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Pm as proved by H. Levine [3] can be stated as follows: Let A be a point of Pm and D a compact domain in Em such that f-l(A) n D consists only of a finite number of points and that f-l(A) does not meet the boundary aD of D. Let n(D, A) be the number of times that A is covered by f(D), counted algebraically, and let v(D) be the volume of f(D). Then n(D, A) - v(D) =
(45)
Jr
A.
ICoD)
The theorem shows the importance of the form A and therefore the summands k 1\ n m - 1 - k • It is desirable to express this summand in terms of powers of n by applying (42), be~ause n is independent of the point A, while depends on A. We have (46)
k 1\
nm-l-~ =
.
1
sm 2k d(Z, A)
(nm - 1-
k ~ cp 1\ (j> I\n m - 2), 2rr
O~k~m-l.
The mapping f induces a dual mapping f* on differential forms. We put (47)
where, by (38), (48)
_ 02 log
a",(l-
IZI
•
at;",at; fl
The a",,. are the elements of an hermitian positive semi-definite matrix. An important consequence of the expressions (48) is that the partial derivatives aa"'fl/at;y are symmetric in a, 'Y. Similarly, denoting by fA the restriction of f to Em - f-1(A), we put (49)
Then (b ..fl ) is an hermitian positive semi-definite matrix, also with the property that ab .. fl/at;y are symmetric in a, 'Y. The differential forms (f1 1 E .. -'" =-E C",.. --f1. a~.. rrr" 2r
(68)
The integral formula (33) therefore leads to the relation _l_{n(r, A) - v(r)} (69)
r''''-'
a 2mf1 = ~ I,-(f1U)®, + )I,U(ar r
-
2",-1 --E C.... ) ®, rrr"
,
where in the integrals on the right-hand side, the argument is rx, with x € ~, and r = const. Notice also that the formula (69) is valid only for those r for which ~T does not meet j-'(A). However, as in the case m = 1, the form of (69) suggests its integration with respect to r. This requires the examination of the improper integrals [(r , A) = (70) J(r, A)
r (-uf1)®" JI,
= J,(-uE I
..
C",,,,)®,
for values of r for which ~T n f-'(A) *- 0. In the first place, the integrands of the integrals in (70) are ~ O. By (57) and the positive semidefiniteness of (Da./l), we have 2"'. d ( 1 -uf1 =< -' -10g sm ) EO'-;: k'-;:",-' . -' k d rrr • ( SIn
)E ", . Q
"
O)CC "'/lS"'~/l '
Similarly, by (61), we have C (-u)" L.."a.
Q;£l',
< = (-log sin
d)("
L.JO ;S: k :;i m.-l
_l_)C CO) . sin2k d
n ~T consists of a finite number of points, as we will suppose, the principal part of the integrand at a singularity is
If f-'(A)
( -log sin d)( EO," k,""'-' sin;k d ) , for the integrals of the functions at the right-hand sides of the inequalities, so that these integrals are convergent. The same is therefore true for the integrals of the functions on the left-hand sides, which proves
270
550
SHIING-SHEN CHERN
that I(r, A), J(r, A) are defined for all values of r. Moreover, the integrals of the functions on the right-hand sides over a small domain of radius e of ~l tend to zero uniformly in r as e - O. From this it follows that I(r, A), J(r, A) are continuous functions in r. In particular, this gives the relation
fr dr[ ~ (-Uf1)®l = Jro Jx) 8r
(71)
for 0 < ro ~ r. For a fixed ro and we put
>0
I(r, A) - I(ro, A) ,
and for r
> ro
we define T(r) by (5)
N(r A) = [r n(t, A) dt . , Jro t 2m - l
(72)
Integrating (69) with respect to r, we then get (73)
N(r, A)
= const + T(r) + S(r, A)
- I(r, A) - 2m
fr
Jro
I(r, A) dr . r
where (74)
S(r, A) = 2m-l[r J(r, A) dr . 11:
Jro
r
Since I(r, A) ;;;; 0, this gives the inequality (75)
N(r, A)
< const + T(r) + S(r, A)
.
Thus, unlike the classical case m = 1, an additional term S(r, A) has to be added to T(r) in order that it majorizes N(r, A) asymptotically. Formula (73) is the integrated form 6f the first main theorem. We will now give a proof of the theorem stated in the Introduction. Under the hypotheses suppose the contrary be true. Let p(A) be the characteristic function of the set f(Em), so that p(A) p(A)
= 1, = 0,
A
€
f(Em),
A ef(Em).
Let dA =,nm be the volume element of Pm. The total volume of Pm being 1, we have, by assumption,
f
f p(A)dA = b < 1. Jp m
Clearly we have
n(t, A)p(A)dA = v(t) .
hm
Integration of the inequality (75) with respect to p(A)dA over Pm gives
271 FIRST MAIN THEOREM IN SEVERAL VARIABLES
T(r)
551
< const + bT(r) + ~PmS(r, A)p(A)dA ~ const + bT(r) + I ~
const
+ bT(r) m
S(r, A)dA
hm
2 - ~r -dr ~ + -n ro l
rPm
dA
~ (-log sin d) ( E Xl
_ 1 _ ) C(Ol®l. O;S .t ::Om-l
sin2,t d
But since both dA and d(Z, A) are invariant under the isometries of P"., the integral
JI on(-log sin d)(Eo:a::;; m-l _l_)dA sin d 2,t
P
is equal to a constant (n/2m-l)h, independent of Z. It follows that T(r)
< const + bT(r) + hiT dr I C (Ol®l. JTo r JXl
The last term in this inequality is, by (64), equal to
2!L Jro IT (vi(r)dr/r2m).
By the second hypothesis of the theorem the latter integral is o(T(r». Hence there is a contradiction and the theorem is proved. UNIVERSITY OF CHICAGO REFERENCES
1.
s.
BOCHNER, and W. T. MARTIN, Several Complex Variables, Princeton 1948, p. 45.
2. H. CARTAN, Notions d 'algebre differentielle ; application aux groupes de Lie et aux varietes 0\1 opere un groupe de Lie , Colloque de Topologie, Bruxelles 1950, pp. 15-27. 3. H. LEVINE, A theorem on holomorphic mappings into complex proiective space, Ann. of Math . 71 (1960) , pp. 529-535.
272 Reprinted from Acta Math. 114 (1965).
HERMITIAN VECTOR BUNDLES AND THE EQUIDISTRIBUTION OF THE ZEROES OF THEIR HOLOMORPHIC SECTIONS BY
RAOUL BOTT and S. S. CHERN Harvard University, Cambridge, Mass., U.S.A., and University of California, Ber/reley, Calif., U.S.A. (')
1. Introduction At present a great dcal is known about the value distribution of systems of mercmorphic functions on an open Riemann surface. One has the beautiful results of Picard,
E. Borel, Nevanlinna, Ahlfors, H. and J . Weyl and many others to point to. (See [1], [2].) The aim of this paper is to make the initial step towards an n-dimensional analogue of this theory. A natural general setting for the value distribution theory is the following one. We consider a complex n·manifold X and a holomorphic vector bundle E over X whose fiber dimension equals the dimension of X and wish to study the zero-sets of holomorphic sections of E. When X is compact (and without boundary) then it is well-known that if the zeroes of any continuous section are counted properly then the algebraic sum of these zero-points is independent of the section and is given by the integral of the nth Chern (2) class of E over X: Thus we have Number of zeroes of
8=
f
x cn(E),
(1.1)
and this formula is especially meaningful for a holomorphic section because the indexes of all the isolated zeroes of such a section are necessarily positive. The central question of the value distribution theory is to describe the behavior of the zeroes of holomorphic sections when X is not compact. (For continuous sections there (') This work was partially supported by a grant from the National Science Foundation . The second author was a professor of the Miller Institute at the University of California (Berkeley) and received partial support from the Office of Naval R esearch. (') With misgivings on tho part of the second author we have adopted a terminology now commonly used .
273 72
RAOUL BOTT .AND S. S. CHERN
are no restrictions in that case, for instance there is always a section which does not vanish at all!) The main results, all concerned with the case dim X
=
1, then take the following form.
One considers a finite-dimensional "sufficiently ample" subspace V of the space of all holomorphic sections of E and shows that under suitable convexity conditions on E and
X "most" of the sections in V vanish the "same number of times". Depending on how "most" and "same number of times" are defined one gets results of various degrees of delicacy and difficulty. For example, the classical Picard theorem asserts that when X is the Gauss-plane, so that E may be taken as the trivial line bundle C, and dim V=2, then at most 2 sections of V in general position can fail to vanish on X. The Borel generalization of this theorem asserts that when dim V =n, then at most n sections in V, in general position, can fail to vanish. Here, as throughout, the term general position is used in the following sense: A set of n elements VI'
... , Vn
of a vector-space V is called in general position,
if any subset of k elements span a k-dimensional subspace of V, for k = 1, ... , dim V.
In the Nevanlinna theory one again deals with X=C, dim V=2, but now a deficiency index O(s) is defined for every sE V -0, which measures the extent to which s behaves unlike the generic section in V. In particular 0 has the properties O(AS) =O(s), if AEC-O;
o,;; o(s) ';; 1; and finally:
o(s) = 1 if s does not vanish on X . The " first main theorem" may
then be interpreted as asserting that 0 considered as a function on the projective space Pl(V) of lines in V, is equal to 0 almost everywhere. Thus " most" sections in the measure
sense behave the same way. The second main theorem yields the much stronger inequality : (l.2)
valid for any system of sections
Si E V
in general position. The Ahlfors generalization deals
with the case dim V =n and again proves among other things that o(s) =0 nearly everywhere, and that now the inequality
is valid for any system of
S i E V,
which are in general position.
Usually these results are stated in terms of maps of X into the Riemann-sphere, (i.e., meromorphic functions) for the Picard and Nevanlinna theory, while the Borel and Ahlfors generalizations deal with m aps of X into complex projective spaces of higher dimensions. The transition to our formulation is quite trivial. Indeed consider the evaluation map: ex: V-'>-Ez which attaches to each section in V, the value of sat x. By definition, a space of sections V will be "sufficiently ample" if and only if:
274 HERMITIAN VECTOR BUNDLES
ex)
ez : V--'>-Ez is onto for each xEX.
f3)
V contains a section which vanishes to the first order at some point of X .
73
Now let k(x) be the kernel of ez • This is then a subspace of a fixed dimension m=dimV-dimE" in V, so that the assignment x--'>-k(x) defines a map ev:X--'>-Pm(V) of X into the Grassmannian of m-dimensional subspaces in V. Now for each sE V, let z(s) be the subvariety of Pm(V) consisting of those subspaces which contain s. Then, for s =1=0, z(s) has codimension n in P m( V), and it is clear that the zeroes of s on X correspond precisely to the intersections of ev(X) with z(s) in P m( V). In particular, when dim Ex is 1, P m( V) is just a projective space, and z(s) is a hyperplane, so that we may reformulate our statements in the terms of the number of hyperplanes which the image of X avoids. Conversely, starting with a map e:X--'>-P1'l(V), one may pull back the quotient bundle of P m(V) (see the end of Section 6) to obtain a bundle E over X, together with a finite dimensional subspace, V, of sections of E, for which e v =e. Indeed, let K c X x V, consist of the subset. (x, v) for which v Ee(x) . Then K is a sub-bundle of the trivial bundle X x V, and the corresponding quotient bundle, X x V/K is the desired bundle E . The constant sections of X x V over X, then go over into the desired subspace, V, of sections of E. Thus these two points of view are completely equivalent. The aim of this paper is to discuss the n-dimensional case and we are able to push to an analogue of thc first main theorem . Thus we obtain the weak equidistribution in the measure sense only. On the other hand this generalization is not quite immediate and in fact depends on a formula in the theory of characteristic classes, which seems to us of independent interest. To formulate this result we need to recall two facts: Namely 1) That the complex structure on X induces a natural "twisted boundary operator", dC, on the real differential forms, A(X), of X, and 2) That a given Hermitian structure on E determines definite representatives, ck(E) EA(X), k = 1, ... , n, of the Chern classes of E . With this understood, we consider a given Hermitian, complex n-bundle E, over X and its Chern form cn(E)EA(X). Also let B*(E)={eEEIO< lei < I} be the subset of vectors in e which are of length greater than 0 and length less than 1, and set n : B*(E)--'>-X equal to the natural projection. Then our first and principal result is expressed by the theorem: THEOREM
1. There exists a real valued form eon B*(E) which is of type (n-1, n-1)
and for which n*cn(E)
=
dd,c 4n e ·
Further if E is non-negative then e may be chosen to be non-negative.
(1.3)
275 74
RAOUL BOTT AND S. S. CHERN
Remark that B*(E) has the homotopy type of the unit sphere bundle S(E), of E , and it is of course well known that cn(E), when lifted to S(E), becomes a boundary. Hence Theorem I refines this result for the complex analytic model B*(X) of S(E). The method which leads to Theorem I also yields the following auxiliary result. PROPOSITION
1.4. Let E be 'iL complex analytic bundle and let c(E) and c'(E) be the
Chern forms of E relative to two different Hermitian structures. Then c(E) -c'(E) =dd c , A. for some A..
In other words, if we define Hk(X) by:
then the class in H*(X) = LHk(X) of the Chern form c(E) , of E relative to some Hermitian structure on E, is independent of that Hermitian structure, so that we may define a "refined Chern class" c(E) EH*(X) . (Cf. Section 3 for definition of A k. k (X).) In fact, Theorem I will follow directly from the following Whitney type duality theorem concerning these refined Chern classes: PROPOSITION
1.5. Let O-+E'-+E-+E"-+O be an exact sequence of holomorphic
vector-bundles over X . Then their refined Chern classes satisfy the duality formula : c(E') ' c(E") =c(E) .
The formula (1.2) is very pertinant for the whole Nevanfuma theory; for instance in the one-dimensional case, e is just a real valued function on B*(E), and is seen to be minus the logarithmic " height" function : e(e)= -log l(e)12,
eEB*(E).
Indeed one may roughly express the situation by saying that the first " main inequality" of the NevanliIma theory is just a twice integrated version of (1.3) . The plan of the paper is as follows: In Section 2 we review the theory of characteristic classes as found in [3], [5]. We then go on to refine this theory for complex analytic Hermitian bundles in Sections 3 to 5. Section 6 is devoted to a proof of the generalized GaussBonnet theorem which fits into the context of this paper. In Section 7 we define the order function, while in Section 8 we formulate and start to prove the equidistribution theorem. Sections 9 and 10 then complete this proof. Our final section brings a leisurely account of the classical Nevanlinna theorem. This Section 11 is included primarily to show how much more will have to be done before an n-dimensional analogue of this delicate theorem is established.
276 75
HERMITIAN VECTOR BUNDLES
2. Curvature and characteristic classes
In this section E will denote a Goo-bundle over the Goo manifold X. We write T
=
T(X)
for the cotangent bundle of X, and A(X) = LA fiX) for the graded ring of Goo complex valued differential forms on X. The differential operator on A(X) is denoted by d. More generally we write A(X; E) for the differential forms on X with values in E . Thus if fiE) denotes the Goo sections of E, then A(X; E) =A(X)®A'('K)f(E). The natural pairing from r(E)®A.(X)r(F) to f(E®F)(l) will often be written simply as multiplication. Our aim here is to give an elementary and essentially selfcontained review of the geometric theory of characteristic classes, as developed by Chern and Well. More precisely, we will describe how the curvature of a cOIillection on the vector bundle E can be used to construct closed differential forms on X whose cohomology classes are independent of the connection chosen and therefore furnish topological invariants of the bundle E. Of the many definitions of a connection we will use the differential operator one. It leads to the simplest local formulae . We will also thereby avoid the possibly less elementary concept of principal bundles. For a more general account of this theory see [3], [4], [5). DEFINITION
2.1. A connection on E is a differential operator D:r(E)-+r(T*®E)
which is a derivation in the sense that for any f EAO(X) :
D(fs) =dj-s +j-Ds, sEr(E).
(2.2)
Rerrw,rks. In general a differential operator from riEl to f(F) is just a C-linear map
which decreases supports. If such an operator is also AO(X) linear, then it is induced by a linear map from E to F, i.e., by a section of Hom (E, F). Thus if Dl and D z are connections then Dl - D z is induced by an element of
r
Hom (E*, T*®E)=Al(X; Hom (E , E)) .
