimR Wu(Ai), since (f*): Hk(X, MT_1) -* E (X, M7_1) is zero in this case. Thus it must be the case that f*: Hk(Mi, Mi_1) -* Hk(Mi, Mi_1) is nilpotent when k > dimR Wu(Ai). q.e.d. If A is a basic set and x e A, we let Ws(x) = Ws(x) n A and WE (x) = WE (x) fl A. While it is true [9] that x e A has a neighborhood homeomorphic to WE(x) x WE (x), it appears to be an open question whether or not dim A = dim WE(x) + dim WE (x). For the cohomological dimension over R however we have the following. Lemma 2. Suppose A is a basic set for f, u = fiber dim Eu(A), and s = fiber dim ES(A). Then (a) dimR WE (A) = dimR WE(x) + u, (b) dimR W1,(A) = dimR WE (x) + s, (c) dimR A = dimR Ws (x) + dimR WE(x), where x is any point of A and e > 0 is sufficiently small.
Proof. We will use the following results from [13, Theorem 2.2 and Lemma 2.1]. If X and Y are compact Hausdorff spaces, then (1) dimR (X X Y) = dimR X + dimR Y, and (2) if n = dimR X, there exists a point p e X such that if U is any sufficiently small neighborhood of p in X, then Hn(X, X - U) # 0. Also if Y is a compact subset of X, then consideration of the exact sequence of the triple (X, Y, A), where A is a closed subset of Y,
Hn(X A) > IIn(Y A)
Hn+i(X, Y)
,
shows that dimR X > dimR Y.
We begin the proof of (a) by showing that dimR W(x) is independent of x E A. If y e A, then using the canonical coordinates [9, p. 781] for A and the fact that WS(orb (y)) is dense in A it is easy to show that WE(x) is homeomorphic to a compact subset of f-(WE(y)) for some m. This implies WE(x) is homeomorphic to a subset of WE(y) since f- is a diffeomorphism. Thus dimR WE(x) < dimR Wa(y) and the same argument shows dimR WW(y) < dimR WE(x).
DIMENSION OF BASIC SETS
439
By results of [6] there is a continuous map cp : A -* Emb (D, M) such that cp(z)(D) = Wa (z) where D is the disk of dimension u. The map i : Wa (x) X D -* Wa (A) given by J(y, t) = o(y)(t) is a homeomorphism onto a compact neighborhood K, of x in Wsu(A). But it is not possible that dimR Wsu(A):> dim K,, because the sets K,, cover Wu(A) and by (2) above together with excision at least one of them must have dimension over R equal to that of Wa (A). Thus dimR WE (A) = dimR Ws(X) + u for all x e A and (a) is proven. Applying this result to ;-1 proves (b). To prove (c) we consider the canonical coordinate map p : Ws (x) X Ws (x) A which is a homeomorphism onto a compact neighborhood J, of x in A. By (1) above dimR J, = dimR Ws (x) + dimR Ws (x). Since J. c A, dimR J,x G dimR A and again using (2) above and excision, it follows that dimR A = dimR Jx
for some x (and hence for all x since dimR Ws (x) and dimR Wa (x) are independent of x). Thus (c) is proven. q.e.d. Lemma 3. If A3 - A2 j - Al is a sequence of vector spaces exact at A2i ai : Ai -* Ai are linear maps commuting with i and j, and 2 is an eigenvalue of a2, then 2 is also an eigenvalue of either a3 or a1.
i
This is [4, Lemma 2] ; the proof is not difficult and will not be repeated here. Lemma 4. If 2 is an eigenvalue of f,* : Hk(M) -* Hk(M), then there is an Mi in the filtration for f such that fk : Hk(Mi, Mi_1) -* Hk(Mi, Mi_1) has .l as an eigenvalue. Proof. Consider the exact cohomology sequence of the triple
Hk(M, M,) - H'(M, M;-1) - Hk(M,, M,-1) There is a map f* induced by f on each of these groups, and these maps commute with the maps of the sequence. We now apply Lemma 1 to this sequence when j = 1. In this case the sequence is
Hk(M, M) _ H1(M) -* Hk(Mi Mo) ,
so either 2 is an eigenvalue of f* on Hk(M1, Mo) or an eigenvalue of f * on Hk(M, M). If the latter we set j = 2 and reapply Lemma 1 to show 2,, is an eigenvalue of f * on either Hk(M2i M) or Hk(M, M2). Continuing this procedure it follows that 2 is an eigenvalue of f* on Hk(Mi, M2_1) for some i, since Hk(M, M) = Hk(M, M) = 0. Proof of Proposition 1. Let k1= maxS. Then by Lemma 1, k1 G dimR WE (Ai) and by Lemma 2, dimR Ws (Ai) = dimR E (x) + u;, where x e Ai and ui =
fiber dim Eu(Ai), so k1 - ui G dimR WE(x). Let k = min S and let M; =
cl(M - M;). Then since f,* : Hk(Mi, M2_1) -* Hk(Mi, Mi_1) has a nonzero
eigenvalue, its adjoint f*,: Hk(Mi, M2_1) -* Hk(Mi, Mi_) has the same eigenvalue. Suppose M is orientable and n = dim M. Then [1, Lemma 4] shows gn_k : Hn-k(Mi 1, Mi) -* Hu k(M;_1, Mi) is similar to either f *k : Hk(Mi, Mi_1) -* Hk(Mi, Mi_1) or to - f *k. In either case gn-k
that if g = f : M -* M, 1
440
JOHN M. FRANKS
has a nonzero eigenvalue. Since g has the same basic sets as f (with Ws(f ; Ai) Wu(g;' Ai)) and M =1110 Ml D ... D11h = 0 is a filtration for g, we can apply to g the argument which showed k1 - ui < dimR W:(x). We have then that (n - k) - fiber dim Eu(g ; Ai) < dimR WE (g ; x) or (n - k) - si < dimR rV (f ; x) where si = fiber dim Es(f ; Ai). Adding this inequality to the one for k1 we have k1
- ui + (n - k) - si < dimR ' (x) + dimR Wu(x)
.
Since n = ui + si, k1 - k < dimR A by Lemma 2. That is, max S - min S < dimR Ai < dim Ai. In case M is not orientable, we let - : M - M be an oriented double cover of M and f : M --+ M a lift of f. If Ai = 7r-'(Ai) and Mi = 2r-'(Mi), then the Ai have all the properties of basic sets for f except they may not be topologically transitive. But f together with the nontrivial covering transformation on M will be transitive, and this is sufficient for everything we have done. So exactly as above, we use the filtration Mi and prove the result for Ai (-,r*: H J(Mi,
Mi_) - HJ(Mi, Mi_1) is surjective-see [1, Theorem 1]). Since dim Ai =
dim A, this completes the proof. Proof of Theorem 1. If 2 # 0 is an eigenvalue of f*: Hk(M) - HI(M) then by Lemma 4 there is an i such that 2 is an eigenvalue of f * : H'(Mi, Mi_) -> Hk(Mi, Mi_). Now if ui = fiber dim Eu(Ai), then from the proof of Proposition 1 we have k - ui < dimR WE(x) and ui - k = (n - k) - si < dimR Wu(x) for x E A. Since dim Ai > dimR Ai = dimR WI(x) + dimR Wu(x)
>max{(k-ui),(ui-k)}=Ik-uiI, the proof is complete. Proof of Theorem 2. If Ai C Mi - Mi_1 is a periodic orbit of period p, then fP fixes each point of Ai and Df2P preserves an orientation on EU(A). Let g = f2P. Since dim Ai = 0, it follows from the proof of Theorem 1 or from [1, Theorem 1] that g,* : Hk(Mi, Mi_1) Hk(Mi, Mi_1) is nilpotent unless k = fiber dim Eu(Ai). Now let L(g) _ k=o (-1)k tr (g,*) _ (-1)1 tr (gu*) where u = fiber dim Eu(Ai). By Lefschetz fixed point theory (see [4, Lemma 3] and [2, Theo-
rem 4.1]). L(g) = Z2,,, I(g; q) where I(g;q) denotes the index of q under g, which by a result of [9, p. 767] is (-1)u. Hence (- 1)u tr (gu)* = L(gm) = (- 1)up for all m > 0. That is, tr (gu)* = p for all m > 0, and it follows that the only nonzero eigenvalue of gu is 1, with multiplicity p. This is because the nonzero eigenvalues with multiplicity of a matrix A are determined by the poles of exp (Em=1 (tr Am)zm/m) (see [1] or [9]) and hence gu has the same nonzero eigenvalues as the p x p identity matrix. Consequently every nonzero eigenvalue of f*: H*(Mi, Mi_1) - H*(Mi, Mi_1) is a root of unity when Ai is
DIMENSION OF BASIC SETS
441
finite. This argument is essentially a reproof of a result of M. Shub [8]. Suppose now that M is orientable. If 2 is an eigenvalue of fk : Hk(M) --> Hk(M) which is not a root of unity, then it follows by Poincare duality (see [1, Lemma 4]) that f* : H,l_k(M) -+ Hn_k(M) has an eigenvalue ±2-1 and hence fn_ : H k(M) -+ H k(M) has an eigenvalue which is not a root of unity. Hence, if A C M, - MS_, is the infinite basic set, then f* : Hj(M8, M,-,) Hj(M,, MS_,) has an eigenvalue which is not a root of unity when j = k and when j = n - k. This follows from Lemma 4 and the fact shown above that f*: H*(Mi, Mti_1) -+ H*(Mi, Mi_1) has only roots of unity and zero as eigenvalues when i s. Thus by Proposition 1, dimA > (n - k) - k if n - k
> kanddimA> k - (n - k) if k> n - k so in any case dimA> In-2kj. If A is an attractor, then the filtration can be chosen such that (Ms, M,-1) = (M1, M° = 0) so f * : H°(Ms, MS _) = H°(M) - H°(MI) is nontrivial and it follows from Proposition 1 that dim A > max {(n - k), k}. This proves the theorem in the case M is orientable. If M is not orientable, let ir : M -+ M be an oriented two-fold covering of M and let f : M -+ M cover f. The map r* : Hk(M) -+ Hk(M) is surjective (see [1, Theorem 1]) so 7r* : Hk(M) , Hk(M) is injective and it follows that f*: Hk(M) , Hk(M) has an eigenvalue which is not a root of unity. Now if Ai = it-I(Ai) it may be that f : Ai -+ Ai is not topologically transitive, but the proof for the orientable case applied to f : M -+ M (using the filtration Mi = 7r-'(Mi)) still shows that if A =7r-'(A) then dim A > I n - 2kI and that if A is an attractor then dim A > max {(n - k), k}. Since dim A = dim A, the result follows. References
R. Bowen, Entropy versus homology for certain diffemorphisms, Topology 13 (1974) 61-67. A. Dold, Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology 4 (1965) 1-8. E. Dyer, On the dimension of products, Fund. Math. 47 (1959) 141-160. J. Franks, Morse inequalities for zeta functions. M. Hirsch & C. Pugh, Stable manifolds and hyperbolic sets, Proc. Sympos. Pure Math., Vol. IV, 1970, 133-163. M. Hirsch, J. Palis, C. Pugh & M. Shub, Neighborhoods of hyperbolic sets, Invent.