Suppose now that E is equipped with a definite connection D . One may then construct the Chern form of E relative to the connection D in the following manner. Let s={St}, i=l , ... , n be a set of sections(Z) of
EI U where U is open in X, such that
the values {Sj(x)} form a base for each Ex. with xE U. (Such a set S will be called a frame of E over U .) In view of (2.2) a formula of the type: (2.3)
(1) The tensor product is over C unless otherwise indicated. (2) We will be dealing with smooth sections throughout.
277 76
RAOUL BOTT AND S. S. CHERN
must then exist and serves to define a matrix of I-forms on U: O(s; D) =
110 Ifll-the so-
called connection matrix relative to the frame s.
In terms of O(s, D) one now defines a matrix K(s, D) =
IIKIIII
of 2-forms on U by the
formula: K II = dO 11- LaO la 1\ Oal. In matrix notation:
K(s, D)"'=dO(s, D) -O(s, D) lIO(s, D) .
(2.4)
This is the curvature matrix of D relative to the frame s. Because even forms commute with one another it makes sense to take the determinant of the matrix I +iK(s, D)/2n and so to obtain an element det {I +iK(s, D)/2n} EA(U). A priori, this form depends on the frame s. However as we will show in a moment, de~
{I +iK(s, D)/2n} is actually independent of the frame s, and therefore defines a
global form, the Chern form of E relative to D, c(E, D) in A(X) . More precisely c(E, D) is defined as follows: We cover X by {Ua } which admit frames
s~
over Ua , and then set
c(E, D) IUa =det {I +iK(sa, D)/2n}. On the overlap these definitions agree because of the asserted independence of our form on the frame s. Consider then two frames sand s' over U. Then there exist elements AijEAO(U) such that s; = LjAijSj and in matrix notation we write simply s' =As. From (2.2) it follows that Ds'={dA+A(!(s, D)}s. Further, by definition, Ds'=(!(s', D)s'. Hence the connection matrices are related by
dA+A(!(s, D)=(!(s', D)A,
s'=As,
(2 .5)
from which one directly derives the important formula :
AK(s, D) =K(s' , D)A,
s' =As.
(2.6)
This transformation law of the curvature matrix, together '''ith the invariance of the determinant under conjugation now immediately implies the desired independence of our form det {I +iK(s, D)/2n} on s. Thus we now have defined c(E, D) explicitly and our next aim is to show that
c(E, D) is closed and its cohomology class independent of D. For this purpose it is expedient to analyse the above construction a little more carefully, and then to generalize the whole situation. Note first of all that the transforma tion law (2.6) is characteristic of the elements of
A(X; Hom (E, E)). Indeed if
~EAD(X;
Hom (E, E)) and if s is a frame for E over U, then
~ determines a matrix of p-forms ~(s) = 1I~(s)ij ll by the formula:
(2.7) and under the substitution s' =As, these matrices transform by the law
~(s')A =A~(s) _
278 HERMITIAN VECTOR BUNDLES
77
The converse is equally true so that in particular the curvature matrix K(s, D) represents a definite element K[E, D]EA2(X; Hom (E, E)). Next we observe that the "determinant construction" really becomes more understandable when formulated in this manner. We let Mn denote the vector-space of n X n matrices over C. A k-linear function (!J on
M n will be called invariant if for all y EGL(n, C): (2.8) The vector-space of all k-linear invariant forms shall be denoted by Ik(Mn). Now given (!JE1k(Mn) and UcX, we extend (!J to a k-linear map denoted by (!Jrr-from Mn®A(U)
to A(U) by setting:
With this understood consider k elements ';;EA(X; Hom (E, E)) and let (!JE1k(Mn). It is then clear that there is a well-defined form (!J(';l' ... , ';k) EA(X), which has the local description: Given a frame
8
over U, thea (2.10)
where the ';;(8) are the matrices of';; relative to 8 and hence elements of A(U)®Mn' We will abbreviate (!J(';,';, .. .,.;) i.e., the case all .;; equal, to (!J((';))' Now given a connection D on E, and a (!JE1k(Mn) we have well-determined forms (!J((K[E, D])) and (!J((1 +iK[E, D]/2:n:)) in A(X), and our Chern form is clearly of the latter type. Indeed we
need only take for (!J the n-multilinear form det on Mn obtained by polarizing the polynomial function x --+det x on M n' to describe the Chern form in the present frame work: c(E, D) =det ((1 +iK[E, D]/2:n:)).
(2.11)
It is now also an easy matter to construct elements (!JkE1k(Mn) so that c(E, D) =
L (!Jk((K[E, D])) .
In short, the two properties of c(E, D) which we are after will follow from the more conceptual assertion that for any (!JE1k(Mn) , the form (!J((K[E, D])) is closed and its homology class independent of D. We will now derive both these properties from the invariance identity (2.8) . Note first that differentiation with respect to y leads to the identity k
L(!J(xl'oo ,,[XhY],oo,Xk)=O, - 1
XhyEM n
(2.12)
279 78
RAOUL BOTT AND S. S. CHERN
and conversely-because GL(n, C) is connected-(2.12) implies (2.8) . This identity now generalizes in a straight forward manner to the extension of gJu and takes the following form in matrix notation. An element xPEMn@AP(U) (called of deg p) is represented by a matrix of p-forms. Matrix multiplication therefore gives rise to a pairing x@y--+x /\ y, of elements of deg p and deg q to elements of deg (p+q). In terms of this multiplication one now defines the bracket [XV, yO] by the usual formula for graded Lie-algebras: (2.13)
In this terminology the following invariance law for any gJ E Ik(M n) follows directly from (2.9) and (2.10) and (2.12): (2.14) whenever the x'" and yare homogeneous elements with q =deg y, and f((X) = Lp>", deg Xp. From the derivation property of d it follows further that, with the x'" as above:
d gJu(xl' . . . , Xk) = L ( - 1 )D(Cl) gJu(xl' ... , dx., . . . , Xk)
(2.15)
where now g((X) = LplI·
6- 652932 A cta mathematica 114 , Imprime le 11 (Lout 1905 .
(3.6)
283 82
RAOUL BOTT AND S. S. CHERN
This is the norm of the frame. Now, let s be a holomorphic frame over U, and let () be a prospective connection matrix for E, relative to s. Then (3.5) applied to all the pairs
<Sf' Sk> implies the relation (}N+N{Jt=dN,
(3.7)
N=N(s).
Hence if () is to satisfy this condition, and is also to be of type (1,0) so as to satisfy (3.4), then we must have:
(}=d'N·N-l
on U.
(3.8)
Thus there is at most one connection with the properties (3 .5) and (3 .6). Conversely, let s = {Sf} be a lwlomorphic frame over U, and set N(s) =
lI <sl> s,>11 as
before. Then the formula: (3 .9)
defines a connection over U, which is seen to be independent of the lwlomorphic frame s chosen, and hence induces a global connection D(N) on E which manifestly satisfies the condition of our proposition. The independence of D on s is proved as follows : Let SI =As be another lwlomorphic frame, over U. Then dA =d'A, because A is a holomorphic matrix. Further Nl =N(SI) =ANjP. Hence
d'N 1 ·N 11 =dA ·A-l +Ad'N·N-IA-l which shows that the matrices (}(s, N)
·~d'N ·N-I,
N =N(s) transform like the connection
matrix of a connection . COROLLARY
3.10. L et E be a lwloTlwrphic bundle with Hermitian norm N, and let
() , K denote the connection and cun'ature matrices of D(N) relative to a holomorphic frame over U. Then on U one has:
() is of type (1,0), and d'() =() l'IfJ .
(3 .11)
K=d"(), whence K is of type (1 , 1) and d"·K = O.
(3.12)
d'K = -[K, ()].
(3.13)
Proof. The first line follows directly from () =d'N ·N-\ where N = N(s) is the norm of s. Indeed d'() =d'd'N · N - l-d'N · d'N - l and d'N-l =
-
N - 1.d'N · N-l. The others are even more
straightforwa,rd. Note that because K is of type (1, 1), the chamcteristic classes of the con-
nection D(N) always are of type (p, pl . These formulae become especially simple when E is a line bundle. Then a holomorphic frame is simply a nonvanishing holomorphic section s, so that, relative to s, () =d' log N(s) and K =d"d' log N(s). Thus in particular, if E admits a global nonvanishing holomorphic section s, then
c1 (E) =
~
-
2n
d"d' log N(s).
(3 .14)
284 83
HERMITIAN VECTOR BUNDLES
The next proposition is a refinement of the earlier homotopy formula (2.19). For simplicity, we will abbreviate K[E, D(N)] to K[E, N]. PROPOSITION
3.15. Con.sider a smooth family of norms Nt> on the holomorphic bundle
E. Then the function (s, s')--+d(s, S'>N,jdt is Hermitian linear over AO(X) and hence determines a section LtEr Hom (E, E), by the formula s,s' EnE). If ({I is any invariant form in Ik(M n ); n=dimE, then
(3.16}
'>
d (s, f f" s>= ff' '&,(s,s; d &
Proof. We have
so that L is well defined. Hence ({I'((K[E, Nt]; L t)) is a global form and it suffices to check the formula (3.17)
locally. We therefore choose a holomorphic frame
8
= {s;} over U, and set N = Ne(s),
K=K(s, D(N t )), 8=8(8, D(N t )) . Then the matrix of L relative to s, is easily computed to
be NN-l, the dot denoting the t-derivative as before. Let us denote this matrix by L also. Finally, we will abbreviate ({I'((K[E, Nt]; L t }} I U to O. Then d'O
=
2. ({I(K, .. . , -
j*;
[K, 8] , .. . ,L , ... ,K) + 2.rp(K, ... ,d'L , ... ,K) . (I)
(j)
j
(3 .18)
(j)
Applying the invariance identity one obtains d'O
=
2.1 ({I(K, ... , d'L+(I) [L,8], ... ,K).
(3 .19)
Finally, one now computes directly from L =NN-l and 8=d'N ·N-l, that 8 =d'L + [L, 8] .
(3.20)
Hence d'O is the form ({I'((K[E, D(Nt)], D(Nt))) of Proposition 2.18 so that (2 .19) implies_ (3.16) . Q.E.D.
285 84
RAOUL BOTT AND S. S. CHERN
This proposition now directly proves Proposition 1.2 of the Introduction. Indeed if
NI and N2 are two Hermitian norms on E, then N t =(I-t)N1 +tN2 defines a smooth family between these two Hermitian norms, so that the formula of the proposition becomes a special case of (3.16). As another direct application we have: COROLLARY
3.21. Suppose E is an n-dimensional complex vector bundle over X,
with Hermitian norm N. Suppose also that E admits n holomorphic sections which span the fiber at each point. Then the refined Chern classes c,(E) are zero for i>O so that: c(E)=1.
Proof. Let
8
(3.22)
be the global frame determined by the sections in question, and define
a Hermitian norm on E, by setting NI(S) =identity. For this norm O(s), K(s) and hence c,(E, N 1 ), i>O clearly vanish. Q.E.D. Remark. The deformation Dt=D(N t ) induced by the variation of Nt is not the linear one encountered earlier. Rather, D t satisfies the differential condition: (3.23) In other words
b t is the D' -derivative of L t , and it is clear that much of the foregoing
depends on just the existence of some LtEr Hom (E, E) for which (3.23) is valid.
In the remainder of this section we will formulate a generalization of (3.16) along these lines. DEFINITION
3.24. A connection D on the holomorphic bundle E over X, is called of
.type (1, 1) if: (3.25) For any holomorphic section s of E I U, D"s =0. (3.26) The curvature matrix K(s, D) of D relative to a frame saver U, are of type (1, 1), i.e., K[E, D]EALl(X; Hom (E, E)).
This is then clearly an extension of the class of connections induced by Hermitian norms on E . Next consider a family of connections D, of type (1, 1) . Such a family will be called bounded by LtEAO(X; Hom (E, Ell if the relation (3.23) holds between btEAl(X; Hom (E, E))
.and Lt. Note that the elements of
r
Hom (E, E) may be thought of as defining degree
286 HERMITIAN VECTOR BUNDLES
85
zero differential operators on A(X; E) and on A(X; Hom (E, E)), the latter action being induced by the composition of endomorphisms. With this understood, the bounding condition (3 .23) may quite equivalently be expressed by: (3.27) In any case it is now easy to check that our earlier argument leading to (3.16) also proves the following more general homotopy lemma. PROPOSITION 3.28. Let D t be a smooth family of connections of type (1,1) on the holomorphic bundle E. Suppose further that D t is bounded by LtEAO(X; Hom (E, E)). Thenfor any g;Elk(M n ), n-=dim E, we have the relation-:
(3.29) We note in conclusion that if D t is related to LtEr{Hom (E, E)} by (3.27), and if Do is of type (1, 1), then D t will be of type (I, 1) for all t. Indeed, D t is of type (1,1) if and only if 0 =O(s, D t ) satisfies the two conditions: OfjEAl.O, d'O =0 1\ 0, whenever s is a. holomorphis frame . Differentiating these conditions with respect to time one obtains: (3.30) Now if (3.23) holds then-setting Lt{s) equal to ~we have d'L+ [L, 0] =(j . It follows that d'(j = [d'L, 0] +[L, d'O] which by resubstituting (3.23) leads to d'O = [11 , 0]. Thus (3.23) together with Oij EAl.O imply the differentiated identities. Q.E.D.
4. The duality formula Consider an exact sequence of holomorphic vector bundles: (4.1) over the base manifold X. We wish to prove the duality formula: c(E) =c(Er)c{Eul for the refined Chern classes of these bundles. For this purpose consider a norm N on E. Such a norm then induces norms N r on
Er and NIl on Ell in a natural manner: The restriction of N to Er defines N r, and the restriction of N to the orthocomplement of Er-denoted by Et---determines NIl' via the Coo isomorphism of Ell and Et induced by (4.1) . Thus (4.1) gives rise to three Chern forms in A(X):c{E)=c{E,N), and c(EbN f ),
287 86 i
=
RAOUL BOTT AND S. S. CHERN
I, I I; and the duality formula will be established once we prove the following proposi-
tion. PROPOSITION
4.2. There exists a form
~
in A(X) such that (4.3)
The proof of (4.3) is based upon a specific deformation of the canonical connection D = D(N). To describe this deformation we need certain preliminaries concerning the geometric implications of the exact sequence (4.1). First we introduce the orthogonal projections (4.2)
which this situation naturally defines. These are then elements of
r Hom (E, E) and there-
fore-interpreted as degree zero operators, they lead to a decomposition of D = D(N) into four parts:
LEMMA
4.3.
In the decomposition just introduced, PjDP j induces the connection
D(N i) on E;. while Pi DPj, i =l=j, are degree zero operators of type (1,0) and (0, 1) respectively: (4.4)
Proof. We first show that P j DPj, i =l=j is AO(X) linear. Consider then Pi DPAts) . Using the derivation properly of D we get PiDPj(fs)=dj-PjP,s+fPiDP,s. Hence as PiP,=O if i=l=j, it follows that PiDPj is a degree zero operator. We next show that PIID"PI=O. This is clearly also a degree zero operator. Hence it is sufficient to show that given e E Ex> there exists some section s of E near x, such that s(x) =e and P//D"PIs =0 at x. Now because
EI is a holomorphic sub-bundle of E we may choose a holomorphic section S1 near x such that the two conditions, S1(X) =PIe; P I S1 = S1 (i.e., S1 Er(E I)) hold ncar x. We may also 0
Q[Eu))).
293 92
RAOUL BOTT AND S. S. CHERN
Assume now that Q[E);;'O. Then Qu[E);;'O and hence by (5.6) Q[Eu);;.t.u[E);;'O. From this it follows that cn.:.1(E u );;' O. The form
L cx- 1 ( -
~
can be written as
1)« det« {Q[Eu); t.u(E)}.
« >0
Hence
~
is an alternating sum of positive terms and therefore neither positive or negative.
However, this is not serious. In fact we can add to ~ a closed form and
~
-
~0 :s:;; O.
~o
so as to make
~ +~o;;.
0
This is done as follows: Let ~o =
L cx-
1
det« «Q[Eu); Q[ElI)))'
(5.8)
« >0
Then by the definition of det«, ~o = Lcx-1(:)cn_1(Eu ), and hence is a closed form. Further note that in view of (5.4), we have
and so our assertion concerning We next replace
~
by
~-~o
~o
is correct.
in (4.1) and use the definition de =i(d" -d').
(5.9)
The formula (4.7) then takes the form: (5.10)
with the bracketed term ;;.0 wherever log N-l(s) >0, i.e., wherever N(s) < 1. Applied to B*(E), this formula therefore precisely proves Theorem I.
6. The relative Gauss Bonnet theorem We already remarked in the introduction that the first main inequality of the Nevanlinn a theory may be thought of as a twice integrated version of the formula (1.2) in Theorem I. The first integral of (1.2) leads to the generalized theorem of Gauss-Bonnet (for the
complex case) and so serves to give a geometric interpretation of the Chern classes cj(E). In this section we will, for the sake of
com~leteness,
briefly derive this development.
The situation we wish to study is the following one: let E be a holomorphic n-bundle with a Hermitian norm N, over the complex n -manifold X with boundary assume that
Sy
ax =
Y, and
is a nonvanishing section of E over Y. The question now arises when
Sy
may be extended to all of X without vanishing, and Theorem I, in the explicit form given by (4.22) may be interpreted as giving an answer to this question.
294 93
HERMITIAN VECTOR BUNDLES
Indeed, let EocE, be the subset {eIN(e»O} complementary to the zero-section in
E, and let no:Eo-+ E be the projection. As· we already remarked, the identity inclusion Eo-+E then induces a nonvanishing section (5.10) gives rise to a definite form
Sf
of nol(E) over Eo, so that the formula
e over Eo, for which
At this stage we will actually only need the form
de 1J(E) = 4n
e,
for which we therefore clearly have the identity (6.1)
In terms of this form, the answer to our question is given by the following proposition. PROPOSITION
6.2. The section Sy of Eol Y may be extended to all of Eo if and only if
The proof of this proposition follows directly from quite elementary obstruction theory, once it is established that the expression y(X; Y; Sy) =
f x cn(E) -
hS:'YJ(E), always
measures the number of times any extension of Sy to X has to vanish. To be more precise we need to recall the topological definition of the order of vanishing of a section
S
of E
at a point p which is an isolated zero of s. This is an integer, denoted by zero(s; p), which is defined as follows: Let Be be a disc of radius e > 0 about p, relative to local coordinates centered at p. Also, let -(x, v), xEP1 (V), of T 1 (V) into Ql(V) , Clearly, if
V
,*,0 then
Sv
vanishes at only one point lEP 1(V) namely at the subspace, [v], generated
by v. Further it is not hard to see that zero[sv, [v]] = l. Thus a) is satisfied. To check the condition fJ) we need a hermitian structure on T 1 ( V) which we of course take to be the trivial one induced by a hermitian structure on V. Thus the curvature form of T 1( V) is equal to 7.ero. Hence by the inequality (5.7) we note that Q{Ql( V)} :;;;. O. Thus in any case f Q,{y)C n {Ql( V)} :;;;. O. Hence it will be sufficient to show that Cn {Ql( V)} is an orientation class for P 1 ( V) to establish fJ). Consider the case n = 1, first. Let VI' v 2 be an orthonormal base for V and let Z->-[Vl + ZV 2 ] be a local parameter near [VI]' Also let SI be the section sv" which is therefore hoI om orphic and ,*,0 on P 1 ( V) - [VI]. Hence on this set
where l SI I(l) is the norm of the section at l . Thus near [VI] we have, ISl I 2([Vl +ZV 2 ]) = IzI2/(1
It follows, again by Stokes, that if BE = ( Iz I < e) then
+ IzI2).
297 96
RAOUL BOTT AND S . S. CHERN
Clearly the second term tends to zero, while the first tends to write z = re
tB
,
+ 1, as is seen directly if we
log z = log r + iO and recall that de = id" - id'. Thus fJ) is true for n = 1.
To get fJ) in gener~l one may use the Whitney duality formula. In the present instance this formula yields: C{Sl(V)}'C{Ql(V)} =c{Tl(V)} = 1.
Thus cn{Q(V)} = [- C1{Sl(V)}r· For n=l, this implies that C1{Sl(V 2 )} is an orientation class of P 1(V 2). Now under the inclusion V 2---7 V, Sl(V) clearly restricts to Sl(V2), Hence C1S1(V)} restricts to an orientation class of P 1(V 2 ). But then C1{Sl(V)} must generate H2(X;Z), X=P1(V) and hence (-ltCl{Sl(V)} must be an orientation class for P1(V)
in general. Q.E.D. An important corollary of (6.5) is the following interpretation of cn(E): COROLLARY
6.13. Let E be a lwlomorphic n-bundle over the complex n-manifold X,
and let s: X ---7 E be a smooth section of E, which is 9=0 on
ax, and which is transversal to the
zero section of X in E. Then zero(s) has a natural structure of a COO manifold of real codimension 2n in X, and the proper orientation class of zero(s) is the Poincare dual of cn(E). Proof. Let y be a smooth singular n -cycle in the interior of X, which is transversal to zero(s), i.e., every singular simplex II which intersects zero(s), meets it in an isolated interior point. Just as in the proof of (6 .4) one now concludes from (6.7) that
f
cn(E) = intersection (II, zero(s)) +
a
He!1ce summing over
II
J
s*'YJ(E).
•
in y , we obtain:
J y
cn(E) = intersection (zero(s) , y).
Q.E.D.
Remark I. It is of course artificial to bring in any assumption of complex analyticity when dealing with the Gauss-Bonnet theorem, and one could modify this account by defining'YJ directly on any smooth hcrmitian bundle. However as we are primarily interested in the complex analytic case here and the more general approach would have taken us even further afield, we only discussed that case. In the next integration the analytic structure plays a vital role. Remark I I. There are two quite straightforward generalizations of the exact sequence
298 HERMITIAN VECTOR BUNDLES
97
over P1(V), for which we will have use later on. Namely, if Pn(V) denotes the Grassmanian of n·dimensional subspaces of V, we have the corresponding sequence O-+Sn(V)-+Tn(V)-+Qn(V)-+O
over P n(V), with T n(V) = P n(V) x V , and Sn(V) being the subset of pairs (A, v) with v EA. Finally this construction makes sense when V is replaced by a vector bundle E over X . That is, one defines Pn(E) as the pairs (A, x) consisting of a point xEX, and an n-
dimensional subspace A in Ex. One lets Tn(E) be the bundle induced from E over Pn(E) by the projection 1\(E)-+X, and then obtains an exact sequence
where Sn(E) consists of the triples (A, x, e) with eC A.
7. The second integration; definition of the order function We are
IlOW
in a position to discuss the generalized first inequality of the Nevanlinna
theory. Just as in Section 6, we will be dealing with a holomorphic hermitian vector bundle E over the complex manifold X, however instead of assuming that X is compact we assume
only that X admits a "concave exhaustion" f. By definition, such an exhaustion is a smooth real valued function, (7.1)
f maps X
f, on X such that
onto R+
(7 .2) f is proper, that is, f-l(K) is compact whenever K is.
(7.3) The (1, 1) form ddcf is
~
0 for large values of f.
With respect to such an exhaustion of X, one defines the order-function of E, by the formula (7.4)
The behavior of T(r) as r-++ the compact case.
00
is then to be thought of as the analogue of
f x cn(E) in
One next defines a corresponding order function for the number of zeroes of a section s on E which is assumed to have only isolated zeroes, by the formula
N(r,s) =
f~ oo zero(s,XT)dr
where zero(s, X T ) = L: zero(s, p), p ranging over the zeroes of s interior to X T • 7- 65 2 932 Acta malhemalica 114. Imprime Ie 11 Bout 1965.
(7.5)
299 98
RAOUL BOTT .AND S. S. CHERN
We note that if the integral along the boundary of Xr could be disregarded, the formula (6.5) would imply that N(r, s) = T(r). This is of course false in general, however we do have the following estimate of this error term under certain circumstances. FIRST MAIN THEOREM. Let E be a positive Hermitian bundle over X where X
has
a concave exhaustion f. Let s be a holomorphic section of E with isolated zeroes, and let N(r, s) be the order function of these zeroes. Then N(r, s) < T(r)
+ constant
(7.6)
where T(r) is the order function of E. In particular if cn(E) > 0 at some point of X, then lim {N(r, s)/T(r)} ':;; 1. Hence the deficiency mea,sure of s, defined by: 6(s) = I-lim {N(r, s)/T(r)} satisfies the inequality
o':;; 6(s) ':;; 1. Proof. Let
"above"
r
rc X
(7.7)
x R be the graph of f, and let W be the region in X x R, which is
and " below" the slice X x r :
W = {(x, t) If(x) .:;; t ':;; r; xEX, tER} . The natural projection W -+ Xr will be denoted by a.
It is then clear that
T(r) =
I
wa"cn(E)dt
with W the orientation induced by the product orientation on X x R, and dt the volume element on R . Suppose now that s =1=0 on X r • Because
lsi < I we may think of s as a section of B*(E)
so that on Xr
where e =e(E) is the form given by Theorem I on B"(E). We may therefore write a"(cn(E) A dt) as d{a"s"dCe A dt}/4n and apply Stokes' formula
to obtain: T(r) =
~
4n
f
a"s"dCe II dt.
(7.8)
ilW
Now the boundary of W clearly falls into the top-face Xr
r" which is the graph of II Xr: aW=(Xr x r) U rr.
X
r, and the bottom face
300 99
HERMITIAN VECTOR BUNDLES
Further, the integrand in (7.8) clearly restricts to zero on the top-face, as dt does. Hence, keeping track of the orientation we obtain so that identifying
-1/4nf rra*s*dce /I dt for this integral,
rr with Xr one obtains: T(r) =
~
4:n:
fx,
s"dCe /I df·
(7.9)
We next use the fact that s is holomorphic. This implies that s"dCe =dCs"e and, furthermore, that s*e EAn-Ln-1(X). Now a direct verification shows that the following identity is valid: PROPOSITION
7.10.
If X is an n-dimensiona,l complex manifold, and fEAO(X),
AEAn-Ln-1(X), then
df /\ dCA = d(dCfA) - Addcj.
(7.11)
When this identity is substituted into (7 .9) and the Stokes formula is used once more in the first term we obtain the relation:
T(r) =
~
4:n:
f
dct- A -
OXr
~
4:n:
f
Addcf,
A = s* e(E)
(7.12)
Xr
and this is the basic integral relation which lies behind the first main theorem when s does not vanish on X r • In the case when s vanishes at isolated points Pi' i
=
I, ..., m , in Xr let X: be obtained
from Xr by deleting e discs D/(e) about the Pi> and let W(e) be W with the solid cylinders C/(e) above these discs removed. Now
T(r) =
~ limf
4:n: constant.
Prool
01 the lemma. We need to estimate the form
J..=s*e near an isolated singularity,
p, of s. For this purpose choose a holomorphic trivialization rp :E-*Ep, of E near p . Then
.'J"e will be close to (rpos)*j;e near p, so that it is sufficient to study this form near p. Our first task is therefore to describe j;e. Let :n;:Ep-*p, ar.d set E=:n;-l(Ep) be the induced bundle over Err The identity map
Ep -* E p, then definr;s a section s of E, which does not vanish on Ep.o = Ep -0, and so geneTates a sub-bundle E] of E there. Let jp:Ep.o-*E be the inclusion. The form j;e is then made out of the curvature forms of E] and Ell = EIE], according to the prescription (5.10). Now as E is clearly the trivial Hermitian bundle over Ep-O, the curvature of E vanishes identically. Hence K(E lI ) has the form 6 i\ 6", where 6 is the degree zero operator PlIDP] of Section 4, and may be computed explicitly. Indeed let u"" '" = 1, ... , n, be an orthonormal frame for E p , and let z'" be the corresponding local coordinates on Ep so that Lz",(q)u",=q,
qEE p ,
302 101
HERMITIAN VECTOR BUNDLES
and let r(q) = (2:lz«(q)12)!. If we interpret the u« as the constant sections of E then the identity section s is given by s(q) = 2:z«(q)u«, and so Ds(q) = 'i,dz«u«. It follows that at a point q, with Zl(Q) = r(q), zp(q) = 0, the frame of Ell determined by the Up, 1 _ 2dz« 1\ dzp, r
f3 = 2, .. . , n, the curvature matrix f3 = 2, .. . , n, is simply given by IX,
f3 =
2, . .. , n.
relative to
(7.18)
In particular then,
With the aid of (7.18) one may estimate all the terms of (5.10) and so conclude that: (7.19) where Wt is bounded on all of Ep.o. The lemma now follows easily from (7.19). Assume first that s is transversal to the zero section at p. Then the Jacobian of rpos is not zero at p. For our convergence questions rpos may therefore be replaced by the identity map. Now let D(e) be the ball of radius e about 0 in
en. Then if e is of the type given by
(7 .19) we clearly have
e1\ O~O,
f
and
IlD('l
for any bounded I -forms () and rp because the volume of the sphere of radius r is of the order r2n-l
and so dominates
r 2(n-ll
log r. The lemma therefore is clear in that case. For a general
isolated zero of s, there exist arbitrarily small perturbations of s with only a finite number
of nondegenerate zeroes near p. Hence our lemma also holds in that case.
8. Equidistrihution in measure In this section we derive "he generalized first equidistibution theorem from the first main theorem with the aid of two essentially known but hard to refer to propositions which are then taken up in later sections. We start with a statement of the theorem we are after: EQUIDISTRIBUTION THEOREM .
L et E be a complex vector bundle of fiber. dimension
n, over the complex connected manifold X, and let V c r( E) be a finite dimensional space of holomorphic sections of E . Assume further that,
(8.1) X admits a concave exhaustion f, in the sense of Section 7. (8 .2) V is sufficiently ample in the sense that:
303 102
RAOUL BOTT AND S. S. CHERN
a) The map
s~s(x),
maps V onto Er for each xEX.
fJ) There is some sEV, and some xoEX, so that s:X-7E, is transversal to the zero-section of E at Xo' Under these circumstances nearly every section in V vanishes the same numher of times. Precisely, a hermitian structure on V defines a hermitian structure on E, and hence a deficiency measure b(s) on the generic sections of V . The assertion is that except for a set of measure 0, the kernel of lir
has dim m. Now it is clear from (8.3) that the induced map e v : X
x~kr'
~ P m( V) ,
defined by
determines an isomorphism of Qm(V) with E: That is (8.4)
A hermitian structure on V induces one on T m( V) and hence on Qm( V) and Sm( V) and hence by (8.4) also on E. Note further that Qm( V) is positive in this structure as T m( V) clearly has zero curvature and "quotient bundles are always more positive" (see Section 4) . Hence E is also positive. Finally, the "height of a section s" in V at any point xEX is clearly bounded by the length of s " qua element" in V. In short we may, after possibly multiplying s by a suitable constant, not only apply the notions of Section 7 to E, but we also obtain the inequality of the first main theorem : N(r, s) < T(r) + constant
valid for sections with isolated signularities. Now condition fJ, of (8 .2) is seen to imply by an explicit check , that e~cn{Qn(V)} is strictly positive near Xo (see remark at end of Section 9). Hence T(r) ~ + 00, so that (8.4)
implies the inequality: ';;: l-;-N(r, s) ';;:1 ..., .
O ..., 1m T(r)
(8 .5)
We now need the following two propositions: PROPOSITION
8.6. Under the assumption (8.2) nearly aU se V, have only isolated
zeroes . In fact nearly all sections sE V are transversal to the zero section of E. (') See the remark at the end of Section 6.
304 HERMITIAN VECTOR BUNDLES PROPOSITION
103
8.7. Under the assumptions (8.2) we have the equality
f
N(r,s)w=T(r),
sE[s] EPI(V)
(8.8)
P,(V)
where w is the volume on PI(V) invariant under the group of isometries of V and normalized by f p, 0 near Xo' 10. Some remarks on integral geometry. The proof of Proposition 8.7 Suppose n : Y --+ X is a smooth fibering of compact manifolds with oriented fiber F. In that situation there is a well-defined operation
called integration over the fiber, which "realizes" the adjoint of n* in the sense that if X and Yare oriented compatibly then for any cpEA(X), 1pEA(Y) we have the identity: (10.1)
The existence of n* on the " form level", suggests the following definition. DEFINITION
(Y, Z, w) where
10.2. L et cpEAk(X) . By an integral representation of cp we mean a triple,
Y~X
is an oriented fibering over X, with projection n, and wEAm(z) is
a volume element(l) on the oriented m-manifold Z, together with a map
(J:
Y --+Z, such that (10.3)
In general the question whether a given closed form cp on X admits an integral repre(') Volume element means a nonvanishing form of top dimernsion, in the orientation class.