Math. 9 (1970) 121-134. W. Hurewicz & H. Wallman, Dimension theory, Princeton University Press, Princeton, 1941. M. Shub, Morse-Smale diffeomorphisms are unipotent on homology, Proc. Sympos. Dynamical Systems, Salvador, Academic Press, New York, 1973.
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747-817. The 0-stability theorem, Proc. Sympos. Pure Math., Vol. IV, 1970, 289-297Stability and isotopy in discrete dynamical systems, Proc. Sympos. Dynamical Systems, Salvador, Academic Press, New York, 1973. E. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. NORTHWESTERN UNIVERSITY
J. DIFFERENTIAL GEOMETRY 12 (1977) 443-460
ISOMETRY OF RIEMANNIAN MANIFOLDS TO SPHERES KENTARO YANO & HITOSI HIRAMATU
1.
Introduction
Let M be a differentiable connected Riemannian manifold of dimension n. We cover M by a system of coordinate neighborhoods {U ; xh}, where and in , n}, and denote the sequel indices h, i, j, k, - run over the range {1, 2, by gji, 17j, Kkjih, Kji and K the metric tensor, the operator of covariant differentiation with respect to the Levi-Civita connection, the curvature tensor, the Ricci tensor and the scalar curvature of M respectively. An infinitesimal transformation vh on M is said to be conformal if it satisfies (1.1)
Yvgji = vjvi + vivj = 2pgji
(vi = gihvh)
for a certain function p on M, where 2v denotes the operator of Lie derivation with respect to the vector field v (see [6]). When we refer in the sequel to an infinitesimal conformal transformation v, we always mean by p the function appearing in (1.1). When p in (1.1) is a constant (respectively, zero), the infinitesimal transformation is said to be homothetic (respectively, isometric). We also denote by YDP the operator of Lie derivation with respect to the vector field pi defined by
pi = gihph = Iip
(1.2)
where (1.3)
I7 = 9ihI h ,
ph = I7hp
gih being contravariant components of the metric tensor. We use gji and gih to lower and raise the indices respectively. The problem of finding conditions for a Riemannian manifold admitting an infinitesimal conformal transformation v to be isometric to a sphere has been extensively studied. For the history of this problem, see [7] and [8]. But in almost all the results on this problem the condition K = constant or Y, K = 0 is not assum0 has been assumed. As results in which the conditon ed, Sawaki and one of the present authors [12] (see also [11]) proved the following two theorems, in which and the remainder of this section, unless stated Communicated July 26, 1975.
444
KENTARO YANO & HITOSI HIRAMATU
otherwise, M will always denote a compact oriented Riemannian manifold of dimension n > 2 admitting an infinitesimal nonhomothetic conformal transformation v. Theorem A. M is isometric to a sphere if v satisfies (1.4)
Y, IY,,(IIGII- n -2 4K" + 2(n n1)(n2 2) 42vK] = 0
where (1.5)
Gji = Kji - 1nKgji
(1.6)
II G112 = GjiGji
,
4 = gjiF"Fj denoting the Laplacian. Theorem B. M is isometric to a sphere if v satisfies (1.7)
I ZII2 - n
+ 24K + 8(n + 1)42 K] = 0 ,
where
Zkjih
(1.8)
1
= Kkjih -
(1.9)
n(n - 1) K(3kgji - ajgki) Zk,,IZk ih .
IIZII2 =
Recently Amur and Hegde [2] (see also [3]) proved the following two theorems.
Theorem C. M is conformal to a sphere if v satisfies 2Dp2VK = 0 and (1.10)
J
(GpipZ
+
12 2v2DPKl dV > 0 ,
n
/
where 2Dp denotes the operator of Lie derivation with respect to pi and dV the volume element of M. Theorem D. M is conformal to a sphere if v satisfies 2Dp2vK=O, 2v2DPK
>0and2vJIGI12=0.
Very recently the present authors [9] proved the following two theorems. Theorem E. M is isometric to a sphere if v satisfies 2, IIG112 = 0 and (1.11)
J
KpipidV
>
1
2n(n - 1) J,'
[2np2K2 + (n + 2)pK2vK + (2vK)2]dV
.
ISOMETRY OF RIEMANNIAN MANIFOLDS
445
Theorem F. M is isometric to a sphere if v satisfies. Yv IJZ III = 0 and (1.11). All the above theorems have been obtained by applying the following Theorem G of Tashiro [5]. The purpose of the present paper is to continue the joint work of the present authors [9] and to prove some propositions on isometry of Riemannian
manifolds to spheres, in which the operator of Lie derivation t plays an important role. In the sequel, we need the following theorems. Theorem G (Tashiro [51). If a complete Riemannian manifold M of dimension n > 2 admits a complete infinitesimal nonhomothetic conformal transformation v such that (1.12)
PA - 1 n4Pgji = 0
then M is isometric to a sphere. Theorem H (Yano and Obata [10]. See also Obata [4]). If a complete Rie-
mannian manifold M of dimension n > 2 admits a nonconstant function p satisfying
(1.13)
V 1p1 - 1JPg11 = 0 , n
2D,K = 0 ,
then M is isometric to a sphere. We remark here that if a Riemannian manifold M of dimension n is isometric to a sphere, then M admits not only an infinitesimal nonhomothetic conformal transformation v satisfying (1.1) and (1.12) but also a nonconstant function p satisfying (1.13). 2.
Lemmas
In this section we prove some lemmas which we need in the next section. M is supposed to be a compact oriented Riemannian manifold of dimension n in all the lemmas except in Lemmas 4, 5, 6, 9 where M is supposed to be only a Riemannian manifold. Lemma 1. If M admits an infinitesimal conformal transformation v, then, for the function p appearing in (1.1) and for an arbitrary function f on M, we have
(2.1)
Proof.
Jpfdv
n J,x 2vfdV .
Since np = Ptvt, by Green's theorem (see [7]) we have
446
KENTARO YANO & HITOSI HIRAMATU
dV
0 = SM V(fvt)dV = SM which proves (2.1). Lemma 2. In M we have
f 'D fhdV = (2.2)
M
J .x
° 'DhfdV
+n
pfdV ,
= f x (Fif)(Fih)dV
=-f JM f4hdV=-f h4fdV M for any functions f and h on M, where IDf denotes the operator of Lie derivation with respect to the vector field Vif on M. Proof. This follows from
0 = f vi(fVih)dV = f (Vif)(Vih)dV + f f4hdV M x
,
J lx
0 = f Mvi(hvif)dV = Lemma 3.
x
(vih)(vif)dV + f h4fdV
.
M
In M we have
f
(2.3)
M
24KdV = -2 fM ppiViKdV
for any function p on M, K being the scalar curvature of M.
Proof. We have (2.3) by putting f = K and h = p2 in (2.2). Lemma 4 (Yano [7]). M, we have (2.4)
S VKkjih
For an infinitesimal conformal transformation v in
= -okV jpi + oJVkpi - Vkplgji + I7jp"gki
(2.5)
Y0Kji = -(n - 2)Vjpi - dpgji
(2.6)
Y,,K = -2(n - 1)4p - 2pK
.
Proof. We can prove these by using (1.1) and the following formulas on Lie derivatives :
v{jhi} = 5j pi + ( pj - gjip" , 2VKk ji" = 1 k2v{ j'ti} - I jSV{k'ti} {jh i} denoting Christoffel symbols formed with gji. Lemma 5. For an infinitesimal conformal transformation v in M, we have
(2.7)
°tvGji = -(n - 2)(1jpi
- 1n dpgji
447
ISOMETRY OF RIEMANNIAN MANIFOLDS
°z' vZk jih = -( I jpi + 3T kpi - vkPhgji + V jP'Lgki (2.8) +I
2
-4P(okgji - 3.% ) n
where Gji and Zkjih are defined by (1.5) and (1.8) respectively. Proof. These follow from Lemma 4. Lemma 6. If M admits an infinitesimal conformal transformation v, then for any function f on M we have
42vf = 1,;4f + 2p4f - (n - 2)pi7if .