306 105
HERMITIAN VECTOR BUNDLES
sentation seems quite difficult. Certainly
f{J
must have integral periods and there are most
probably much more subtle conditions which also have to be satisfied. For our purposes it will however be sufficient to show that the characteristic class cn(E) of a hermitian bundle which is ample in the sense of (9.2) CIC) always has an integral representation. Note that if
f{J
has an integral representation, then any pull-back
/*f{J
also has an integral repre-
sentation. Hence it will be sufficient to get a representation theorem for cn[Qn( V)] over Pn(V)· In the next proposition we describe a quite general representation theorem for the
Grassmann-varieties Pn(V). We will first simplify the notation as follows: V will denote a fixed hermitian vector space of dimension d; and we write simply Pm Qn etc., for Pn(V), Qn(V) etc. The bundle Qn is always considered in the hermitian structure induced on Qn by the trivial structure on Tn; so that the Chern forms c(Qn) are well-defined.
Now let 0 1,
hold for arbitrarily large values of c. We can clearly conclude from (11.21) and these two lemmas that lim A(c)/T(c) =0, so that the Nevanlinna theorem is a direct consequence of (11.22) and (11.23). Both Lemmas are well·known, see for instance [2]. The first one follows from an integral geometry argument, while the second one is a purely real variable inequality.
313 112
RAOUL BOTT AND S. S. CHERN
References This paper links classical function theory with differential geometry; it is difficult to give an adequate bibliography satisfactory to readers from both fields. We will restrict our· selves in giving some standard literature from which other references can be found: [1]. Classical value distribution theory: NEVANLINNA, R., Eindeutige analytische Funktionen. Berlin, 1936. [2]. Holomorphic curves in projective space: .Am.FORS, L., The theory of meromorphic curves. Acta Soc. Sci. Fenn., Ser. A. 3, no. 4 (1941). WEYL, H., Meromorphic Functions and Analytic Curves. Princeton, 1943. [3]. Differential geometry of connections: KOBAYASm, S. & NOMIZU, K., Foundations of differential geometry. Interscience, 1963. [4]. Characteristic classes: HmZEBRUCH, F., Neue topologische Methoden in der algebraischen Geometrie. Springer 1962. [5]. Curvature of connections and characteristic classes: CHERN, S., Differential geometry of fiber bundles. Proc. International Congress 1950, 2 (1952) 397-411. GRIFFITHS, P. A., On a theorem of Chern. IUinois J. Math., 6, 468-479 (1962).
Received August 17, 1964
314
. On the Kinematic Formula In Integral Geometry SHIING-SHEN CHERN 1. Introduction. The kinematic density in euclidean space was first introduced by Poincare. In modern telminology it is the Haar measure of the group of motions which acts on the space. One of the basic problems in integral geometry is to find explicit formulas for the integrals of geometric quantities over the kinematic density in terms of known integral invariants. An important example is the kinematic formula of Blaschke, as follows [1]: Let E3 be the euclidean three-space, and let dg be the kinematic density, so normalized that the measure of all positions about a point is 87r2 _ (In other words, the measure of all positions of a domain D with volume V such that D contains a fixed point is 87r 2 V.) Let Di , i = 1, 2, be a domain with smooth boundary, of which Vi , M ~i), M~i) , Xi are respectively the volume, the area of the boundary, the integral of mean curvature of the boundary, and the Euler characteristic. Then, if Dl is fixed and D2 moves with kinematic density dg, we have
where xeD} n gD.) is the Euler characteristic of the intersection Dl n gD 2 • Formula (1) contains as special or limiting cases many of the formulas in integral geometry in euclidean three-space. It was generalized to euclidean n-space by C. T. Yen and the present author [3] and to the non-euclidean spaces by Santalo [4].
This paper will be concerned with a pair of compact submanifolds (without boundary) M P , M· of arbitrary dimensions p, q in euclidean n-space E" and with the integration with respect to dg of certain geometrical quantities of the submanifold M P n geM"). The latter depend only on the induced riemannian metric of M P n geM') and are defined as follows: Let X be a riemannian manifold of dimension k. In the bundle B of orthonormal frames over X we have the coframes, which consist of k linearly independent linear differential forms CPa , such that the riemannian metric in X is (2) 101 Journal of Mathematics and Mechanics, Vol. 16, No.1 (1966) .
315
s.
102
S. CHERN
(In this and other formulas in this section we have 1
~ Ci, (3, 'Y,
0
~ k.)
Let
(3)
be the connection forms of the Levi-Civita connection. They are linear differential forms in B and satisfy the" structural equations" dtpa
=
L: f{J{J /I. f{J{Ja , L: f{Ja"/ /I. f{J"l1l + .pall , (J
(4) dtpall
=
"/
where (5)
The coefficients
(6)
Safl"l!
are functions in
and have the symmetry properties
B
Sall"/!
=
-Sall!"I
=
Sall"l!
=
S"llall ,
SaP"/!
+ Sa"/!P + Sa!P"I
-Spa"/! ,
=
o.
From these functions we construct the following scalar invariants in X: (7) ~
where e is an even integer satisfying 0
ofCil
~
e
k,
••• Ci.) (3.
\81 ...
+ 1 or -1 according as CiJ , • •• , Ci . is an even or odd permutation of {3, , .. . , {3. , and is otherwise zero, and the summation is over all Ci\ , ..• , Ci. and (3\ , , fl. , independently from 1 to k. When X is oriented and compact, we let
is equal to
(8)
iJ..(X) =
L
I. dv,
where dv is the volume element. Thus J1.. (X) are integral invariants of X, with J1.o(X) equal to the total volume. If k is even, then the Gauss- Bonnet formula says that [2] (9)
(k -
? )k12 ( ~7r
1)(k -
3) . . . 1 X
(X) ,
where x(X) is the Euler-Poincare characteristic of X . The numerical coefficient before the summation in (7) is so chosen that I,(X) = 1, when X is the unit k-sphere in Ek+l with the induced metric. These integral invariants appear in a natural way in Weyl's formula for the
316
103
THE KINEMATIC FORMULA
volume of a tube [8]. In fact, suppose X be imbedded in En and suppose T. be a tube of radius p about X, i.e., the set of all points at a distance ~ p from X. Then, for p sufficiently small, Weyl proved that the volume of T p is given by the formula (10)
_ V(Tp) - 0 ...
~
(e - 1)(e - 3) .,. 1 "'h 2) ... m /l,(X)p ,
"7' (m + e)(m + e _
o~ e ~
e even,
k,
m = n - k,
where 0", is the volume of the unit sphere (of dimension m - 1) in Em, and its value is given by (11) This formula is also valid for large values of p, provided that the volume is counted with multiplicities for domains where Tp intersects itself. The main result of this paper is the kinematic formula
(12)
J}J.,(M n P
L
gMQ) dg =
Q c.}J..(W)}J.,-.(M ),
O:!I ~t!
, even
e even,
o~ e ~ p +
q - n,
where c. , to be given below (formula (79», are numerical constants depending on n, p, q, e. 2. Preliminaries. Let En be the euclidean space of dimension n, and xel .•. en an orthonormal frame, or simply a frame, in En, so that x I: En and e1 , • • • , eft are vectors through x, whose scalar products satisfy the relations 1 ~ A, B, C ~ n.
(13)
The kinematic density is the volume element in the space of all frames, and is given by (14)
dg
=
A
A.B.C B+r
dz k A dz k
335
122
CHERN, LEVINE, AND NIRENBERG
and choose Cas before.
Then we have
r)6 (ddcv)r A '1fr ~ r ::( dd'v)r A Y -v )6-V
dV r =
(-v)
- r
~4
1
dv A d'v A (dd'v),-' C'1fr - ~ dC A d'v A (dd'v),-' (_V)'+l
6
(-v)'
where the last equality follows from Green's theorem. dC Ad'v -dv Ad'C = i(iJ +aK A (a -iJ)v- i(iJ+a)v A (a -
Av,
Since
oK
= 2i(ac A av - iJC A iJv) ,
it contains no term of type (1,1).
r
L
It follows that
r
dC A d'v A (dd'v),-l A '1fr = dv A d'C A (dd'v),-' A '1fr , (_v)r )6 (-v)'
and by Green's theorem this is equal to _
1 r -1
r
L
dd'C A (dd'v),-l A V . (_V)'-l
On the other hand, one sees easily that dv A d'v A (dd'v)r-l A V is a non-negative multiple of d V. Hence we get Ir ~
1
r - 1
r
L
dd'C A ( dd'v )r-, A '1fr ,
-v
and the desired inequality (7) follows from induction hypothesis. COROLLARY. Let the polydiscs ~, ~, be defined as in the lemma. Let u be a CZ-plurisubharmonic function in ~ with o < u < 1. Then there is a constant B independent of u such that for any r x r minor of (U ik ) we have the estimate
To deduce this corollary from the lemma we set v = u - 1. Then 0 < - v < 1 and abs.
~ _ l _ abs .
-
(-v)'
336
123
INTRINSIC NORMS
Thus the corollary follows from the Lemma and (7'). For r = 1 the corollary was proved by P. Lelong [9]. The lemma proved above has a real analogue whose proof is similar. Let v be a negative convex function of class C2 in a domain g) in R", i.e., a function whose hession matrix is positive semidefinite,
Ei,j V.i.jei~j. ~ 0
0
Fo?" any subdomain K with compact closure in constant A, independent of v, such that
rJK{I v.v.
k
I ~ + -l-Iany 7' x Ivl r
3.
r minor of (v
g),
the're is a
j)l}dV ~ A.
i %%
Semi-norms and their properties
Using the definition (2) we shall prove the theorem. THEOREM 1. Let M be a complex manifold and / a homology class with real coefficients. Then N{t} is finite. To prove the theorem, let T be a closed current belonging to /0 By a theorem of de Rham [10, §. 15], there exist operators RT, AT whose supports belong to an arbitrarily small neighborhood of the support of T such that (9)
RT = T
+ bAT + AbT ,
where b is the boundary operator of currents. The operator R is a regularizing operator, constructed by convohtion with a smooth kernel, which is given by (10) where"", is a closed C~-form with support in a neighborhood of the support of T. Since bT = 0, there exists in every homology class a regular current and it suffices to show that upper bound independent of u, where
IP = dCu A (ddCu)k-l , IP = du A dCu A (ddCu)k-l , Consider first the case dim / = 2k - 1. we have
LIP A "'" has a finite dim / = 2k - 1, dim / = 2k .
By Green's theorem
337
124
CHERN, LEVINE, AND NIRENBERG
Since..;r is a fixed C--form, it follows from our Corollary in § 2 that this integral is bounded in absolute value by a constant independent of u E 1"; If dim 'Y = 2k, we have, since dy = 0,
L
0 when 'Y *- o. Remark 2. The family, 1", of plurisubharmonic functions on M with values between 0 and 1 can be used to define another seminorm which assigns to a homology class 'Y the number {13a)
N'{'Y} = suP.. ;€~ infT€T I T[dCu o 1\ ddcu l 1\ ... 1\ ddCu k ]
I,
if dim 'Y = 2k
+
1
338
125
INTRINSIC NORMS
N'{'Y} =
sUP.,e~
inf Ter I T[du o /\ d Cu , /\ dd cu 2
/\
•••
/\
ddCu k] I ,
if dim'Y = 2k • The proof of finiteness of N'('Y) requires a slight modification of the preceding arguments. The analogue of the corollary needed here is Let the polydiscs ~, ~, be defined as in lemma of § 2. Let u" u 2 , •• ' . Ur be C 2 plurisubharmonic functions in ~ with 0 < u, < 1. Then there is constant C independent of the u,' such that if J = (j" . ", ir) and K = (ku "', k r) 1 ~ i, < ... < ir ~ nand 1 ~ k, < ... < kr ~ n, LEMMA.
where UJK is the coefficient of dZ;, /\ dZ k, /\ ••• /\ dZ;r 1\ dZ kr in dd cu , /\ ..• /\ ddcu" and dV is the element of 'Volume in C". To prove this we note first that the matrix with (J, K)'h entry, U JK is positive semi-definite (by induction on r), and so we need only consider the case J = K = (1, 2, "', r). Then using the notation and
technique of the lemma of § 2, we have
rj.l, I UJJ I d V = rj
6,
~
dd cu , /\ ... /\ ddcu r /\
L(·ddCu , /\ ... /\ ddCu r /\ ,y
= - )~ d( = -
=
,y
/\ d Cu , /\ dd"u 2
/\
•••
ddcu r /\
,y
L
du , /\dd Cu 2 / \ · · ·/\dd"ur/\d"(/\,y
~ 6 U, . dd"u 2 / \
•••
/\
ddCu r /\ dd"( /\
,y -
Theorem 2 and Remark 1 following it are true for N', and N';:;:; N. The equations replacing (13) and (13a) are their multilinear versions. (13')
(IS'a)
dd"u o /\ ... /\ ddcUk = 0 , for N' on H 2 k+l(M, R) . c duo /\ dd u , /\ '" /\ dd"u k = 0 , for N' on H 2k (M, R) .
Another, possibly larger, semi-norm results if we change the definition of N' by allowing U o to be any C2 function with 0 < U o < 1, but still requiring u" .. " Uk to be in ~. The other norms which we introduce may also be modified in a similar manner with the aid .of k + 1 functions in place of one.
339 126
CHERN, LEVINE, AND NIRENBERG
Remark 3. Another seminorm can be defined by the consideration of a different family of functions. Let ~I be the family of negative C 2-functions, defined locally up to a multiplicative positive constant, which are plurisubharmonic. For such a function v the forms (14)
dv
dd"v
v
v
are well defined on M. With the aid of the functions of fine, to a homology class 'Y, N {'Y} = sup I
.e~,
~I
we de-
infTer I T[ d'v /\(_V)k (dd'v )k-l ] I ' if dim 'Y = 2k-- 1 ,
(15)
N {'Y} = sup '
ve~,
inf
Ter
I T[ dv /\ d'v /\ (dd'v)k-'] I (_V)k+' ' if dim 'Y = 2k .
By applying the Lemma in § 2 it can be proved that N,{'Y} is always finite, and is hence a semi-norm in the homology vector space HI(M, R) (l = 2k - 1 or 2k). Unfortunately we know no example for which N,{'Y} is not zero. In particular, if dim 'Y = 1, then necessarily N,{'Y} = O. It may be observed that in the proof of Theorem 1, only the property of local boundedness of the functions u is utilized. We will therefore introduce wider families of functions and thereby generalize the semi-norms introduced above. Let 61.[ = {U;} be a locally finite open covering of M. We denote by ~(GU) the family of plurisubharmonic C2-functions U;: U; - R defined in each member of the covering which satisfy the following conditions: ( 1) the oscillation of U; in U; is less than one; ( 2) du; = du; in U; n U; *- 0. The latter defines a closed real one-form in M. Similarly, d"u; and dd'u; are also well defined in M. Without ambiguity, we can denote them without the indices. Analogous to (2) we define N{'Y, GU}
=
SUPue~(ql)
N{'Y, GU}
=
SUP"e~lql)
(16)
infTer I T[d'u /\ (ddCu)k-l] I if dim 'Y = 2k - 1 , inf Ter I T[du /\ d'u /\ (dd"u)k-l] I if dim 'Y = 2k •
340 127
INTRINSIC NORMS
Let 7r: M -+ M be the universal covering manifold of M and let be a fundamental domain on M. We denote by 1"( U) the family of plurisubharmonic C2-functions on M such that their oscillation in is less than one and their differentials are well defined on M. We define
a
a
N{'Y,
a} =
SUP"E~(UI infTET I T[dCu 1\ (dd"u)k-I]
I,
if dim 'Y
(17)
N{'Y,
a}
= 2k -
= SUP.. e:T
D.
be holomorphic mappings which satisfy the conditions
z EnD) , j;(D)
We choose
Zo
= Z, Zi
Let
ai'
I;:
E
fk(D) ,
n fi+l(D)
=F 0 ,
Zll ••• , Zk_lI Zk
E
j;(D)
= (,
n fi+l(D)
such that
,
bi E D be points satisfying Zi_l = j;(a;), Zi = f i(b;) ,
Then the Kobayashi pseudo-distance is defined by d(z, 1;:) = inf El&i:>k h(a i , b;) ,
(37)
where the infimum is taken with respect to all the choices made. Kobayashi proved that c(z, () ~ d(z, (). We will establish the theorem. Between the pseudo-distances the following inequalities are valid THEOREM 71 •
(38)
< IT c·.( z, ("V) ="2
< 7r P( w, " Y) < Pc ( z, '".V) ="2 ,,=
d( Z, '"r) •
1 In our original proof we showed c ~ (2 '::)p ~ d. Kerzman observed that our argument could be used to prove the more general result (38).
348
135
INTRINSIC NORMS
The first inequality becomes an equality if M is simply connected.
By the conformal mapping T
(39)
we map u + iv. and the induced ( 40)
= i - exp (rriw) i
+ exp (rriw)
,
the unit disk D onto the infinite strip S: 0< u < 1, w = Under (39) the real axis of D corresponds to the line! + iv origin of D to the point w = t. S has a hyperbolic metric from that of D by the mapping (39), which is given by ds Z
=
---,--_dT_d_f=-2(1 - cos 2rru)
(1 - n)'
Thus the hyperbolic distance on S between the points t and t + vi is (41)
Since S admits the group of hyperbolic motions, we can normalize the holomorphic mappings M -- S such that the image of z is the point ~ and the image of ~ lies on the line Re w = t. Hence the Caratheodory pseudo-distance can be redefined as follows. (42)
where g runs over all holomorphic mappings g: M -- S such that =!. If a denotes the segment joining g(z) to
g(z) = k and Re g«() g«(), we have
1m g(O =
L L
dCu =
dv =
~: dcu
.
where u = Re g is a pluriharmonic function which belongs to the family Y. Since T[d Cu] = 1m g«() for any chain T this proves that c(z, ()
~
-%- Pc(z, 0 .
To prove equality consider a pluriharmonic function u in :T. If among all chains T, infT I T[d'u] I 0, then the integral of dcu around any closed path vanishes. But then the function
*
v(z) =
r' dOu
J
'0
is well defined independent of the path of integration and is there-
349
136
CHERN, LEVINE, AND NIRENBERG
fore a conjugate pluriharmonic function of u. Hence w = u defines a holomorphic mapping 10: M -> S. By (40) we have
; I dv i ~
+ iv
ds .
Since dv = d'u, it follows that ;
p,(z, () ~ c(z, ~) ,
and hence
If M is simply connected then to every pluriharmonic function u in J" there exists a conjugate pluriharmonic function v defined up to an additive constant. It follows that for any chain T bounded by ( and z, T[d'u] equals v(~) - v(z) and is therefore independent of the chain T. Consequently p,(z,
0
= p(z, () .
To prove the last inequality in (38) we use the fact that ,0 is non-increasing under a holomorphic mapping. We will also follow the above notation in the definition of the Kobayashi pseudo-distance. Let li be the straight segment in D joining a i to bi and let fi(l;) = L i , 1 ~ i ~ k. Let u be a pluriharmonic function on M, with 0< u < 1. Then we have
..!!..- I r 2
JL;
d'u
I~ ~ po(a 2
i ,
b;) = h(a;, b;) ,
where Po is our metric in D. The last equality follows from what we just proved, as D is simply connected. It follows that
+IEi ~L; dc~~1 ~ Ei
h(a;, b;) ,
l~i~k.
Now the right-hand side may be chosen as close to d(z, () as we like, while the left-hand side is not smaller than (rr /2)(p(z, (»). This implies the desired inequality. Remark. We do not know when (7':/2)p(z, () = d(z, (), nor how (rr/2)p(z, (; U) compares with d(z, (). Using chains one may also introduce pseudo-metrics p ,(z. (; 011) and ,o,.(z, (; U); but on a compact manifold these are zero.
350
137
INTRINSIC NORMS
7. Remarks on the differential equations
The differential equations (13) and (13a) are, in general, overdetermined systems of non-linear differential equations. For k =1. equation (13), ddcu = 0,
asserts that u is pluriharmonic, while in general, equation (13) means that the rank of the hessian U;j is less than k. Almost nothing is known about the solvability of these equations. The case k = n reduces to a single equation which is the complex analogue of the Monge-Ampere equation det {U;i} = 0 ; it is non-linear degenerate elliptic in view of our requirement that the matrix U ;j be positive semi-definite. It would be interesting to formulate boundary value problems for these equations. We remark that the equation (ddCu)" = 0 also arises as the Euler equation for a stationary point of the functional I(u) =
(43)
L
d11, 1\ d"u 1\ (ddcu)"-l
under, perhaps, some boundary conditions. Consider for example the class B of C2 plurisubharmonic functions which are required to equal one on some components of the (smooth) boundary of a compact manifold M, and zero on the others. If v is a member of B, let 'Y denote the (2n - I)-dimensional homology class of the level hypersurfaces v = constant. Then we observe that for T E 'Y, if v satisfies (dd' v)" = 0, (44)
\ dv 1\ (dd' v)" -' = \ dv 1\ d'v 1\ (ddCv)"-' = I(v) .
J7'
)M
It is not difficult to verify that the functional I is convex and one is therefore tempted to conjecture that (45)
N {/ }
= infveB I(v)
.
If this is the case then N{'Y}, which is defined as the supremum of a functional would also be characterized as the infimum of another, a situation that often arises in, so called, dual variational problems in the calculus of variations. The problem of minimizing l(v) seems an interesting one. In the case of the annulus 1 < I z I < a,
351 138
CHERN, LEVINE, AND NIRENBERG
the function Vo = log I z Iflog a is indeed the minimizing function, since the convex functional I is stationary at Vo. The differential equation (13) has a real analogue, which is a2u ) ;£ k , rank ( -.~-. ax'ox'
(46)
1 ;£ i, j ;£ n ,
where u(xt, ... , x") is a real-valued Cqunction in the real variables xt, "', x". Equation (46) and its generalizations have been studied in connection with some geometrical problems (cf. [4], [5], [11]). In fact, if u = U(XI, " ' , xn) is considered as the equation of a nonparametric hypersurface in the eu~lidean (n+1)-space E"+t, the left-hand side of (46) is called the index of relative nullity, being the rank of its second fundamental form. Hartman and Nirenberg [5, p. 912] proved that for n = 2, k = 1 (in which case condition (46) means that the surface has zero gaussian curvature) the surface is a cylinder if U(XI, x') is defined for all (Xl, x2) E R2. For higher dimensions a similar result is not true, as shown by an example of Sacksteder [11]. In this respect we wish to refer to a general theorem of Hartman [4] concerned with sufficient conditions for an isometrically immersed submanifold in an euclidean space to be cylindrical. UNIVERSITY OF CALIFORNIA, BERKELEY BRANDEIS UNIVERSITY COURANT INSTITUTE. NEW YORK UNIVERSITY.
REFERENCES [1] [2]
[3]
[4]
R. D. M. ACCOLA, Differentials and extremal length on Riemann surfaces. Proc. Nat. Acad. Sci. USA 46 (1960), 540-543. L. V. AHLFORS and L. SARlO, Riemann Surfaces, Princeton University Press, Princeton , 1960. C. CARATHEODORY, Uber eine spezielle Metrik, die in der Theorie der analytischen Funktionen auftritt. Atti. Pont. Acad. Sci. Nuovo Lincei 80 (1927), 135-141. P. HARTMAN, On isometric immersions in euclidean space of manifolds with non-negative sectional curvatures. Trans. Amer. Math. Soc . 115 (1965),
94-109. [5 J - - - and L . NIRENBERG, On spherical image maps whose jacobians do not cha.nge sign. Amer. J. ~ath. 81 (1959), 901-920 . [6] H . HUBER, Uber analytische Abbildungen vvn Ringgebieten in Ringgebiete. Compos. Math . 9 (1951), 161-168. [7] S. KOBAYASHI, Invariant distance s on complex ma.nifolds and holomorphic mappings. J. ~ath . Soc. Japan 19 (1967), 460-480.
352 INTRINSIC NORMS
139
H. J. LANDAU and R. OSSERMAN, On analytic mappings of Riemann surfaces . J. Anal. Math. 7 (1959-60), 249-279. [9] P. LELONG, Sur les derivees d'une fonction plurisousharmonique. C. R. Acad. Sci. Paris 238 (1954), 2276-2278. [10] G. DE RHAM, Varietes Diiferentiables, Actualites Sci. et Ind. No. 1222,. Hermann, Paris, 1955. [11] R. SACKSTEDER , On hypersurfaces with no negative sectional curvatures. Amer. J. Math. 82 (1960) , 609-630. [12] M. SCHIFFER, On the mod1tlus of doubly connected domains. Quart. J. Math. 17 (1946), 197-213.
[8]
(Received September 9, 1968)
353 Reprinted from Essays on Topology and Related Topics, Springer Verlag, 1970.
Some Formulas Related to Complex Transgression RAOUL BOTT
and SHIING S. CHERN 1
1. Introduction Let X be a complex manifold of complex dimension nand rr : E -+ X a holomorphic vector bundle whose fiber dimension is also n. On E we introduce a positive definite hermitian norm N and denote by B*(E)={eEEIO 17') onto itself. An clement ~EA(VEf> V) is called r('al if ~=~. The exterior algebra A(VR) of the underlying real vector space J'R of V can be identified with the real subalgebra of rca I elements of A(VEf> V). In particular. JI R itself can be identified with the real vector space of the real clements of V EB V. The space Ann(VEf> V) is one-dimensional, and it has a generator in (', /\ ('n /\ defined up to a positive factor. It therefore makes sense to talk about the sign of a non-zero real element of A"nl VEB V). If W is a second vector space, we have a direct sum decomposition
e, /\ .. . /\
en
(4) where each summand has four degrees. the superscripts being the bidegrees relative to the first factor and the subscripts those to the second factor. If (,¢'EA(VEf> j;"),
I/,I]'EA(WEB W),
we define a multiplication by
(5)
(( ®,,) ,\ (¢' ® ,,') = (~ /\
n ® (" /\ ,,').
By linearity this is extended to a multiplication in A( V Ef> V) ® l II Ef> W). Having several kinds of products, we will, in later formulas. frequently drop all multiplication signs whenever there is no danger of confusion. We observe that, if
(6) then
(7) where
(7 a)
k=(p+q)(P' +q')+(r+s)(r' +s').
3. Affine Connection Let X be a complex manifold of complex dimension m and Ti: E-+X be a holomorphie vector bundle of fiber dimension n. At a point XEX let Ex be the fiber of E and T: the cotangent space of X . We consider the COO -bundle
355 50
RAOlil. BUTT
and
SIIIIN(;
S.
CIII:Rl'
U A (E., EEl £..)0 A(T;EEl T;)~ X
(8)
xeJ
and the bundles
(9) XE.\'
whose fibers arc respectively the summands in the decompositions of the fibers of (8) according to (4) (with V and W replaced by Ex and T; respectively). Let A~: be the space of ex' sections of the bundle (9). Then A88, which we will denote by AO(X) or simply by AO, is the ring of Coo complex-valued functions on X and A~sq is a module over A O More generally. A?sO is the space of CO-forms of type (r,s) and exterior differentiation in X defines the mappings d'" . A-oO rs
(10)
-+
A- r,oOS + 1 •
An affine connection in the bundle E is a structure which allows us to extend the operators in (10) to the more general spaces A~sq . An affine connection in E is defined to be the operators (11 ) ( 12)
d=d'+d",
which have the properties : (13)
d(v+w)=dv+d\\" d(jv)=df®v+ fdr,
v,wEA68,
[EA O
An element l'E A68 (i. e., a vector field) is called holomorphic if the corresponding mapping l': X -+E is a holomorphic mapping. The connection d is said to be of type (1,0), if d" v = 0 for holomorphic v. We have the following theorem: Theorem. If d is an affine connection in E, there exists a unique collection of operators (14) (15)
d". Apq rs
-+
Apq . r+l,5'
d'" . Apq rS
-+
Apq r.s+l'
d=d'+d",
which have the following properties: lX)
/3)
d(¢+'1)=d¢+d'1, ¢,'1EA~sq, d({()=d~ (+( -1)'+s¢dC (EA{t,
y) d commutes with the bar operation and coincides with the given affine connection on .,168 and with the exterior differentiation on .,1~o.
356 Some Formulas Related to Complex Transgression
51
The proof of this theorem parallels that of Theorem 7.1 in [3] and is straightforward. We will omit it here. The local properties of an affine connection arise most readily from the consideration of a frame field. By this we mean elements ejE .1))::. 1~ i ~ 11. defined locally in a neighborhood U of X, such that e •.. .. 'C n #0. (Notice that by our convention the latter is the exterior product.) Then we .:-an write (16)
where w{ are complex-valued one-forms in U. Differentiating (16) according to our Theorem, we get (17) where
Q{ = dwi -
(18)
I
W: wi. , 1\
1 ~ i,j, k~n.
k
The matrices (19)
w=(w{),
Q=(Q{)
are called the connection and curvature matrices respectively (relative to the frame field). It can be verified (cf. [2]) that under a change of the frame field Q goes into a similar' matrix. Therefore the 2n-form
(V2~ 1)" det
Q is globally defined in X; it is called the n-th Chern form
of the affine connection. Since E is a holomorphic vector bundle, the bundle space E is a complex manifold and the projection n: E-+X is a holomorphic mapping. Over E we have the induced holomorphic bundle n* E-+E and the above results apply. We denote by (x, v), vEE;:e, a point of E and by r(..f) the cotangent space to E at (x,v). Analogous to (8) and (9) consider the bundles (20)
U
A(E;:e Ef) E;:e) ® A( r~.v) EB r~,v)) -+ E
(x.I ') eE
and (21)
U
A~s~(;:e,v) -+ E.
(x,f')eE
Let M~sq be the space of CO sections of the bundle (21). The projection n induces the mapping (22)
n *·.
Apq -+ Mpq rs rs ,
which permits us to identify A~sq with a subset of M~r
357 RAOUL BUTT and SIIIING S . CIiERN
52
The affine connection in E induces an affine connection and gives rise to the operators (23)
" M- rspq --+
(/ .
'-1I'q
n
r
.
+ l.s·
d" : M:': ---. M:'.~ +
III
If·
E
I '
with d=d' +d". By assigning v to (x. dE E. I'E Ex. we can consider I" as an clement of Albg. Similarly, we have rEM~~(\. If the connection is of type (1.0). we have dl'EM:~~ and drEAlg:. Then (24)
rx=dvdv=cx
is real and belong to
M:::
~ will play an important role later on.
4. Hermitian Structure Suppose an hermitian structure be given in our holomorphic vector bundle E, i.e., a scalar product (1',11' ) .. , XEX, v,wEE.. , which is positive definite hermitian and is C oo in x . Let e j , 1 ~ i ~ n, be a frame field and let
1 ~i,j~n .
(25)
In order to simplify the formulas which follow, we introduce the matrices (26)
so that (27)
I
~i,j,k~n.
The hermitian structure defines uniquely an admissible connection of type (1,0), whose connection matrix relative to our frame field is given by (cf. [2, p. 45]) (28)
w=d' H · H- 1
It follows from (18) that the curvature matrix is (29)
Q= -d' d" H ' H-I +d'H ' H-
1
/\
d"H·H- 1 •
Thus the elements of the matrix Q are two-forms of type (1,1) and the n-th Chern form is a form of type (n, n). Algebraically the curvature is perhaps best described by the element (30)
358 53
Some Formulas Related tll.complex Transgression
which is independent of the choice of the framt: field ej and is globally ddined in X. It can be verified thai the Bianchi identity is t:+ dim base, P(QI) = 0 and TP(8) defines a real cohomology class in the bundle. Our object here is to give some geometrical significance to these classes. In § 2 we review standard results in connection theory. In § 3 we construct the forms TP(8) and derive some basic properties. In particular we show that if deg P = n and the base manifold has dim 2n - 1 that the forms TP(8) lead to real cohomology classes in the total space, and, in the case that P(QI) is universally an integral class, to R/Q characteristic numbers. Both the class above and the numbers depend on the connection. In § 4 we restrict ourselves to the principal tangent bundle of a
* Work
done under partial support of NSF Grants GP-20096 and GP-31526.