(2.9)
Proof.
For an infinitesimal conformal transformation v, we have (see [71)
9kjvkvjvh + Kzhvi + n - 2 Vh(vtvt) = 0
(2.10)
n
Thus we obtain (2.9) by using (2.10) and the identity
gjiFjvtivhf - KhZvif = vh(4f) , which holds for any function f on M. Lemma 7. If M admits an infinitesimal conformal transformation v, then YVYDPKdV
(2.11)
m
n f n+2 if
n
n+2 fm (2.12)
fm p42vKdV ,
$M
and consequently (2.13)
fm
J
Y[v,D,]KdV
n n 2 J M per" 4 KdV + 2(n -} 2)
JM
p4LvKdV ,
where Dp denotes the vector field pi, and [v, Dp] the commutator of vector fields v and Dp. Proof. Using Lemmas 1, 3 and 6, we have
fm pYv4KdV = fm p4Y,KdV - 2 fm p24KdV + (n - 2) fm ppTTiKdV
=
f M
p4Y,KdV + (n + 2) f pYD,KdV
448
KENTARO YANO & HITOSI HIRAMATU
= f p4Y KdV - n
n
'V 'DPKdV ,
2 SM
which proves (2.11). (2.12) follows immediately from Lemma 2. Lemma 8. In M we have, for any function p on M, (2.14)
(2.15)
fm KjipjpidV = - 2 fm Kjipjp'dV 4JM
Proof.
(2.16)
fx p(YDpKji)gjidV
pYDPKdV -
1f
4 Jar p(
jidV DKkjih)gkg p
From the definition of K it follows that fm pYDpKdV = fm
pyDp(Kjigji)dV
pKjiYDpg'idV
fm p(YDpKji)gjidV + SM
.
On the other hand, since pi is a gradient, we have (2.17) (2.18)
YDpgji = -2Vjpi
YDpgji = 2Vjpi ,
Vj(ppiKji) = Kjipjp2 + pKjiV'pi + 2pp'ViK
where we have used FjK ji = IF jK. Using (2.16), (2.17) and (2.18), we have (2.14). We also have
(2.19)
fm p2DPKdV = fm pYDp(Kk jingkhgji)dV
= fm p(YDpKkjih)gkhgjidV - 4 f IV pKjiVjpidV from which and (2.18), (2.15) follows immediately. Lemma 9. In M we have, for any function p on M,
Kjipjpi + n (dp)2 + 2Dp4p(2.20)
Proof.
-(Fjpi -
n
Jpgji)(vjpi
14YDPp
-
Using Ricci formula we have 42',,p = gkivkv j(pipi) = 2gkiV k(piV Jpi)
n4pgiil 1
449
ISOMETRY OF RIEMANNIAN MANIFOLDS
= 2gk'(VkVJpi)pi + 2(v>Pi)(V1pi)
=
2gki(VivkP> - Kkijhph)pi + 2(v1Pi)(VJpi)
from which we find (2.20). Lemma 10. In M we have, for any function p on M,
SM Kp1p1dV + n n 1 fm YDPJpdV (2.21)
-f
(vp,
-
4Pgji(Vipi
n
4Pgii)dV ,
-n
or
fm KlipjpidV - n (2.22)
-J a2
Proof.
(v3Pi
n
1 fm (dp)W
- n i pgji
k7i
1
P
- n zPgui)dV
.
These follow from Lemmas 2 and 9.
Lemma 11. A sphere S" of dimension n > 2 admits a nonconstant function p such that (2.23)
v j pi
=0 - -Jpgji n
and consequently (2.24)
(2.25)
42p +
n
1
1
Kd p= 0,
v;vid p+
n
1
1
KV,pi = 0
vivizp - 1 42Pgii = 0 . n
Proof. It is known [11] that Sn admits a nonconstant function p such that (2.23) holds. This shows that the vector field pi defines an infinitesimal non-
homothetic conformal transformation on Sn with the associated function (1 /n)d p. Since K is a positive constant, using (2.6) in which v and p are replaced by ph and (1 / n)d p respectively we have the first equation of (2.24) and
therefore J p + (1 /(n - 1)) pK = c (c: constant), which implies the second equation of (2.24). From (2.23) and (2.24) we obtain (2.25). 3.
Propositions
In this section, we prove a series of propositions in which the operator of Lie derivation YDP plays an important role. M is supposed to be a compact
450
KENTARO YANO & HITOSI HIRAMATU
oriented Riemannian manifold of dimension n admitting an infinitesimal conformal transformation v in all the propositions and corollaries except : in Pro-
position 4 where M is supposed to be a complete Riemannian manifold of dimension n > 2, in Propositions 5, 7 and Corollary 5 where M is supposed to be a complete Riemannian manifold of dimension n > 2 admitting a complete infinitesimal nonhomothetic conformal transformation v, in Propositions 6, 12 and 13 where M is supposed to be only a Riemannian manifold, and in Propositions 8, 10 and Corollaries 1, 3 where M is supposed to be a compact oriented Riemannian manifold of dimension n. Proposition 1. For M we have (3.1) fm
1
G,ip5pidV -I 1 fM
f"'
0
2n
n2
The M of dimension n > 2 admits a nonhomothetic v such that the equality in (3.1) holds if and only if M is isometric to a sphere. Proof. By using (1.5), (2.6), Lemmas 1 and 2 and the identity
f Vi(pp'K)dV = fly KpipidV + f pK4pdV + f ppiViKdV = 0 ,
(3.2)
we have
f
M
K,ip3 pidV - n n 1 fm
= fM G,ipjpidV + =
fM G,i pjpidV
n-1 n
M
-
n
n
p)ZdV
f m KpipidV f
n n
pYDPKdV M
n
1 fM
p)2dV
fm pKJpdV
(4p)2dV
= fMGjip'pidV +n21 fM
= f G,ip'pidV +
1
f"'
n2
12n f 12n f
m
(d
M
Thus from Lemma 10 we obtain
f
M
(3.3)
G,ip'pidV + -1
n2
f YvYDPKdV - i f YDPY KdV 2n M
-fm (Fjpi - n d pgji} (V ipi
M
- n d pg'i)dV
ISOMETRY OF RIEMANNIAN MANIFOLDS
451
which implies (3.1). If the equality in (3.1) holds, then from (3.3) and Theorem G it follows that M is isometric to a sphere. Conversely, if M is isometric to a sphere, M admits an infinitesimal nonhomothetic conformal transformation v such that the equality in (3.1) holds because, for a sphere, G;i = 0 and K is a positive constant. Proposition 1 is a generalization of Theorem C. Proposition 2. If the dimension n of M is greater than 2, then
SM2v2MG dV - (n - 2) J
(3.4)
0.
,
ar
The M of dimension n > 2 admits a nonhomothetic v such that the equality in (3.4) holds if and only if M is isometric to a sphere. Proof. First of all we have -Tv G I = 2(-TvG.')G'i - 4p I GII'. 2
Substituting (2.7) in the above equation we find
-Tv 11GII2 = -2(n - 2)G,iVjpi - 4p GII2 ,
because of G;igjz = 0 or (3.5)
KjiV'p2
=- n
2 2 p I G 1I2 -
2(n
I
1
2)
-T,, G I I2 + 1 Kd p.
Using (2.18) and (3.5) we have P'(ppiKji) = K.jip'p2
-
2
P' IIGII'
--2(n
1
n
2
2) p-Tv I I G I12 + 1 p-TDPK + pKd p .
Integrating both sides of the above equation over M and using (2.6) and Lemmas 1 and 2, we obtain JMK;ipipzdV - n n
2
n
+
M(dp)'dV
2 Jx p' IIGII' dV -
2n J M
2
n
1J
2JM
2n(n
1
2)JM Y,Y, IIGIII, dV
YvYDPKdV - 1 f"' pKJpdV - n n 1 f p)'dV m (d p' G j j' dV - 2n(n1-
2)
f"' Yvtv
11 G I IZ dV
452
KENTARO YANO & HITOSI HIRAMATU
+ 2n SM YvYDPKdV - 2n fm
YDPYvKdV
or, by Lemma 10, II G IIZ dV - (n - 2) fm 2-'[V,DP]KdV
fm
= 2n(n - 2) f"' (v1pi +4nfm p2I
- n 4Pg;i) (V ipi - n d pgii )dV
G I IZ dV,
which together with Theorem G gives the proposition. Remark 1. Proposition 2 is a generalization of Theorem D. Using (2.13) and Lemma 1 we have SM (3 6)
f Y,Yv4KdV
1
n+2JM
- 2(n + 1) f n(n+2) M
Therefore Proposition 2 is essentially equivalent to Theorem A. Using (2.6), (3.2) and Lemmas 1 and 2 we have n SM KpipidV
SM Y[ (3.7)
- 2(n
1
1)
f
[2np2K2 + (n + 2)pKtvK + (IvK)Z]dV , M
which implies that Proposition 2 is essentially equivalent to Theorem E. Proposition 3. For M we have (3.8) SM
Yv1v
IZIIZ
dV - 4
fm
0.
The M of dimension n > 2 admits a nonhomothetic v such that the equality in (3.8) holds if and only if M is isometric to a sphere. Proof. First of all we have
yv IIZII2 = 2(YvZkj )Zkiih - 4p IIZIIZ Substituting (2.8) in the above equation we find yv IIZIIZ = -8G5iV'pi - 4p IIZIIZ
because of Zk,ik = G,i and G;ig'i = 0, or
453
ISOMETRY OF RIEMANNIAN MANIFOLDS
(3.9)
2 p II Z II2 - 1 Y,, it Z I2 + n K4 p .
KjiV''pi
8
Using (2.18) and (3.9) we have
V'(pp Kji) = Kj,pipi - 1 p2 JZIJ2
- 1 Pyv Al, + 1 pYDPK + 1 pKdp
.
Integrating both sides of the above equation over M and using (2.6) and Lemmas 1 and 2, we obtain fm KjipjpidV
- n n 1 fm (4p)zdV JarY"Y'IZI zdV
farp2jjZIj2dV-
2
1
+ 2n Jar
YVYDPKdV -
2n
far YDPYVKdV
,
or, by Lemma 10, p
f if
2' 2v II Z II2 dV - 4 Jar'[,,,DP]KdV 8n far
(Fjpi -
ndpgji)(V3p. - n4pgi )dV + 4n f
p2 JJZJI2dV ,
M
which together with Theorem G gives Proposition 3. Remark 2. Using (3.6), (3.7) and (3.8) we see that Proposition 3 is essentially equivalent to Theorems B and F. Proposition 4. M admits a nonconstant function p satisfying (3.10)
2'DPgji = 2pgji
I
eL DPK = 0
cp being a function on M, if and only if M is isometric to a sphere. Proof. If M admits a nonconstant function p satisfying (3.10), then, by Theorem H, M is isometric to a sphere because (3.10) is equivalent to (1.13). Conversely if M is isometric to a sphere, then M admits a nonconstant function p satisfying (2.23) and hence (3.10) because K is a positive constant for a sphere. Proposition 5. M admits a transformation v such that YDPgji = 2pgji cp being a function on M, if and only if M is isometric to a sphere.
454
KENTARO YANO & HITOSI HIRAMATU
Proof. This follows immediately from Theorem G. Ackler and Hsiung [1] proved this proposition for a special case in which the manifold M is compact and oriented and both SfVK = 0 and SfDPK = 0 hold. Proposition 6. For any function p on M we have (3.11)
Kjipjp1 + n (4p)2 + SfDPQp -
2
J- D,ap < 0 .
The complete M of dimension n > 2 admits a nonconstant function p such that the equality in (3.11) holds and SfDPK = 0 if and only if M is isometric to a sphere. Proof. This follows from Theorem H and Lemma 9. Proposition 7. M admits a transformation v such that the equality in (3.11) holds if and only if M is isometric to a sphere. Proof. This follows from Theorem G and Lemma 9. Proposition 8. For any function p on M we have (3.12)
J p(SDpKji)gjidV +
2(n
-
1)
n
f pJ2pdV > 0 . x
The M of dimension n > 2 admits a nonconstant function p such that 9DPK = 0 and the equality in (3.12) holds if and only if M is isometric to a sphere. Proof. Using Lemmas 2, 8 and 10 we have tit
p(YDpKji)gjidV +
(3.13)
2 rM (Fjpi
2(n
-
1)
fm
n
pd2pdV
- n d pgjz)(vjp2 - n d pgii)dV
,
which together with Theorem H gives Proposition 8. Corollary 1. M of dimension n> 2 admits a nonconstant function p such
that 9DPK = 0 and (3.14)
YDpKji
= - 2(nn- 1) d2pg;i 2
if and only if M is isometric to a sphere. Proof. If M is isometric to a sphere, then M admits a nonconstant function p such that (2.23) holds. Therefore using (2.24) we have 1
2
n
n
YDpKji = -KYDpgji = 2 n2
Kdpg;i = -
KV jpi
2(n - 1) dZpg;i . n2
455
ISOMETRY OF RIEMANNIAN MANIFOLDS
The "only if" part of the corollary is an immediate consequence of Proposition 8. Remark 3. By (2.25) in Lemma 11, (3.14) in Corollary 1 can be replaced by
2(n - 1) PjVidp .
YDPKji
(3.15)
n
Proposition 9. For M we have (3.12), and the M of dimension n > 2 admits a nonhomothetic v such that the equality in (3.12) holds if and only if M is isometric to a sphere. Proof. This follows from (3.13) and Theorem G.
Corollary 2. M of dimension n > 2 admits a nonhomothetic v such that (3.14) holds if and only if M is isometric to a sphere. Proof. This follows from Lemma 11 and Proposition 9. Remark 4. By (2.25) in Lemma 11, (3.14) in Corollary 2 can be replaced by (3.15). Proposition 10. For any function p on M we have p(yDPKkjih)gkhgjidV + fm
SM
(3.16)
P
+ 4(n - 1) j pdzpdV > 0
DpKdV .
JM
n
The M of dimension n > 2 admits a nonconstant function p such that IDPK = 0 and the equality in (3.16) holds if and only if M is isometric to a sphere. Proof. Using Lemmas 2, 8 and 10, we have SM pl
DpKk jih )gkhgjidV + SM p
DPKdV +
4(n
- 1) n
pd2pdV
SM
(3.17)
4 f", (17jpj - n dpgji)(P'pi - n dpgji)dV , which together with Theorem H gives the proposition. Corollary 3. M of dimension n > 2 admits a nonconstant function p such
that fDPK = 0 and (3.18)
o 7 °L
4
n2
2
p(gkhg ji - gjhgki)
if and only if M is isometric to a sphere. Proof. If M is isometric to a sphere, then M admits a nonconstant function p such that (2.23) holds. Since K is a positive constant and
456
KENTARO YANO & HITOSI HIRAMATU
Kkjih = n(n
1-
K(
1)
(gkhgji - gjhgki)
for a sphere, using (2.24) we obtain / + gkhvjpi - vjphgki - gjhvkpi) - n(n2- 1) K(Fkphgji
n
(VkFhd pgji + gkhvjviJp - vjvh4pgki - gjhvkvid p)
which together with (2.25) gives (3.18). The "only if" part of the corollary is an immediate consequence of Proposition 10. Remark 5. As is seen in the proof of Corollary 3, (3.18) in Corollary 3 can be replaced by ° 'DpKkjih
(3.19)
2
n
(vkvhdpgji + gkhvjvidp - vjvhJPgki - gjhvkvidp)
Proposition 11. For M we have (3.16). The M of dimension n > 2 admits a nonhomothetic v such that the equality in (3.16) holds if and only if M is isometric to a sphere. Proof. This follows from (3.17) and Theorem G. Corollary 4. M of dimension n > 2 admits a nonhomothetic v such that ° 'DpKkjih
(3.20)
_ - (
1
)
1
[DPK
4(n -
n
Zp
gkhgji -gjhgki)
holds if and only if M is isometric to a sphere. Proof. This follows from Lemma 11 and Proposition 11. Remark 6. In Corollary 4, we see, by using Lemma 11, that (3.20) can be replaced by DpKk jih 1
(3.21)
hgji - gjhgki)
n(n - 1) n
(Vkvhdpgji + gkhvjviJp - vjvhdpgki - gjhvkvidp)
Proposition 12. If M of dimension n > 2 admits an infinitesimal conformal transformation v, then (3.22)
(
Dp°L vGji)gji < 0 .
ISOMETRY OF RIEMANNIAN MANIFOLDS
457
The complete M of dimension n > 2 admits a complete infinitesimal nonhomothetic conformal transformation v such that the equality in (3.22) holds if and only if M is isometric to a sphere. Proof. By using (2.7) we have
(YvGji)gji = 0
,
and consequently
(IDp2VGji)gji = -(YvGji)YDpgji = 2(2vGji)Fjpi
_ -2(n - 2)(Fjpi - 1 dpgjiFjPi n
_ -2(n - 2)(Fjpi
-
dpgji)(V'pi
-
n
dpgii)
n which together with Theorem G gives the proposition. Proposition 13. For M of dimension n > 2 we have
(3.23)
(2'Dp2VZkjih - 2pYDpZkjih)gkhgji < 0
The complete M of dimension n > 2 admits a complete nonhomothetic v such that the equality in (3.23) holds if and only if M is isometric to a sphere. Proof. From (2.8) it follows that
YvZkjih = -gkhvjpi + gjhvkpi - Vkphgji + vjphgki 2 n
+ dp(gkhgji - gjhgki) + 2pZkjih
,
and therefore that (YvZkjih)gkhgii = 0 . Using this we obtain (YD yVz"jih)gkh9ji
= 4(yvZkjih)gjipkph 1 i = -4(n - 2) Fjpi - -dpgjiFjp
n
+ 8pZkjing'ipkph .
On the other hand, since Zkjingkhgji = 0 we have (YDpZkjih)gkhgji = 4Zkjingj'Fkph
Thus
-
1
n
apgii)
458
KENTARO YANO & HITOSI HIRAMATU
(YDPYVZkjih - 2'PYDpZkjih)gkhgji
_ -4(n -
2)(Pjpi
-
4Pgji )('Pi P
-
n
4pgii)
n
which together with Theorem G gives the proposition. Corollary 5. M admits a transformation v such that eZ. DpeZ. VGji = 0
(3.24)
or
2p°Z' DpZkjih = 0
(3.25)
if and only if M is isometric to a sphere. Proof. This follows from Propositions 12 and 13. Proposition 14. For M we have (3.26)
JM
P(YDpGji)gjidV - n
J
Y[v,Dp7KdV > 0
The M of dimension n > 2 admits a nonhomothetic v such that the equality in (3.26) holds if and only if M is isometric to a sphere. Proof. We have, by using Gjigji = 0,
p(1DpGji)gji = -pGjiyDpgji = 2pGjiVjpi (3.27)
= 2pK jiV jpi -
2
n
PK4P
or, using (2.18), 2
p(
DpGji)gj' = Vj(ppiKji) - Kjipjpi - 2 pYDPK
- n pKdp
.