364 CHARACTERISTIC FORMS
49
manifold and show that if 0, 0', Q, Q' are the connection and curvature forms of conformally related Riemannian metrics then P(QI) = P(QII). Moreover, if P(QI) = 0 then TP(O) and TP(O') determine the same cohomology class and thus define a conformal invariant of M. In § 5 we examine the question of conformal immersion of an n-dim manifold into R'Hk. We show that a necessary condition for such an immersion is that in the range i> [k/2] the forms Pt(Q2i) = 0, and the classes {(1/2) TPt(O)} be integral classes in the principal bundle. Here Pt is the ith inverse Pontrjagin polynomial. In § 6 we apply these results to 3-manifolds. In a subsequent paper, [3], by one of the present authors and J. Cheeger, it will be shown that the forms TP(O) can be made to live on the manifold below in the form of "differential characters". These are homomorphisms from the group of smooth singular cycles into R/Z, subject to the restriction that on boundaries they are the mod Z reduction of the value of a differential form with integral periods evaluated on a chain whose boundary is the given one. These characters form a graded ring, and this ring structure may be exploited to perform vector bundle calculations of geometric interest. We are very happy to thank J. Cheeger, W. Y. Hsiang, S. Kobayashi, J. Roitberg, D. Sullivan, and E. Thomas for a number of helpful suggestions. 2. Review of connection theory*
Let G be a Lie group with finitely many components and Lie algebra §. Let M be a C~ oriented manifold, and {E, M} a principal G-bundle over M with projection n: E ->- M. Rg: E ->- E will denote right action by g E G. If {E', M'} is another principal G-bundle and cp: E ->- E' is a C~ map commuting with right action, cp is called a bundle map. Such a map defines rp: M ->- M', and the use of the same symbol should lead to no confusion. Let {Ea, Ba} be a universal bundle and classifying space for G. Ba is not a manifold. Its key feature is that every principal G-bundle over M admits a bundle map into {Ea, Bal, and any two such maps of the same bundle are homotopic. If A is any coefficient ring, U E Hk(Ba, A), and a = {E, M}, then the characteristic class u(a)
E
Hk(M, A)
is well-defined by pulling back u under any bundle map. Since G is assumed to have only finitely many components it is well-known that (2.1) We finally recall that Ea is contractible.
* This
chapter summarizes material presented in detail in [7].
365
50
S-S CHERN AND J. SIMONS I A
Let §l = § (8) § (8) ••• (8) §. Polynomials of degree l are defined to be symmetric, multilinear maps from §l -+ R. G acts on §I by inner automorphism, and polynomials invariant under this action are called invariant polynomials of degree l, and are denoted by JI(G). These multiply in a natural way, and if PEI1(G), QEJl'(G) then PQEP+l'(G). We set I(G) = E(8)II(G), a graded ring. These polynomials give information about the real cohomology of B a• In fact, there is a universal Weil homomorphism (2.2)
such that W: I(G) -+ H eVen (Ba , R) is a ring homomorphism. If {E, M} is a principal G-bundle over M we denote by Ak .l(E) k-forms on Etaking values in §I. We have the usual exterior differential d: Ak.l(E)-+ Ak+l .I(E). If ep E Ak,Z(E) and 'ifF E Ak·.l·(E) define ep /\ 'ifF ep /\ 'ifF(x" ••• , Xk+ k')
=
E
A k+k' .Z+/ ' (E)
Errshufflea(n)ep(x"l' ••. , X"k ) (8) 'ifF(x"k+l' ..• , Xr.k+k') •
If ep E Ak.1(E) and 'ifF E Ak·.l(E) define [ep,
'ifFl E Ak+k·.l(E)
[ep, 'ifF](x" ••• , XHk') = E rrshuffl e a(n)[ep(x"l' ••• , Xr.k )' 'ifF(X,Tk+l' ••. , Xr.k+k' )] •
Let P be a polynomial of degree land ep E Ak.l(E). Then P(ep) = po ep is a real valued k-form on E. The following are elementary
'ifFl
epl
(2.3)
[ep,
(2.4)
([ep, ep], ep] = 0
(2.5)
d[ep,
(2.6)
d(ep /\ 'ifF)
(2.7)
d(P(ep»)
(2.8)
P(ep /\ 'ifF /\ p) = (-l)kk'P('ifF /\ ep /\ p)
= (_l)kk '+l[y,
'ifFl =
[dep,
'ifFl +
(_l)k[ep, d'fl
= dep /\ y + (-l)kep /\ d'ifF
= P(dep)
where epEA k.!, 'ifFEA k·. l ', pEAk". l" and in the first three lines l
= lf = 1.
If P E [l(G) then differentiating the invariance condition shows (2.9)
E:=l (-l)k'+" '+k; P('ifFl /\ ••• /\ ['ifF;,
epl/\ ••• /\
'ifFl)
= 0
where 'ifFi E Aki.1(E) and ep E A' .l(E). For e E E, let T(E), denote the tangent space of E at e and V(E), = {x E T(E), I dn(x) = O}. V(E), is called the vertical space, and may be canonically identified with §. If x E V(E), we let x E 9 denote its image
366
51
CHARACTERISTIC FORMS
under this identification. A connection on {E, M} in a § valued I-form, 8, on E satisfying R;(8) = ad;'o 8, and 8(v) = if for vertical v. If 8 is a connection, setting H(E), = {x E T(E). I 8(x) = O} defines a complement to V(E), called the horizontal space; i.e., T(E), V(E), EB H(E), and dR.(H(E),) = H(E)RaW The structural equation states
=
(2.10)
d8
=
1
Q - -[8 8] 2
'
where Q is the curvature form. Q E N"(E) and is horizontal, i.e., Q(x, y) = Q(H(x) , H(y)), H(x), and H(y) denoting the horizontal projections of x and y. (2.4) and (2.5) show dQ = [Q,O].
(2.11)
An element cp E Ak,l is called equivariant if R;(cp) = ad g-' 0 cpo A connection is equivariant by definition, and so is its curvature by (2.10), as equivariance is preserved under d, wedge products, and brackets. If rp E Ak'I(E) is equivariant and PE Jl(G) then P(cp) is a real valued invariant
. I
!dorm on E. In particular, Ql = Q /\ ••• /\ Q is equivariant, and so P(QI) is real valued, invariant and horizontal, and so uniquely defines a 2l-form on M whose lift is P(QI). We will also denote this form on M by P(QI). Formulae (2.11) and (2.9) immediately show this form is closed. THEOREM 2.12 (Weil homomorphism). Let a = {E, M, 8} be a principal G-bundle with connection, and let P E II(G). Then
P(QI) Le., P(QI) represents the Weil image of P.
E
W(P)(a) ;
chara(,"~eristic
class corresponding to the universal
For some of the calculations in the sections that follow it will be convenient to have classifying bundles equipped with connections. To do this we use a theorem of Narasimhan-Ramanan [10]. We introduce the category c(G). ObJ"ects in c(G) are triples a = {E, M, 8} where {E, M} is a principal G-bundle with connection O. Morphisms are connection-preserving bundle maps; i.e., if a = {E, M, 8} and a = {E, M, e}, and cp: {E, 1If} -+ {E, M} is a bundle map, then cp: a -+ a is a morphism if cp*(B) = O. An object A E c(G) is called n-classifying if two conditions hold: First for every a E c(G) with dim M ~ n there exists a morphism cp: a -+ A. Second, any two such morphisms are homotopic through bundle maps. We do not require the homotopy to be via morphisms.
367
52
S-S CHERN AND J. SIMONS
THEOREM 2.13 (Narasimhan-Ramanan). For each integer n there exists an n-classifying A E s(G).
3. The forms TP(O)
Let a = {E, M, O} E s(G). The bundle {n*(E), E} is trivial as a principal G-bundle, and so all of its characteristic cohomology vanishes. Thus n*(P(Ql») = P(Ql) is exact when considered as a form on E. Set CPt = tQ + 1/2(t2 - t)[O, 0], and set (3.1)
TP(O)
= l ~:P(O /\ cp:-')dt •
P E Jl(G), and TP(O) is a real-valued invariant (2l - l)-form on E. It is of course not horizontal. PROPOSITION
3.2. dTP(O)
Proof. Set f(t)
= P(cpD.
= P(Ql). Then f(O)
(3.3)
P(Ql)
=0
and f(l) =-= P(Ql), Thus
= ~:f'(t)dt •
We claim (3.4)
1'(t)
= ldP(O /\ cpl-') •
We first observe !i(CPt) dt
=
Q
+ (t
-
~)[O, 0]
•
2
Using (2.3)-(2.8) we have
l' = lP(;/CPt) /\ cpl-')
= lP(Q /\ cp:-') + l(t - ~)P([O, 0]/\ cp;-') • On the other hand, ldP(O /\ cp:-l) = lP(dO /\ cp:-') - l(l - l)P(O /\ dcpt /\ cpl-2) = lP(Q 1\ cpl-') -
~lP([O 0] /\ CP:-') - l(l - l)P(O 1\ dcpt 1\ cpl-2) 2
'
by the structural equation (2.10). Now using (2.10), (2.11), and (2.4) dcpt = t[ CPt, 0] •
Plugging this into the formula above and using the invariance formula (2.9) on the last piece we get
368 53
CHARACTERISTIC FORMS
ldP«(} /\ ~l-I)
lP(Q /\ ~l-I)
=
- ~
lP([B, B] /\ ~:-I)
+ ltP([B, B] /\ ~l-I) = ff
by the computation above. This shows (3.4) and the proposition follows from (3.3). The form TP«(}) can of course be written without the integral, and, in fact, setting Ai = (_1)il! (l - 1)!/2i(l
+ i)! (l
- 1 - i)!
one computes TP(B)
(3.5)
=
:L::~ AiP(B /\ [B, B]i /\ QH-I) •
The operation which associates to a E e(G) the form TP(B) is natural; i.e., if ~: a-+a is a morphism, since ~*(O) = B and thus ~*(n) = Q, clearly 9*( TP(O)} = TP(ff). This naturality characterizes T up to an exact remainder: PROPOSITION 3.6. Given PE ]I(G), let S be another functor which associates to each a E e(B) a (2l - I)-form in E, SP(B), which satisfies dSP(B) = P(QI). Then TP(B) - SP(B) is exact.
P1·oof. Let a = {E, M, B} with dim M = n. Choose a = {E, ii,O} E e(G) so that is m classifying with m sufficiently greater than n. Let~: a -+ be a morphism. Now in E we have dSP(O) = dTP(O) ==> SP(iJ) - TP(iJ) is closed. But since E is an approximation to Ea its 2l - 1 cohomology vanishes for sufficiently large m. Thus SP(O) = TP(O) + exact. So by the naturality assumption on S, SP(B)=~*SP(iJ)=~*TP(O)+~* exact = TP(B) + exact. 0
a
a
3.7. Let PE JI(G) and QE ]'(G). (1) PQ(QI+.) = P(QI) /\ Q(Q')
PROPOSITION
(2) TPQ(B)
=
TP(B) /\ Q(Q')
+ exact =
TQ(B) /\ P(Ql)
+ exact.
Proof. (1) is immediate. To prove (2) we may use naturality and work In a classifying bundle. But there, d( TP(B) /\ Q(Q'») = P(Ql) /\ Q(Q') = PQ(Q l+.) = d(TPQ(B»). Similarly d(TQ(B) /\ P(Ql») = d(TPQ(B»). (2) then
follows by low dimensional acyclicity of the total space of the classifying bundle. 0 We are interested in how the forms TP(B) change as the connection changes. PROPOSITION 3.8. Let B(s) be a smooth 1-parameter family of connections on {E, M} with s E [0, 1]. Set B = B(O) and B' = (d/ds)(B(s») 1.=0' For PE]I(G)
369
54
S-S CHERN AND :s(TP(8(S)))
1.=0
J.
SIMONS
= lP(8' /\ Ql-1)
+ exact.
Proof. Building on the theorem of Narasimhan-Ramanan it is not difficult to show that one can find a principal G-bundle {E, M} which classifies bundles over manifolds of dim ~ m ~ dim M, and to equip this bundle with a smooth family of connections lJ(s) , and to find a bundle map cp: {E, M} --+ {E, M} such that cp*(IJ(s») = 8(s) s E [0, 1). It thus suffices to prove the theorem in {E, M}, and by choosing m large enough E will be acyclic in dimensions ~2l-1. We now drQ.p all "hats" and simply assume H2l-1(E, R) = O. Thus it is sufficient to prove both sides of the equation have the same differential. Now d(:s(TP8(S»)
18=0)
=
~(dTP(8(S»)) 1.=0
= ~ (P(Q(S)I) 1.=0) = lP(Q' where
Q'
/\ Ql-l)
= (d/ds)(Q(s») 1.=0' Also
d(lP(8' /\ Ql-1»)
= lP(d8' /\ Ql-1) - l(l - 1)P(8' /\ dQ /\ QI-2)
= lP(d8' /\ Ql-l) - l(l - 1)P(8' /\ [Q, 8) /\ Ql-2) by = lP(d8' /\ Ql-1) + lP([ 8'. 8)/\ Ql-l) by (2.9) .
(2.11)
Now d8' = d( (d/ds)(8(s) )1,=0) = (d/ds)(d8(s») 1.=0 = (d/ds)(Q(s) - (1/2)[ 8(s), 8(s)]) 1.=0= Q' - [8', 8). Putting this in the calculation above shows
and this with the first calculation completes the proof.
o
If P E Jl(G) and P(Ql) = 0 then TP(8) is closed in E and so defines a cohomology class in E. We denote this class by {TP(8)} E H 21-1(E, R).
THEOREM 3.9. Let a = {E, M,8} with dim M = n. If 2l - 1 ·= n then TP(8) is closed and {TP(8)} E H"(E, R) depends on 8. If 2l - 1 > n then TP(8) is closed and {TP(8)} E H 2l-1(E, R) is independent of 8. Proof. P(Ql) is a horizontal 2l-form. If 2l - 1 ~ n then 2l > nand since the dimension of the horizontal space is exactly n, P(Ql) = O. Thus TP(8) is closed, and {TP(8)} is defined. We will see in a later section that when 2l - 1 = n, {TP(8)} depends on the connection. However, suppose 2l - 1 > n. Since any two connections may be joined by a smooth I-parameter family, it is sufficient to show, using the notation of the previous proposition that
370 CHARACTERISTIC FORMS
!(TP(O(S»))
1.=0 =
55
exact.
By that proposition it is sufficient to show P(O' 1\ QH) = O. Since 0' is the derivative of a family of connections, all of which must agree on vertical vectors, O'(v) = 0 for v vertical. Thus P(O' 1\ QH) is a horizontal (2l - 1)0 form, and thus must vanish for 2l - 1 > n. The equation in E, dTP(O) = P(QI), implies that TP(O) 1 Em is a closed form, where Em is the fibre over mE M. Formula (3.5) shows that TP(O) 1 Em is expressed purely in terms of 01 Em' which is independent of the connection. More specifically, let c:v denote the Maurer-Cartan form on G, which assigns to each tangent vector the corresponding Lie algebra element. Set TP
(3.10)
=
(_1)1-1 P(c:v 1\ [c:v, c:v]H) •
21Cl ~ 1)
TP is a real valued, bi-invariant (2l-1)-form on G. It is closed and represents an element of H2H(G, R). For mE M and eE Em let A.: G--Em by A.(g) = R g(e). Then A. *(8) = c:v, and (3.5) shows A.*(TP(O») = TP.
(3.11)
The class {TP} E H2I-1(G, R) is universally transgressive in the sense of [1]. In fact, recalling Borel's definition of transgressive ([1], p. 133), a class hE Hk(G, A) is called transgressive in the fibre space {E, M} if there exists e E Ck(G, A) so that c 1 G E hand oe is a lift of a cochain (and thus a cocycle) from the base. It is called universally transgressive if this happens in the classifying bundle. In this case the transgression goes from {TP} via TP(8) to P(QI). One can do this over the integers as well as the reals, and if we set
I;(G) = {PE J1(G) 1 W(P)
E
H21-1(Bc, Z)}
one can easily show (3.12)
PE I;(G)
{TP}
=
E
H 2!-1(G, Z)
and (3.11) shows this is equivalent to (3.13)
PE I;(G)
=
TP(8) 1 Em E H 21-1(Em' Z)
where in all these equations we mean the real image of the integral cohomology. The following proposition will provide a proof of this, but also will give us some extra understanding of the form TP(8) when PE I;(G). For a real number a let aE RjZ denote its reduction, and similarly for
371
56
S-S CHERN AND J. SIMONS
any real cochain or cohomology class The Bockstein exact sequence (3.14)
~
will denote its reduction mod Z.