Integrating both sides of the above equation over M and using (2.6), we find
fm K jipjpidV - n n 1 fm (d p)2dV 2 Jar
p(YDpGji)gjidV - 2 f"' P2DpKdV
- 1n J pKJ pdV M
2 J a1
n-1 n
f (d p)2dV Jar
p(YDpGji)g'idV + 2n
f" YVYDpKdV
ISOMETRY OF RIEMANNIAN MANIFOLDS
+ 2n
459
SM (4p)2vKdV ,
or, by Lemmas 2 and 10,
1 SM 2'[v,DP]KdV (17jpi - n 4Pgji)(ViPi - n 4Pgii)dV 2 SM
SM p(2DPGji)gjidV (3.28)
which together with Theorem G gives the proposition. Corollary 6. M of dimension n > 2 admits a nonhomothetic v such that 1
p22DPGji =
(3.29)
nZ
if and only if M is isometric to a sphere. Proof. This is an immediate consequence of Proposition 14. Corollary 7. M of dimension n > 2 admits a nonhomothetic v such that
'DPGji = -
(3.30)
1
n(n
2)
[4K - 2(n n 1) dYvK]gji
if and only if M is isometric to a sphere. Proof. This follows from Lemma 7 and Proposition 14. Proposition 15. For M we have (3.31)
P(yDpZkjih)gk''gjidV -
fm
n JM
'[v,Dp]KdV > 0 .
The M of dimension n > 2 admits a nonhomothetic v such that the equality in (3.31) holds if and only if M is isometric to a sphere. Proof. We have, by using Zkjih,gk't = Gji and Gjigji = 0, p(YDPZkjih)gkhgji = -2pGjiyDpgji which together with
p(YDPGji)gji = -pGji2Dpgji implies
p(2'DpZk jih)gkhgji = 2p(yDPGji)gji
Integrating both sides of the above equation over M and using (3.28), we obtain M
P(YDpZkji3a)gkhgjidV
-2
n SM
Y[v,DPI KG.Y
460
KENTARO YANO & HITOSI HIRAMATU
4 J" (vpi
- n Jpgji )(17ip1 - n JPgji" Idv
,
which together with Theorem G gives the proposition. Corollary 8. M of dimension n > 2 admits a nonhomothetic v such that (3.32)
D,Zkjih =
p
2
nz(n - 1)
(_T[v,Dp]K)(gkhgji - gjhgki)
if and only if M is isometric to a sphere. Proof. This is an immediate consequence of Proposition 15. Corollary 9. M of dimension n > 2 admits a nonhomothetic v such that cfDoZkjih
(3.33)
2
n(n - 1)(n + 2) kvJK
- 2(n n 1) d-TvKj (gkhgji - gjhgki) ,
if and only if M is isometric to a sphere. Proof. This follows from Lemma 7 and Proposition 15. Bibliography [ 11
L. L. Ackler & C. C. Hsiung, Isometry of Riemannian manifolds to spheres, Ann.
Mat. Pura Appl. 99 (1974) 53-64. K. Amur & V. S. Hedge, Conformality of Riemannian manifolds to spheres, J. Differential Geometry 9 (1974) 571-576. [ ] , Some conditions for conformality of Riemannian manifolds to spheres, Tensor 28 (1974) 102-106. [ 4 ] M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962) 333-340. [ 5 ] Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965) 251-275. [ 6 ] K. Yano, The theory of Lie derivatives and its applications, North-Holland, [2]
Amsterdam, 1957.
[ 7 ] -, Integral formulas in Riemannian geometry, Marcel Dekker, New York, 1970. [8]
, Conformal transformations in Riemannian manifolds, Differentialgeometrie im Grossen, Berichte Math. Forschungsinst., Oberwolfach, Vol. 4, 1971, 339351.
K. Yano & H. Hiramatu, Riemannian manifolds admitting an infinitesimal conformal transformation, J. Differential Geometry 10 (1975) 23-38. [10] K. Yano & M. Obata, Conformal changes of Riemannian metrics, J. Differential Geometry 4 (1970), 53-72. [11] K. Yano & S. Sawaki, Riemannian manifolds admitting a conformal transformation group, J. Differential Geometry 2 (1968) 161-184. [12] , Riemannian manifolds admitting an infinitesimal conformal transformation, Kodai Math. Sem. Rep. 22 (1970) 272-300. [ 91
TOKYO INSTITUTE OF TECHNOLOGY KUMAMOTO UNIVERSITY, JAPAN
J. DIFFERENTIAL GEOMETRY 12 (1977) 461-471
SOME ALMOST HERMITIAN MANIFOLDS WITH CONSTANT HOLOMORPHIC SECTIONAL
CURVATURE LIEVEN VANHECKE
B. Smyth proved in [3] Theorem A. Let M be a complex hypersurface of a Kahlerian manifold M of constant holomorphic sectional curvature p. If M is of complex dimension > 2, then the following statements are equivalent :
(i) M is totally geoddsic in Al-, (ii) M is of constant holomorphic sectional curvature, (iii) M is an Einstein manifold, and at one point of M all sectional curwhen p > 0 (resp. G 0). vatures of M are > 4I p (resp. G 1p) 4 Considering nearly Kahler manifolds, S. Sawaki and K. Sekigawa proved in [2] the following generalization of this theorem. Theorem B. Let M be a complex hypersurface of a nearly Kahler manifold M with constant holomorphic sectional curvature p. If M is of complex dimension > 2, then the following statements are equivalent :
(i) M is totally geodesic in M, (ii) M is of constant holomorphic sectional curvature, (iii) at every point m e M all the sectional curvatures of M satisfy K(x, y) > 4p{1 + 3g(x, Jy)z} , where x, y are any orthonormal vectors of T,,,,(M). An almost Hermitian manifold with J-invariant Riemann curvature tensor is called an RK-manifold [6]. RK-manifolds with pointwise constant type form a particularly nice class of almost Hermitian manifolds, and many properties for Kahler manifolds can be generalized to this class [4], [5], [6], [7]. An RKmanifold with pointwise constant type and pointwise constant holomorphic sectional curvature is an Einstein manifold. The main purpose of this paper is to generalize the theorem of Smyth to complex hypersurf aces of such manifolds satisfying an interesting condition. This is done in § 3 following the same arguments as in [2], [3]. In § 1 we give some generalizations of theorems for RK-manifolds [6] to almost Hermitian manifolds. In § 2 we state some differential-geometric proCommunicated by K. Yano, July 26, 1975.
462
LIEVEN VANHECKE
perties of a complex hypersurf ace of an almost Hermitian manifold satisfying a certain condition, and finally in § 4 we give some properties for the holomorphic bisectional curvature [1]. We remark that, if necessary, the complex hypersurface is supposed to be connected. 1. Let M be a C°° differentiable manifold which is almost Hermitian, that is, the tangent bundle has an almost complex structure J and a Riemannian metric g such that g(JX, JY) = g(X, Y) for all X, Y E y(M) where y(M) is the Lie algebra of C`° vector fields on M. We suppose that dim M = n = 2m, and we denote by F the Riemannian connection on M. Let R be the Riemann curvature tensor, S the Ricci tensor defined by
(1)
n
S(x, Y) _
R(x, e21 y, ez) i=1
where x, y r= T, ,,(M), m r= M and {e1} is an orthonormal local frame field, and
K(x, y) the sectional curvature for a 2-plane spanned by x and y. We denote by H(x) the holomorphic sectional curvature of the 2-plane spanned by x and Jx. The sectional curvature of the antiholomorphic plane spanned by x and y, where g(x, y) = g(x, Jy) = 0, is called the antiholomorphic sectional curvature. An almost Hermition manifold such that the Riemann curvature tensor R is J-invariant, that is,
(2)
R(JX, JY, JZ, JW) = R(X, Y, Z, W)
,
yX, Y, Z, W e y(M)
is said to be an RK-manifold [6]. For such a manifold we have
(3) (4)
K(x, y) = K(Jx, Jy) S(x, Y) = S(Jx, Jy)
,
,
K(x, Jy) = K(Jx, y)
S(x, Jy) + S(Jx, y) = 0
We say further that an almost Hermitian manifold is of constant type at m r= M provided that for all x e Tm(M) we have
(5)
A(x, Y) = A(x, z)
with
(6)
A(x,y) = R(x,y,x,y) - R(x,y,Jx,Jy)
whenever the planes defined by x, y and x, z are antiholomorphic and g(y, y) = g(z, z). If this holds for all m r= M, we say that M has (pointwise) constant type. Finally, if X, Y E y(M) with g(X, Y) = g(X, JY) = 0, 2(X, Y) is con-
stant whenever g(X, X) = g(Y, Y) = 1, then M is said to have global constant type. The following theorems are generalizations of theorems given in [6]. The proofs are easy verifications.
ALMOST HERMITIAN MANIFOLDS
Theorem 1. Then
463
Let M be an almost Hermitian manifold and x, y e
R(x, y, x, y) = 32{3Q(x + Jy) + 3Q(x - Jy) - Q(x + y)
- Q(x - y) - 4Q(x) - 4Q(y)}
6 y) - 32(Jx, Jy)} + 1{13,(x,
(7)
+
6
{2(x, JY) + 2(Jx, Y}
,
where Q(x) = R(x, Jx, x, Jx). Theorem 2. Assume M is almost Hermitian, and let x, y e that g(x, x) = g(y, y) = 1 and g(x, Jy) = cos 8 > 0. Then
be such
K(x, y) = 8 {3(1 + cos 8)'H(x + Jy) + 3(1 - cos 8)''H(x - Jy)
- H(x + y) - H(x - y) - H(x) - H(y)}
(8)
+
6
{131(x, y) - 32(Jx, Jy)} +
6
{2(x, Jy) + 2(Jx, Y)}
,
if g(x, y) = 0. Theorem 3. Suppose M has constant holomorphic sectional curvature P at with g(x, x) = g(y, y) = 1 and g(x, y) = 0. a point m e M, and let x, y E Then
{1 + 3g(x, Jy)Z} +
K(x, y) = (9)
16{13,1(x, Y) - 32(Jx, Jy)}
4
+
16{2(x, Jy) + 2(Jx, Y)} .
Theorem 4. Let M be an almost Hermitian manifold with pointwise constant holomorphic sectional curvature p and pointwise constant type a. Then M is an Einstein manifold with (10)
2S(x,
x) = (m + 1)i + 3(m - 1)a
for g(x, x) = 1, and M is a space of constant holomorphic sectional curvature if and only if M has global constant type a. The definition of a in the theorem is given by (11)
2(x, y) = a ,
if g(x, x) = g(y, y) = 1 where x and y span an antiholomorphic plane.
464
LIEVEN VANHECKE
The following theorem is proved in [6]. Theorem 5. Assume M is an RK-manifold. Then M has (pointwise) constant type if and only if there exists a C°-function cr such that (12)
A(X, Y) = a{g(X, X)g(Y, Y) - g(X, Y)2 - g(X, JY)2}
for all X, Y e x(M). Furthermore, M has global constant type if and only if (13) holds with a constant function x. 2. For our purpose we need some considerations on complex hypersurfaces of an almost Hermitian manifold. We follow the notation of [2] and refer to that paper for the proofs of the given properties. See also [3]. Let M be an almost Hermitian manifold of complex dimension m + 1, and denote the almost complex structure and the Hermitian metric of M by J and g respectively. Moreover, let M be a complex hypersurface of M i.e., suppose that there exists a complex analytic mapping f : M -+ M. Then for each m e M we identify the tangent space Tm(M) with f*(Tm(M)) C T f1x,(M) by means of f Since f * o g = g' and J o f * = f * 0 J' where g' and J' are the Hermitian metric and the almost complex structure of M respectively, g' and J' are respectively identified with the restrictions of the structures g and J to the subspace f*(Tm(M)).
As is known, we can choose the following special neighborhood '&(m) of m for a neighborhood lC(f(m)) of f(m). Let {f/ ; m } (i = 1, 2, , 2m + 2) be a system of coordinate neighborhoods of M. Then {Qe ; mi} is a system of coordinate neighborhoods of M such that m2m+1= m2m+2 = 0 where mi = mi of.
By P we always mean the Riemannian covariant differentiation on M, and by N a differentiable unit vector field normal to M at each point of °I?(m). If X and Y are vector fields on the neighborhood Qe(m), we have (12)
I1Y = VXY + h(X, Y)N + k(X, Y)JN ,
where VSY denote the component of FXY tangent to M, V is the covariant differentiation of the almost complex Hermitian manifold M, and h and k are symmetric covariant tensor fields of degree 2 on °&(m). We have further (13) (14)
IXN = -AX + s(X)JN , Px(JN) = -BX + t(X)N ,
where AX and BX are tangent to M. A, B, s and t are tensor fields on 1&(m) of type (1,1) and (0,1) respectively, and A and B are symmetric with respect to g and satisfy (15)
(16)
h(X, Y) = g(AX, Y) k(X, Y) = g(BX, Y)
Now let M be a complex hypersurface satisfying the condition
ALMOST HERMITIAN MANIFOLDS
465
h(X, Y) = k(X, JY)
(17)
for any vector fields X and Y on QI(m) at every point m E M. It is easy to verify that this condition is independent of the choice of N. For such a hypersurface we have (18)
JA = -AJ ,
JB = -BJ ,
where JA and JB are symmetric with respect to g. Condition (17) is equivalent to (19)
B=JA.
Moreover we have Lemma 6 [2]. In a complex hypersurface M of M satisfying (17), at any , m) of point p E 0&(m) there exists an orthonormal basis {ei, Jez} (i = 1, 2, T,(M) with respect to which the matrix A is diagonal of the form
where Aez = 2 ez and AJe2 = -2jei. Lemma 7 [2]. If R and R are the Riemannian curvature tensors of k and a complex hypersurface M of k satisfying (17) respectively, then for any vector fields X, Y, Z, W on 0&(m) we have the following Gauss equation : R(X, Y, Z, W) = R(X, Y, Z, W) (20)
- {g(AX, Z)g(AY, W) - g(AX, W)g(AY, Z)} - {g(JAX, Z)g(JAY, W) - g(JAX, W)g(JAY, Z)}
.
Lemma 8 [2]. Let M be a complex hypersurface of k and satisfy condition (17). (i) If {x, y} is a 2-plane tangent to M at a point of 01'(m), then (21)
K(x, y) = K(x, y) - {g(Ax, x)g(Ay, y) - g(Ax, y)Z} - {g(JAx, x)g(JAy, y) - g(JAx, y)2}
where x, y form an orthonormal basis of the 2-plane.
LIEVEN VANHECKE
466
(ii)
(22)
If x is a unit vector tangent to M at a point of Gll(m), then H(x) = H(x) + 2{g(Ax, x)2 + g(JAx, x)2}
.
Proposition 9 [2]. Let M be a complex hypersurface of 111 of (pointwise) constant holomorphic sectional curvature u. If M is of complex dimension > 2 and satisfies condition (17), then at each point of M there exists a holomorphic plane whose sectional curvature in M is u, and therefore if M is of (pointwise) constant holomorphic sectional curvature u, then p = u. Finally this proposition gives Theorem 10 [2]. Let M be a complex hypersurface of M of constant holomorphic sectional curvature. If M is of complex dimension > 2 and satisfies condition (17), then the following statements are equivalent:
(i) M is totally geodesic in M, (ii) M is of constant holomorphic sectional curvature. 3. Let 1V1 be an almost Hermitian manifold, and M a complex hypersurface of 111 satisfying condition (17). It follows at once from (20), (19) and (18) that (23)
R(JX, JY, JZ, JW) - R(X, Y, Z, W) = R(JX, JY, JZ, JW) - R(X, Y, Z, W)
for any vector fields X, Y, Z, W on Old(m). Hence Theorem 11. Let M be an almost Hermitian manifold, and M a complex
hypersurface of k satisfying condition (17). If k is an RK-manifold, then M is also an RK-manifold. Further we have also (24)
R(X, Y, Z, W) - R(X, Y, JZ, JW) = R(X, Y, Z, W) - R(X, Y, JZ, JW)
for any vector fields X, Y, Z, W on Gll(m). Hence (25)
. (X, Y) = 2(X, Y) ,
and from (25) and Theorems 5, 11 we obtain Theorem 12. Let 1i21 be an RK-manifold of (pointwise) constant type a, and M a complex hypersurface satifying condition (17). Then M has (pointwise) constant type a. We need only this theorem for RK-manifolds, but it is easy to prove that this is still valid for a general almost Hermitian manifold. With the help of Theorem 4 we obtain an equivalent version of Theorem 10 for manifolds with (pointwise) constant type. Theorem 13. Let M be an almost Hermitian manifold of (pointwise) constant type, and M a complex hypersurface of complex dimension > 2 satisfy-
ALMOST HERMITIAN MANIFOLDS
467
ing condition (17). Then the following statements are equivalent:
(i) M is totally geodesic in k, (ii) M has global constant type and pointwise constant holomorphic sectional curvature. The following theorem is an immediate consequence of (9) and (12). Theorem 14. Let M be an RK-manifold with (pointwise) constant holomorphic sectional curvature p and (pointwise) constant type a. If x, y E T (M), M E M and g(x, x) = g(y, y) = 1, g(x, y) = 0, then (26)
K(x, y) = pf l + 3g(x, Jy)}z + -3,a{1 - g(x, Jy)2}
We prove now the main theorem of this paper. Theorem 15. Let M be a complex hypersurface of an RK-manifold M with constant holomorphic sectional curvature / and constant type a. If M is of complex dimension > 2 and satisfies condition (17), then the following statements are equivalent :
(i) M is totally geodesic in M, (ii) M is of constant holomorphic sectional curvature (or equivalently, M has global constant type and pointwise constant holomorphic sectional curvature), (iii)
(27)
at every point m E M, all the sectional curvatures of M satisfy
K(x, y) > p{1 + 3g(x, Jy)'} + 4a
,
if a > 0 ,
K(x, y)
0, from (27) and the expression (26) for M we obtain
- 4 ag(x, JY)' > 22i ,
468
LIEVEN VANHECKE
which implies 2i = 0 (i = 1, , m). It follows then from Lemma 6 that A is identically zero at each point of M, so that M is totally geodesic in Al. In the same way we can treat the case a G 0. Following the same arguments we obtain Theorem 16. Let M be a complex hypersurface of an RK-manifold Al with pointwise constant holomorphic sectional curvature p vnd vanishing con-
stant type. If M is of complex dimension > 2 and satisfies condition (17), then the following statements are equivalent :
(i) M is totally geodesic in Al, (ii) M has pointwise constant holomorphic sectional curvature, (iii) at every point m e M, all the sectional curvatures of M satisfy (31)
K(x, y) > 4p{1 + 3g(x, Jy)Z}
where x, y are orthonormal vectors which span the 2-plane of Consider again an almost Hermitian manifold M of constant holomorphic sectional curvature p and (pointwise) constant type a. We know from Theorem 4 that k is an Einstein manifold with (32)
S=pg,
2p=(m+ 1)p+3(m-1)a.