- . H;(X, Z) ~ H;(X, R) ~ H;(X, R/Z) ~ H;+l(X, Z) - .
shows that a real class, U, is an integral class if and only if [j = O. For X any manifold and A any coefficient group we let C(X, A) denote the cochain group with respect to the group of smooth singular chains. If cp is a differential form on X then cp E C(X, R), and by ;p E C(X, R/Z) we mean its reduction mod Z as a real cochain. 3.15. Let a = {E, M, O} E €(G). C2l-1(M, R/ Z) so that
PROPOSITION
exists u
E
,,-....;
Then if P E I;(G) there
+ coboundary •
TP(O) = n*(u)
Proof. Let a = {E, £1, B} E €(G) be k-classifying with k sufficiently large. Since PEl; we know that P(fil) represents an integral class in £1. ~
Thus the R/Z co cycle P(fil) vanishes on all cycles in
£1,
and so is an R/Z ,,-....;
coboundary; i.e., there exists U E C21-1(£1, R/Z) such that OU = P(fil). Thus ,,-....;
on*(u) = n*(ou) = n*(p(fil)) ~
= n*(p(fil)) = So on*(u) and so
=
~
...--..
~
dTP(B)
= oTP(B) = o(TP(B)) .
o(iP(B)). Since we have chosen k large, Eis acyclic in dim 2l-1, ,,-....;
TP(B)
= n*(u) + coboundary .
The proposition then follows in general by choosing a morphism cp: a and taking u = cp*(u).
a 0
We note that (3.13) and hence (3.12) follow directly from this. We also note that for these special polynomials, the classes {TP(O)}, when they exist, have the property that their mod Z reductions are already lifts. That is THEOREM 3.16. Let a = {E, M, O} E €(G) and let P E I;(G). P(QI) = O. Then there exists UE H2l-1(M, R/Z) so that
{TP(8)}
= n*( U)
Suppose
.
Proof. Choose u E C 2l-1(M, R/ Z) as in Proposition 3.14. The assumption P(QI) = 0 implies n*(ou) = O. Since every chain in M comes from one in E this means ou = O. Thus u is an R/ Z cocycle in M, and Proposi-
372
57
CHARACTERISTIC FORMS ~
tion 3.14 shows n*(u) ~ TP(O). Letting U E H21-1(M, R/ Z) denote the class represented by u the theorem follows. Characteristic numbers in R/Q. An interesting special case of this theorem occurs when M is compact, oriented, and dim M = 2l - 1. Then for each PE J;(G) we know that P(Q Z) = 0 and {TP(O)} E H21-1(E, R) depends ~
on the connection. On the other hand, reducing mod Z, {TP(O)} = n*(U) for some U E H2Z-1(M, R/ Z) ~ R/ Z. Thus U is determined up to an element of ker n"'. Now, either ker n* = H2Z-1(M, R/Z), or ker n* is a finite subgroup of H21-1(M, R/Z). In the second case, since all finite subgroups of R/Z lie in Q/Z, U is determined uniquely in R/Z /Q/Z ~ R/Q. Let f.L denote the fundamental cycle of M. Define SP(O) E R/Q by if ker n'" = H2Z-1(M, R/Z) SP(O) = u{f!)/Q otherwise. SP(O)
=0
Examples in the last section will show that these numbers are nontrivial invariants. * COROLLARY
3.17. Suppose dim M {TP(O)}
E
< 2l-
1. Thenfor PE J;(G)
H21-1(E, Z) .
Proof. Since dim M < 2l - 1, H2Z-1(M, R/Z) = 0 and so {T.P(8)} = Thus from (3.14) {TP(O)} is the image of an integral class.
o.
4. Conformal invariance
In this section we suppose G = Gl(n, R). § consists of all n x n matrices, and we define the basic invariant polynomials Q" ••• , Q"
It is well known that the Qi generate the ring of invariant polynomials on a = {E, M, O} is a principal G bundle then 0 = {O;;} and Q = {Q;,}, matrices of real valued 1 and 2-forms respectively. One verifies directly that for any cP = {cp;j} E Ak .1(E)
§. If
(4.1)
Q . . 1\ Q . . 1\ •.. 1\ Q . . QI( Q I) = ,"," L..iil'···.il = l '1"2 '2" 3 'l"l
(4.2)
These polynomials have different properties. In particular the Wei! map
* This construction was made in discussion with J . Cheeger. producing the mod Q reductions of R/Z invariants developed in (3).
It is an easy way of
373 58
S-S CHERN AND J. SIMONS
(see (2.2») takes the ring generated by {Qzl} isomorphically onto the real cohomology of BOICn.Rl = B OCn )' while the kernel of the Wei! map is the ideal generated by the {Q21+1}' PROPOSITION 4.3. Let a = {E, M, O} E c(Gl(n, R»). Suppose 0 restricts to a connection on an O(n) subbundle of E. Then Q2IH(Q2!+1) = 0, and TQ2l+1(0) is exact.
Proof. The first fact is well known and is one way to prove Q21+1 E Ker W. Our assumption on 0 is that there is an O(n) subbundle F ~ E such that at each fE F, H(E)f ~ T(F)f' or equivalently that at all x tangent to F, O,;(x) = - O;,(x). It easily implies that at all points of F, Q;; = -Q;, as a form on E. Now if A is a skew symmetric matrix then tr (A 2 1+1) =::- 0 and by polarization we see Q2t+,(A, @ ••• @ A Zl+ 1) = 0 when all A, are skew symmetric. Since QZIH(Q21+') is invariant, we need only show it vanishes at points in F, but at these points the range of Q2t+1 lies in the kernel of Q2l+" Thus Q21+,(Q21+') = O. The same argument shows that TQ2l+1(0) F = O. (Here we mean the form restricted to the submanifold, F, and not simply as a form on E considered at points of F.) Thus TQ2t+'(0) is a closed form in E whose restriction to F is O. Since E is contractible to F, TQ2t +'(0) can carry no cohomology on E and hence must be exact. 0 1
Let us now specialize to the case where of the tangent bundle of M. Points in E (m; e" "', en) where mE M and e" "', en equipped with a natural set of horizontal, defined by dn:(x) =
E = E(M), the bundle of bases are (n + I)-tuples of the form is a basis of T(M)m' E comes real valued forms W,' "', W,,'
L:?=,wi(x)e,
where x E T(E)., and e = (m; e" "', en)' Now let g be a Riemannian metric on M, and let 0 be the associated Riemannian connection of E(M). Let E" "', E" be horizontal vector fields which are a dual basis to w,' "', w". Let F(M) denote the orthonormal frame bundle. F(M) ~ E(M) is the O(n) subbundle consisting of orthonormal bases, and since 0 is the Riemannian connection, 0 restricts to a connection on F(M). Let h be a C~ function on M, and consider the curve of conformally related metrics SE
[0, 1] •
Let O(s) denote the curve of associated Riemannian connections on E(M). Let 0 = 0(0), 0' = (djds)(O(s») 1.=0' and F(M) the frame bundle with respect to g = g(O).
374
CHARACTERISTIC FORMS
59
LEMMA 4.4. At points in F(M) O~;
= o,;d(h n) + E;(h n:)w; - E;(h n:)w; • 0
0
0
Proof. This is a standard computation, and is perhaps most easily done by using the formula for the Riemannian connection in terms of covariant differentiation (cf. [7]). It is easily seen how the connection changes under conformal change of metric, and one then translates this result back into bundle terminology. THEOREM 4.5. Let g and {j be cdnformally related Riemannian metrics on M, and let 0, Q, 8, {i denote the corresponding connection and curvature forms. Then for any PE I'(Gl(n, R») (1) TP(8) = TP(O) + exact, (2) P({il) = P(Ql) . COROLLARY. P(Ql) = 0 implies that the cohomology class {TP(O)} E 2 l H - 1(E(M), R) is a conformal invariant. The corollary follows immediately from the theorem, and (2) follows immediately from (1) and Proposition 3.2. So it remains to prove (1). Since the Qi generate I(Gl(n, R») we can assume P is a monomial in the Q,. Using Proposition 3.7, an inductive argument shows that it is sufficient to prove (1) only in the case P = Ql. Proposition 4.3 shows that for any Riemannian connection Q2l+,(Q2l+1) = 0 and TQ2l+1(O) is exact, so we can assume l is even. Any two conformally related metrics can be joined by a curve of such metrics, with associated connections O(s). By integration it is sufficient to prove :S(TQ21(O(S»)) = exact.
(*)
Since each point on the curve is the initial point of another such curve, it is enough to prove (*) at s = O. By Proposition 3.8 it will suffice to prove (**)
We use the notation and formula of Lemma 4.4, and work at f E F(M). Set a = (o;;d(f n:») (3 = (E;(f 0 n:)w; - E;(f 0 n:)w;) • 0
Then 0' = a
+ (3.
Now (4.2) shows Q2l(a 1\ Q2l-')
= d(f
by Proposition 4.3. Also using (4.2),
0
n:) 1\ Q,l_l(Q21-')
=0
375
60
S-S CHERN AND J. SIMONS
But, since 8 is a Riemannian connection, the Jacobi identity holds. This states
and shows Q2I((3/\ Q2H) = O. Thus at points in F(M), Q2I(8' /\ and (**) follows by invariance.
Q21-') =
0,
0
5. Conformal immersions
Let G = U(n). Let A be a skew Hermitian matrix and define the i'h Chern polynomial Ci E J;( U(n») (5.1)
where Ci is extended by polarization to all tensors. Let ei denote the i'h integral Chern class in B u(.. ). Then ei E H2i(BG' Z), and letting 'r(e i ) E H2i(BG' R) denote its real image, W(Ci )
(5.2)
=
r(e i )
•
We also define the inverse Chern polynomials and classes Cl and eiL
(5.3)
(1 (1
+ C~ + ... + Ct + ... ) (1 + C, + ... + C.. ) = 1 + ei + ... + ef + ... ) U (1 + e, + ... + e,,) = 1 . l
These formulae uniquely determine Cf and ef, and since W is a ring homomorphism (5.4)
W(Cf) =
et .
The inverse classes are so named because, for vector bundles , they are the classes of an inverse bundle. That is, if V, Ware complex vector bundles over M with V EEl W trivial, then using the product formula for Chern class, cf. [9], one knows (5.5)
ei ( W)
=
e;'(V) •
Let G"je) denote the Grassmann manifold of complex n-planes in C.. H, and let E ". k(e) denote the Stiefel manifold of orthonormal n-frames in C"H, with respect to the Hermitian metric. Then {E ...k(e), G" .k(e) } is a principal U(n) bundle. There is a natural connection in this bundle most easily visualized by constructing it in the associated canonical n-dim vector bundle over G"je). Let i(t) be a curve in G" .k(e) and let p(t) be a curve in the
376
61
CHARACTERISTIC FORMS
vector bundle with rr p = 'Y. So for each t, 'Y(t) is an n-plane in C"+k, and p(t) is a vector in C"+k with p(t) E 'Y(t). Then p'(t) = (d/dt)(p(t)) is a vector in C,,+k, and the covariant derivative of p(t) along 'Y is obtained by orthogonally projecting p'(t) into 'Y(t). We let 0 denote this connection and 0
set a".k(C) = {E".k(C), G".k(C), O} • PROPOSITION
5.6. For i
>k
(1)
C/(Qi) = 0
(2)
{TCt(On E H
2H
(E".k(C),
Z) .
Proof. Since the n-dim vector bundle associated to {E".k(C), G".k(C)} has a k-dim inverse, (5.5) shows that ct(a".k(c») = 0 for i > k. Thus the form CNQi) is exact on G".k(C). Now G".k(C) is a compact, irreducible Riemannian symmetric space, and it is easily checked that the forms P(QL) are invariant under the isometry group. Thus CHQi) is invariant and exact, and therefore must vanish. So the class {TCHOn E H 2H (E".k(C), R) is defined. Since W(Cn = ct E H 2'(Bu ("" Z) we see that Ct E 1;( U(n»). Using Theorem ~ 3.16 we see that {TCl (O)} is a lift of a 2i - 1 dimensional R/Z cohomology class of G".k(C). But the odd dimensional cohomology of this space is zero ~ (for any coefficient group), and thus {TCt(On = o. The Bockstein sequence (3.14) then shows that {TCHOn E H 2H (E" .k(C), Z). 0
i
Now let G = O(n). Let A be a skew symmetric matrix and define for ' .. , [n/2] the i th Pontrjagin polynomial Pi E I~'(O(n»)
= 1,
(5.7)
det (r.J - (1/2rr)A) =
2, I:::':J P,(A ® ... ® A)","-2i + ..---'-----.
Q(", ,,-Odd)
where we ignore the terms involving the n-odd powers of "'. Also let Pi E H " (B o("" Z) denote the i th integral Pontrjagin class. Then W(Pi)
Let p: O(n)
~ U(n)
=
r(Pi) •
be the natural map. Then p induces p*: I( U(n») -+ -+ H*(O(n»), and p: B o(", -+ B u (", . Using Theorem 2.12
I(O(n») , p*: H*( U(n»)
one easily sees W(p*(Q»)
=
p*(W(Q))
for any Q E [L( U(n»). The definitions of Pi and Pi are such that (5 .8)
P*(C2i ) = (-I)iPi
,
p*(c2.) = (_I)ipi .
We also define the inverse Pontrjagin polynomials Pl(5.9)
(1
+ P + ... + P("/2])(1 + P t + ... + l
Pt
+ ... ) =
1
377
62
S-S CHERN AND J. SIMONS
and note that Pi that
E
Io2i(O(n») since p*(ct)
E
H 2i (B oc .. ), Z), and one easily sees
W(pn = (-1)ir(p*(cli)) • Formula (5.9) shows PiL = -Pi - P i - 1Pt - ••. - P,Pf-,. Proposition 3.7 shows that TPt(8) = - TPi (8) + terms involving curvature. Thus for any
a
=
{E, M, 8} E s(O(n»)
(5.10)
TPt(8) 1 Em
= -
TPi (8) 1 Em •
We now define the real Grassmann manifold, G... k , the real Stiefel manifold E ... k , and the canonical connection 8 on {E ... k' G... k } exactly as in the complex case. We set a ... k = {E... k , G...k , 8} E s(O(n». PROPOSITION
5.11. For i > [k/2]
(1)
Pt(Q2i) = 0
(2)
{(1/2) TPl(8)} E H"-'(E... k , Z) .
Proof. The natural map R" -+ C" induces the commutative diagram 'P
E ... k ---> E"'k(C)
1
'P
1
G... k ---> G... k(c) • It is straightforward to check that Pt(Q2i) = (_1)i k, (1) follows from Proposition 5.6, and from (2) of that proposition we see that {TPt (8)} = (_1)i [k/2]. Proof. Let cp: M" --+ R"H be a conformal immersion. By Theorem 4.5 we may assume cp is an isometric immersion. Let F(M") denote the orthonormal frame bundle of M", and consider the Gauss map F(M")~E... k
1
Mn
1
~Gn.k
which is defined as usual by mapping a point into the tangent plane at its image. Letting 0 denote the canonical connection on En .• , it is a standard fact that *(0) = 0, the Riemannian connection on F(Mn); i.e., : {F(Mn), Mn, O}
~
a n •k
is a morphism. Thus by naturality and the previous proposition, in F(M") , Pt(Q2i) = 0 and ((1/2) TPt(O)} E H4H(F(M"), Z) for i> [k/2]. By invariance, Pt(Q2i) = 0 in all of E(M") , and since ((1/2) TP{{O)} E H4i-I(E(Mn), R) it must actually be an integral class there since its restriction to the retract F(Mn) is integral. 0 Remark. This theorem is probably of interest only for the codimension k ~ n/2. This is because if k > n/2 our condition i > [k/2] already implies Pt(Q2i) = 0 for dimension reasons, and the corresponding class, {TPi'(O)}, is
independent of connection (see Theorem 3.9). At the same time Corollary 3.17 already shows that {TPt(O)} E Ht"- +g -+ "'f"-+g- -
'; oz"
ow
L..
near the manifold , M. We describe the manifold v= 2). Equations (2.14b) are equivalent to
+ 2-w which leads to (3.9). (b) Returning to (3.9) it remains to satisfy the relations tr
F22 =
0,
(tr)
2 F32 =
0,
(tr)3
F33 =
0
which give rise to a set of differential equations for the curve y and for the associated frame. We begin with the condition (tr)
2 F32 =
0 which gives rise to a differential equation
of second order for the curve y, where the parametrization is ignored. For this purpose we assume that the parametrization is fixed, Bay by Re F32
q(~) =~
on p(~). According to Lemma 3.4 the coefficients of
F32
and study the dependence of are analytic functions of p, p
and their derivatives up to order 5. But if the hypersurface is in the form (3.9) then depends on the derivatives of order
~2
F32
and is of the form (3.14)
where K 32 , B depend on p, p, p', p' analytically, and B is a nonsingular matrix for small lui. To prove this statement we recall that (3.9) was obtained by a transformation z ->- p(w) +O(w)z + ... , w->-q(w) + ...
We choose Re q(u) =u fixing the parametrization; 1m q(u) is determined by p, p. To study the dependence of
F32
z
at U=U o we subject (3.9) to the transformation =
8(W*) +z* + ... , w
=
q(w* +uo)
(3.15)
which amounts to replacing p(u) by p*(u*) =p(uo+u) +O(Uo+U)8(U). Considering p and p' fixed at u = germ of
8
U o we require 8(0) = 0, 8'(0) = 0 and investigate the dependence of F 32 on the at u=auo. We choose the higher order terms in (3.15) in such a way that the
form of (3.9) is preserved as far as terms of weight
~5
is concerned. This is accomplished
by the choice z = Z*+8(W*)+2i O.
Thus we have constructed a holomorphic transformation taking M into the normal form, and the existence proof has been reduced to that for ordinary differential equations. The choice of the initial values for p'(O) Eon, U(O) and Re q'(O), Re q"(O) allows us to satisfy the normalization condition (2.9) of § 2. In fact, these 2n +n2 + 1 + 1 = (n + 1)2 + 1 real parameters characterize precisely an element of the isotropic group H. Thus we have shown THEOREM
3.5. If M
i8
a real analytic manifold the unique formal transformation of
Theorem 2.2 taking M into a normal form and 8ati8fying the normalization condition is given by convergent 8eries, i .e. defines a holomorphic mapping. Two real analytic manifolds Ml> M2 with distinguished points PIEMl> P2EM2 are holomorphically equivalent by a holomorphic mapping", taking PI into P2 if and only if <Mk,Pk) for k=l, 2 have the same normal forms for some choice of the normalization conditions. Thus the problem of equivalence is reduced to a finite dimensional one. The arbitrary initial values for the differential equations tr (tr)3
F33 =
F22
= 0, (tr)2 F32 = 0,
0 have a geometrical interpretation: At a fixed point P EM they correspond to
(i) a normalized frame e"ETc, <e", ep) =h"fj (ii) a vector en +l E TR - Tc corresponding to the tangent vector of the curve y, and (iii) a real number fixing the parametrization, corresponding to Re q"(O). With the concepts of the following section this will be viewed as a frame in a line bundle over M. As a consequence of these results above we see that the holomorphic mappings taking
a nondegenerate hypersurface into themselves form a finite dimensional group. In fact, fixing a point the dimension of this group is at most equal to that of the isotropy group H, i.e. (n+l)2+l. Adding the freedom of choice of a point gives 2n+l +(n+l)2+1 =(n+2)2-1 as an upper bound for the dimension of the group of holomorphic self mappings of M. This upper bound is realized for the hyperquadrics. The above differential equations define a holomorphically invariant family of a parametrized curve y transversal to the complex tangent bundle, with a frame e" propagating along y. The parameter ~/(IX~+{J)
({J=FO) keeping
~=O
~
is fixed up to a projective transformation
fixed. Thus cross ratios of 4 points on these curves are
invariantly defined. We summarize: (i) tr F22 =0 represents a first order differential
412 246
S. S. CHERN AND J. K. MOSER
equation for the frame ea , (ii) (tr)2 Fa2 =0 defines a second order differential equation for the distinguished curves y, irrespective of parametrization and (iii) (tr)a Faa =0 defines a third order differential equation for the parametrization. (c) The differential equations tr
F22
=0, (tr)2 Fa2 =0, (tr)a Faa =0 remain meaningful
for merely smooth manifolds. Indeed, if M is six times continuously differentiable one can achieve the .above normal forms up to terms of order 6 inclusive, simply truncating the above series expansions. Clearly the resulting families of curves and frames are invariantly associated with the manifold under mappings holomorphic near .ill. Indeed since the differential equations are obtained by the expansions of § 2 up to terms of weight ,;; 6 at any point one may approximate M at this point by a real analytic one and read off the holomorphic invariance of this system of differential equations. In this case the distinguished curves yare, in general, only 3 times continuously differentiable but the normal form (see (2.11) via a holomorphic map, cannot be achieved, not even to sixth order in z,
z.
This would require that the function f(z, u), g(z, u) defining the transforma-
tion and which can be taken as polynomials in z admit an analytic continuation to complex values of u. If the Levi form is indefinite one has to require an analytic continuation to both sides which can happen only in the exceptional case of analytic curves y . If, however, the Levi-form is definite, i.e. in the pseudoconvex case one has to requirc only that f(z, u), g(z, u) admit one sided analytic continuations. However, we do not pursue this artificial question but record that the structure of differential equations for the curves y and their associated frame is meaningful in the case of six times differentiabl, manifolds. (d) In the case n = 1 the normal form has a simpler form since the contraction (tr)
becomes redundant. For this reason F 22 , F 2a , F a2 , Faa all vanish and the normal form can be written
v = zz+ C42 Z 4 Z2+ C24 Z 2Z4 +
L:
Ck1i'
Zl
(3 .18)
k+I~7
where again min (k, l) ;;;' 2. This normal form is unique only up to the 5 dimensional group H given by
z ~ J.(z+ awl 15-\ w ...... 1J.1 2 wt5- 1
(3.19)
with O+J.EC, aEC, rER. It is easily seen that the property C42 (0)+0 is invariant under these transformations. If C42 (0) = 0 we call the origin an umbilical point. For a nonumbilical we can always achieve c42 (O) = 1 since z~J.z leads to c42(0) ...... J.alc42(0) . By thib normalization J. is fixed up to sign.
413 247
REAL HYPERSURFACES IN COMPLEX MANIFOLDS
For a nonumbilical point we can use the parameters a, r to achieve
so that the so normalized hypersurface can be approximated to order 7 in z, Z, u by the algebraic surface (3.20) where j EC, k E R, and j2, k are invariants at the origin. The above statements follow from the fact that (3.19) with A=I, r=O leads to
so that j = C52 (0) +2C43(0) is unchanged. We fix a so that C43 (0) =0 and consider (3.19) with A=I, a=O which gives rise to
Choosing Re C~2(0) = 0 we obtain (3.20), where we still have the freedom to replace z by -z. Thus j2 and k are indeed invariants. The above choice (3.20) distinguishes a special frame at the origin, by prescribing a
a/au transversal to the complex tangent plane and a complex tangent ±a/oz in the complex tangent plane. These pairs of vectors can be assigned
tangent vector vector pair
to any point of M which is non-umbilical. These considerations clearly are meaningful for seven times differentiable M. The above vector fields, singular at umbilical points, can be viewed as analogous to the directions of principal curvature in classical differential geometry. This analogy suggests the question: Are there compact manifolds without umbilical points1 Are there such manifolds diffeomorphic to the sphere 8 3 1 Clearly the sphere 1z 12 + 1W 12 = 1 consists of umbilical points only as, except for one point, this manifold can be transformed into v =zz (cf. (1.4)) . Therefore we can say by (3.18): Any 3-dimensional manifold M in 0 2 can at a point be osculated by the holomorphic image of the sphere 1z 12 + 1W 12 = 1 up to order 5 but generally not to sixth order. In the latter
case we have an umbilical point. For
n;;' 2
the analogous definition of an umbilical point is different: A point p on
M is called umbilical if the term F 22 in the normal form vanishes. Again, it is easily seen that this condition is independent of the transformation (1.23) and we can say: Any nondegenerate manifold M of real dimension 2n + 1 in Cn +1 (n;;' 2) can at a point be osculated by the holomorphic image of a hyperquadric v= ( z, z) up to order 3, but generally not to order 4.
In case one has fourth order osculation one speaks of an umbilical point.
414 248
S . S . CHERN AND J. K . MOSER
(e) The algebraic problems connected with the action of the isotropy group on the normal form are prohibitively complicated for large n . But for a strictly pseudoconvex 5-dimensional manifold in C3 we obtain an interesting invariant connected with the 4th order terms F 22 . We assume n=2 and
2
, and we will introduce new ones by intrinsic conditions, so that the total number equals the dimension of Y and they are everywhere linearly independent. The condition that our G-structure is integrable implies (4.16)
where cf>/, cf>" are not completely determined. We shall study the consequences of the equations (4.10), (4.16) by exterior differentiation. To be in a slightly more general situation the gap's are allowed to be variable. It will be convenient to follow the practice of tensor analysis to introduce g"fJ by the equations (4.17)
and to use them to raise and lower indices. It will then be important to know the location of an index and this will be indicated by a dot, thus (4.18)
The exterior differentiations of (4.10), (4.16) give respectively
i(dgap- cf>"p- cf>P"+ gapcf» " wet 1\ wP + (- dcf> + iwp" cf>P + icf>pA wP)" w= 0,
(4.19)
(dcf>l - cf>p.Y1\ cf>Y~ - iWfJ" cf>") " wfJ + (dcf>" - cf> 1\ " - fJ " p~) 1\ w = O.
(4.20)
419 253
REAL HYPERSURFACES IN COMPLEX MANIFOLDS LEMMA
4.1. There exist 4>l. which satisfy (4.16) and
dgap + g,,:[34> - 4>ap - 4>pa = 0,
or
(4.21)
4>PI< = 4> pa,
dga./J - ga./J 4> + 4>a/J + 4>/ia. =
o.
(4.21 a)
Such 4>,t are determined up to additive terms in w. In fact, it follows from (4.19) that the expression in its first parentheses is a linear combination of wIp~
be exterior quadratic differential forms, satisfying (4.27) modw,
Then we have
(4.28)
where SaiQo has the symmetry properties: (4.29) (4.30)
Computing mod w, we have, from the first equation of (4.27), <J>p~==Xp~yA w
y,
where Xp«y are one-forms. Its complex conjugate is
y <J>pa == X[3ay A w •
420 254
S. S. CHERN AND J . K. MOSER
By the second equation of (4.27) we have
The first term, XaPy II wY, is therefore congruent to zero mod w, wU • But it is obviously congruent to zero mod we. Hence we have the conclusion (4.28). The symmetry properties (4.29) and (4.30) follow immediately from (4.27). Thus Lemma 4.2 is proved. Equation (4.20) indicates the necessity of studying the expression (4.31)
Using (4.21) we have (4.32) It follows that
since Using the differentiation of (4.21), we get (4.33)
TIpa+ TIap=gpad.
By (4.20), (4.26), (4.33), it is found that (4.34)
or
(4.34a)
fulfill the conditions of Lemma 4.2. For such II> the conclusions (4.28)-(4.30) of the Lemma are valid. The forms
fJ~
I
, a, 1JI fulfilling equations (4.16), (4.21), and (4.26) are defined up to
the transformation
fJ~=.pp .a+DfJ~w, a ='a+D/I~w/l+Eaw
1JI
=
where G is real and LEMMA
4.3. The
(4.35)
1JI' + Gw+i(Eawa-Eawa), (4.36)
Dl
can be uniquely determined by the conditi0n8 - - gaPsaQ/I,,-· -- - 0 S(Il1-
(4.37)
der
To prove Lemma 4.3 it suffices to study the effect on SaP"iU when the transformation (4.35) is performed. We put
421 REAL HYFERSURFACES IN COMPLEX MANIFOLDS
S=yaPsaf!, -
D=D:: .
255 (4.38)
Since yaP and SaP are hermitian and DaP is skew-hermitian, S is real and D is purely imaginary. Denoting the new coefficients by dashes, we find (4.39) (4.40)
It follows that
Since we wish to make one set of
S;u =
DI. satisfying
0, the lemma is proved if we show that there is one and only
(4.36) and
- iSga= ygUD+ (n+ 2) DiU'
(4.41)
In fact, contracting (4.41), we get
2(n+I)D
=
-is.
(4.42)
Substitution of this into (4.41) gives
(n+ 2) DiU =
-
iSiU+ 2 (n~ I) SYiU'
(4.43)
It is immediately verified that the DiU given by (4.43) satisfy (4.36) and (4.41) . This proves
Lemma 4.3. By the condition (4.37) the
rpi
are completely determined and we wish to compute
their exterior derivatives. By (4.34) we can put (4.44) where
Ai
are one-forms. Substituting this into (4.20), we get
drpa -rp II rpa -rpP II rpft -
Ap~ II
w P = flN\ W,
(4.45)
fta being also one-forms. From (4.44), (4.33), and (4.26), we get (APa+Aap) II W=YPa wll "P, (4.46)
or
To utilize the condition (4.37) we shall take the exterior derivative of (4.44). We will need the following formulas, which follow immediately from (4.16), (4.45), (4.21):
dW a = d(Yapo./i) = - w PII rpaP + Wa II rp + W II ,foa,
(4.47)
drpa = d(YaPrpP) = rpap II rpP + Aya II wY + fta II w.
(4.48)
We take the exterior derivative of (4.44) and consider only terms involving we lIo.i, ignoring those in w. It gives
422 256
S. S. CHERN AND J . K. MOSER
dS p:'" - ST:''' 4>1.- Sp:'" 4>~~ + Sp:'" 4>i. - Sp:'" 4>a: =.i(A.I g~+ A.!.gp,,- fJl A.aQ - fJ/ Aup)
mod w, w"', ai
(4.49)
and by contraction
When (4.37) is satisfied, the left-hand side, and hence also the right-hand side, of (4.50) are congruent to zero. The congruence so obtained, combined with (4.46), gives ,--
1
-
1L~=-~g~1ji
or
,u_
1.1lu
lLe.=-~UQ1ji,
"ii mo d w,w,w.
Hence we can put (4.51) or
(4.510.)
Substituting into (4.46), we get
v~p+ W"QP=O.
(4.52)
We can therefore write (4.44) in the form
-
=
Sp:'"ofi /\ wO" + Vlewe /\ w -
..
V:pawO" /\ w,
(4.53)
which is the formula for dJ.. Formula (4.53) defines /. completely; it is consistent with earlier notations in Lemma 4.2 and in the subsequent discussions where /. are defined only mod w. Substituting into (4.20), we get
where
11"
are one-forms. Notice also that (4.49) simplifies to
dSp:,,,-ST:'''4>I.-Sp!.a4>e~+S~a4>i.-SPl.. 4>J.='O, mod w, wt%, w ii
(4.55)
on account of (4.51) or (4.51 a). Consider again the transformation (4.35) with Dp~=O. The 4>P~ are now completely determined. From (4.53) its effect on
V/'II is given by
VIQ= Vp:~-i{fJ/Ep+!fJlEe} · Contracting, we have This leads to the lemma:
VllI =
VP~e- i{n+ HEp.
(4.56) (4.57)
423 257
REAL HYPERSURFAOES IN COMPLEX MANIFOLDS
LEMMA 4.4. With (4.21) and (4.37) fulfiUed as in Lemmas 4.1 and 4.2 there is a unique set of '" satisfying
(4.58) To find an expression for dtp we differentiate the equation (4.26). Using (4.16), (4.47), and (4.54), we get w II (-dtp+ IItp+2iP JI.p-iw/lllvp-iv PIIwp)
=
o.
Hence we can write (4.59) where
e is a
one-form.
With this expression for dtp (and expressions for other exterior derivatives found above) we differentiate (4.54) mod wand retain only terms involving w ll II w IF • By the same argument used above, we derive the formula
-- S pp.u." "-.l.P • - ",+' JI " +~Q/'+4>,
n,,"+l
=
Jt ClO
=
2iwa ,
-ic/>a, )
(5.30)
429 REAL HYPERSURFAOES IN COMPLEX MANIFOLDS
263
The :n;A B are one-forms in Y, and the matrix (5.31) l~
au-valued, i.e., (:n;) (h) + (h) t(n) = 0,
Tr (:n;)
=
o.
(5.32)
Moreover, restricted to a fiber of Y, the non-zero :n;'s give the