Now let M be a complex hypersurface of M which satisfy condition (17), and consider further the basis {ei, Jei} of Lemma 6. Then it follows with the help of (9), (21) and (22) that (33)
(34)
H(ei) = p - 222 , S(ei, ei) = 2(m + 1)p + 2(m - 1)a - 222
If M is an Einstein manifold, then we have (35)
S=pg,
p=p-222 A2 = 221
(36)
where (37)
422=422=(m+
1)p+3(m-1)a-2p
Moreover (38)
H(ei) = p - m
2 1(p
+ 3a) = p - 2(m -
1)v ,
denoting the antiholomorphic sectional curvature. Hence Theorem 17. Let M be an almost Hermitian manifold with constant holomorphic sectional curvature p and (pointwise) constant type a, and let M be
469
ALMOST HERMITIAN MANIFOLDS
a complex Einstein hypersurface satisfying condition (17). If p is the Ricci curvature of M, then
(i) P 2 satisfies condition (17) and is totally geodesic in M, then the following statements are equivalent : (i) the antiholomorphic sectional curvature of M (or of k on M) is zero,
(ii) k=konM,
(iii) k(x, x) = 1 for g(x, x) = 1, where k (resp. k) denotes the Ricci tensor of M (resp. M). 4. Let a (resp. a') be a holomorphic 2-plane defined by the unit vector x (resp. y). Then the holomorphic bisectional curvature H(a, a') is defined by
[1]
H(a, a') = R(x, Jx, y, Jy)
(39)
It is easy to verify that H(a, a) depends only on a and (40)
Using (6) we obtain
H(a, a) = R(x, Y, X, Y) + R(Jx, Y, Jx, y) - 2(x, y) - 2(Jx, y)
which together with (7) gives Theorem 20. Let M be an almost Hermitian manifold, and a (resp. a') a holomorphic 2-plane in m E M defined by a unit vector x (resp. y). Then
H(a, a) =
6{Q(x + Jy) + Q(x - Jy) + Q(x + Y)
+ Q(x - y) - 4Q(x) - 4Q(y)}
(41)
-
{2(x, Y) + 2(Jx, JY) + .l(Jx, Y) + 2(x, Jy)} . 8
470
LIEVEN VANHECKE
If g(x, Jy) = cos B and g(x, y) = cos 0, then
(42)
H(a, a') = -{(1 + cos B)'H(x + Jy) + (1 - cos 0)2H(x - Jy) + (1 + cos ¢)2H(x + y) + (1 - cos ¢)2H(x - y)
- H(x) - H(y)} - s{2(x, Y) + A(Jx, Jy) + A(Jx, Y) + 2(x, Jy)} .
Using (12) we obtain Theorem 21. Let M be an RK-manifold with pointwise constant holomorphic sectional curvature la and pointwise constant type a. Then (43)
(44) (45)
H(a, a') = 2 (a - a) + I (a + a) (cos' B + cos' ¢)
2(p-a) p,
if p+a>0, if p+a 0 and M is minimal in MZn(c) or (ii) M2n(c) is flat, i.e., c = 0. Proof.
Since Tr AH = IH I2, the parallelism of H implies that either H = 0 is a parallel isoperimetric section. If H = 0, then M. is minimal in MZn(c), and the sectional curvatures of M is G 4c. Thus by the hypothesis we have c > 0. If H # 0, and H/CHI is an isoperimetric section, then Theorem 3 implies that k111(c) is flat. Remark 2. If M2n(c) is flat, then there exist compact submanifolds of MZn(c) which satisfy the assumptions of Theorem 3 and also admit parallel isoperimetric section. For example, let S' be a unit circle in the complex plane C'. Then S' X S' is a such totally real surface in C2. In view of Theorem 3, it is interesting to study totally real submanifolds or H / I H I
of the complex number space Cn which admits a parallel isoperimetric section. The proofs of the following two theorems are similar to that of Theorem 2 in [6]. So we just only give the necessary outlines of the proofs. Theorem 4. Let M be a compact n-dimensional totally real submanifold
imbedded in C. If M has nonnegative sectional curvature, and it admits a parallel isoperimetric section , then M is a product submanifold M, x . X MT, where M, is a compact v,-dimensional totally real submanifold imbedded in some Cvt, and Mt is contained in a hypersphere of Outline of proof. The assumption of the theorem implies that PAF = 0. C2t.
Thus the distinct eigenspaces T , TT of A, define parallel distributions of M. By the de Rham decomposition theorem, M is a product of Riemannian X MT, where the tangent bundle of MS corresponds to T. manifold M, x By Lemma 6 and a lemma of Moore [5] we see that M = M, x x MT
480
BANG-YEN CHEN, CHORNG-SHI HOUH & HUEI-SHYONG LUE
is a product submanifold imbedded in Cn = C"1 X X C. Moreover, Lemma 6 implies that each of Mt's is a totally real submanifold imbedded in some Cvt :
M, X ... X M,
imbedding
r
3,
Cv1 X ... X Cvr
is a parallel Let be the component of in the subspace C. Then normal section of M, in Cv,, and Mt is umbilical with respect to art( ). From these it follows that Mt is contained in a hypersphere of Cv° (see, for instance, [2])
Theorem 5. Let M be a compact n-dimensional totally real submanifold imbedded in Cn. If M has nonnegatve sectional curvature and parallel mean x M,r, where curvature vector H, then M is a product submanifold M, x Mt is a compact vt-dimensional totally real submanifold imbedded in some Cvt, and Mt is also a minimal submanifold of a hypersphere in Cvl. Outline of proof. Since the mean curvature vector H is parallel and there exists no compact minimal submanifold in Cn, H/IHI is a parallel isoperimetric X M,r such section. By Theorem 4, M is a product submanifold M, x
that Mt is totally real in some Ct't and Mt is umbilical with respect to the component 7rt(H) of H in the subspace Cvl. Since 7rt(H) is parallel and is the mean curvature vector of Mt in Cvt, Mt is a minimal submanifold of a hypersphere in Cv,. References T. E. Cecil, Geometric applications of critical point theory to submanifolds of complex projective space, Nagoya Math. J. 55 (1974) 5-31. [ 2 ] B. Y. Chen, Geometry of submanifolds, M. Dekker, New York, 1973. [ 3 ] B. Y. Chen & K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974) 257-266. , Two theorems on Kaehler manifolds, Michigan Math. J. 21 (1974) 225[4] [1]
229.
[51 J. D. Moore, Isometric immersions of Riemannian products, J. Differential Geometry 5 (1971) 159-168. [6] B. Smyth, Submanifolds of constant mean curvature, Math. Ann. 205 (1973) 265-280.
[71 K. Yano, Totally real submanifolds of a Kaehlerian manifolds, J. Differential Geometry 11 (1976) 351-359.
MICHIGAN STATE UNIVERSITY WAYNE STATE UNIVERSITY NATIONAL TSINGHUA UNIVERSITY, TAIWAN
J. DIFFERENTIAL GEOMETRY 12 (1977) 481-491
GEOMETRY OF HOROSPHERES ERNST HEINTZE & HANS-CHRISTOPH IM HOF
1.
Introduction
Let M be a Hadamard manifold, i.e., a connected, simply connected, complete riemannian manifold of nonpositive curvature. To be more precise, as-
sume that the sectional curvature K of M satisfies -b2 < K < -a2, where 0 < a < co and 0 < b < oo. If p E M and z is a point at infinity (cf. EberleinO'Neill [4], which we give as a general reference for Hadamard manifolds), there exists a horosphere through p with center z. This is defined as follows : Denote the geodesic ray from p to z by r, and consider the geodesic spheres through p with center 7(t), t > 0. As t goes to infinity, these spheres converge to the horosphere. More precisely, the horospheres are the level surfaces of the Busemann function F = lim F, where Ft is defined by F,(p) = d(p, 7(t)) - t. In the flat case (a = b = 0), horospheres are just affine hyperplanes, and in the case of constant negative curvature, using the Poincare model we see that horospheres are euclidean spheres internally tangent to the boundary sphere, minus the point of tangency. The main purpose of this paper is to
show that, to a certain extent, the geometry of horospheres in M may be compared with that in the spaces of constant curvature - a2 and -b2, respectively. We give two examples : 1. (Theorem 4.6). If ° is a horosphere and h denotes the distance in
° with respect to the induced metric, then for all p, q E Ye
a sinh ad(p, q) < h(p, q) < b sinh b d(p, q) , 2
2
where d is the distance function of M. 2. (Theorem 4.9). If r is a geodesic tangent to a horosphere X', and if p, q are the projections of r(± co) onto , then
b M be a geodesic ray, and let V E Mp, p = 7(0). Then there existis a unique stable Jacobi field Y along r with Y(0) = v,
and we shall denote this stable Jacobi field by Y. Proof. (i) Uniqueness follows immediately from the fact that in a Hadamard manifold the length of a Jacobi field is a convex function. (ii)
Denote by Y, the unique Jacobi field along r with YJ0) = v and
Y,(n) = 0. Applying Rauch's comparison theorem to Y,,, with the flat case), we get II Yn(0) -
it
n0. (i) Let p e K and n > no. Then IIIZ,n - ZHI (p) = 1t,,.(0) - tPz(0)JI goes
to zero uniformly on K, if the angles n, VvZ,
- Yv I (p) = Fv grad F,n - Y'(0) = Yp-L(0) - Y,(0) I
where v = V(p), YPP,: is the Jacobi field along lPPn with Y,,,(0) = v and YPP (d(p, p.n)) = 0, and L denotes the component orthogonal to also depends on n. But an easy computation shows 11 Y,
(0)
- YPPn(0) = 11 V - vj-11 < d(p, P-..)
I1 v I
which
I
d(P, Pa)
which goes to zero uniformly on K as n tends to infinity. Thus it is enough to show
Y' .(0) - Yv(0)1 - 0 uniformly on K. For T > 0 let X Pn be the
Jacobi field pp along rPPn with TPn(0) = V(p), X(T) = 0, and define XP analogously. Then
GEOMETRY OF HOROSPHERES
485
I YPP.(0) - Yv(0) II < II YPp'(0) - XppJ0) II + II XT"(0) - XP (0) II
+ II XT'(0) - yv(0) By Rauch's comparison theorem II Y,p (o) - XPPn(o) II
T. The same argument yields Xpi(0) - Yv(0) II 5 I V(p) T. Thus the problem is reduced to show that, for fixed T, the difference Xpz(0) II goes to zero uniformly on K, I
as n tends to infinity. Using a lower curvature bound on KT = {p E M I d(p, K) < T} it is clear that d(q,(p), q(p)) -- 0 uniformly on K, where qn(p) = rpp, (T), q(p) = rpz(T). By the differentiable dependence of Jacobi fields and their derivatives on the boundary values, the result now follows. The radial flow. Now we want to study the flow generated by the vector field Z, which we call the radial flow (with respect to a fixed z E M(oo)) and
denote by * or {*J. Since the geodesics going to z are the integral curves of Z, this vector field i s obviously complete, and Jr is given by Jr _ ;70 (P o (1 R X Z) :
R X M -* M, where (P denotes the geodesic flow, and 'r the canonical projection. Proposition 3.1 implies immediately that i is C'. The following properties of Ia* are infinite versions of the lemma of Gauss and the comparison theorem of Rauch. Proposition 3.2. (i) If a vector u E MP is parallel to Z(p), then i,,k(u) is parallel to Z(* ,(p)), and III,*(u)II = IIuII (ii) If a vector v E MP is orthogonal to Z(p), then i,,k(v) is orthogonal to Z(* ,(p)), and the following inequalities hold vi
e-" < I*,,(v)II < IIv1I e a`
fort>0.
Proof. (i) It is enough to show iJr (Z(p)) = Z(* ,(p)), but this is true, since the geodesics going to z are the integral curves of Z.
We recall that Z = -grad F, where F is a Busemann function at z, and that the horospheres centered at z are the level surfaces of F. Therefore the complements MP = {v E MP v L Z(p)} are the tangent spaces of the horospheres, and , maps horospheres onto parallel horospheres. This implies the first part of (ii). In order to prove the inequalities, we now compute ]t,K(v) for v E MP explicitely. By definition *,,(v) = 7r* o (pt,k o Z,F(v). We use the (ii)
identification TSM = SM +Q TM Q+ TM given by 7rs X 'r,F x K, where SM de-
notes the unit tangent bundle, 'rs : TSM SM is the canonical projection, rr,: TSM - TM is the differential of 7r: SM M, and K : TSM TM is the connection map. Then Z,k(v) = (Z(p), v, V7,Z) = (Z(p), Y2(0), Y;;(0)), where Yv is the stable Jacobi field along rpz with initial value Y,(0) = v, (compare Proposition 3.1). Therefore we get coz oZ*(v) = (Z(*,(p)), Y7,(t), Y'(t)) and
486
ERNST HEINTZE & HANS-CHRISTOPH IM HOF
*t,(v) = Y,,(t). By the comparison theorem for stable Jacobi fields we conclude
IIv1Ie-ac < Il*tx(v)II
0,
(2 aresinh 2 h(O)e-11 < d(t) < h(0)e-at As H. Karcher remarked, this can be improved by a different method to
d(0)e-"' < d(t) < (2 sinh
2
d(0) Ie-at
487
GEOMETRY OF HOROSPHERES
Two estimates for the Busemann function with geometric applications. We
consider a Busemann function F at an infinite point z. To compare F with Busemann functions in spaces of constant curvature, we study the restriction f = F o r for a given geodesic r. While f measures the deviation of r from a fixed horosphere with center z, the derivative grad F> measures the angle between r and the horospheres centered at z. In the following, f,, and f b denote functions defined analogously in the spaces Ha and Hb, respectively. Lemma 4.2. Given that f , la, f b are as described above. Assume f (O)
f .(O) = f b(0) and f'(0) = f a'(0) = f(0). Then f' (s) < f'(s) < f '(s) for s > 0 and f a(s) < AS) < f b(s) for s E R. Proof. For s > 0 consider the triangle 4 determined by p = r(0), q = r(s)
and z. The angles a = p(q, z) and p = 1 1(p, z) satisfy cos a = - f'(0) and cos R = f'(s). Let a be the geodesic ray from p to z, and denote by 4(t) the triangle determined by p, q and 6(t). The angle p(t) = fb(s)). Lemma 4.3. Given that f, fa, fb are as before. Assume f (O) = f a(0) = fb(O) and f (l) = f a(l) = f b(l) . Then fb(s) < f(S) < fa(S)
for
s E [0, 1]
.
Proof. Fix s E [0, 1] and look at the triangles 41 = (r(0), r(s), z) and 42 = (r(1), r(s), z). In one of thet riangles, say in 41f the angle R at r(s) is not smaller than the corresponding angle Pa in Ha. Suppose for the moment that R equals Pa. Then Lemma 4.2, applied to 41, implies f (s) G f a(S). This is a fortiori if
R>Pa-
The proof of the inequality on the left hand side is similar. Remark. Since in the flat case f o is linear, the above lemma gives another proof of the convexity of F. For the geometric applications consider triangles 4 with two vertices p, q E M and one vertex z at infinity. Such a triangle gives rise to the following data: I = d(p, q), a = p(q, z), and R = fa(l)
These give the estimates for a and A, since f'(0) _ -cos a and f'(l) = cos f3. Distances on horospheres. Our next aim is to compare the '-distance h(p, q) of two points p, q on a given horosphere ' with their usual distance d(p, q). If moreover p and q lie on a different horosphere `, then their distance h'(p, q) may be different from h(p, q). However, the following theorem gives estimates independent of the chosen horosphere. and denote their-distance by h(p, q). Theorem 4.6. Assume p, q E Then
a sinh 2 d(p, q) < h(p, q)
0 sin (3 1( 1 sin R < h(s) < l(s) < 1 a ` coth as + cos (3 f b ` coth bs + cos (3 1I
Proof. First we prove the inequality on the right hand side. Consider the same data as above in H, and fix s > 0. Lemma 4.2 implies f (s) > fo,(s) and (3(s) < Pa(s), where f denotes the restriction F o r, and (3(s) the angle between r(s) and grad,,,) F. Now we compute 11 ,u(s) lI. We decompose (s) into cos n(s) grad,,, F and an orthogonal part tJ-(s) of length sin n(s). Then a(s) _ 7)*(r(s)) = 77x(rJ-(s)) = irt,(rJ-(s)) for t = f (s). Therefore Proposition 3.2 implies 1I a(s) J < sin (3(s) e-afts'. Similarly we get II Fla(s) 11= sin Pa(s) e-of-11), which together with AS) > f a(s) and (3(s) < Ms) yields !I,u(s) 1I < 11 lia(s) 11 and, by integration, h(s) < ha(s), where ha,(s) is the corresponding function for Ha, as usual. Now
an easy computation in Ha gives ha(s) = a-1 sin R (coth as + cos p)-1. Next we prove the inequality on the left hand side. Consider the same data as
above in Hb, and assume h(s) < hb(s) for a certain s > 0. By Lemma 4.2 we have f(s) < fb(s). In Hb there is a unique point qb with Fb(gb) = fb(s) and rjb(gb) = pb(h(s)). Denote by rb the geodesic segment from pb(0) to q, and by sb its length. The assumption h(s) < hb(s) implies sb < s. Now consider the curve r' in M lying over an .XP-geodesic from p(O) to p(s) with For' _
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Fb o rb, and denote its length by s'. Proposition 3.2 implies (as in the proof of Theorem 4.6) s' < sb. By its construction the curve r' joins p(0) to a point q
with the properties F(q) = fb(s) and (q) = p(s). Since f(s) < fb(s), the convexity of the distance function d(p(0), ) implies s < d(p(0), q) < s', which contradicts s' < sb < s. Hence h(s) > hb(s). From now on we assume a > 0, i.e., the curvature of M is bounded away from zero. In this case the point p(co) is defined to be the intersection with ' of the unique geodesic from r(co) to z. Denote by l the length of p, and by h the -distance between p(O) and p(cc). Corollary 4.8. Assume p < it/2 as before. Then 1
sing
sin (8
1
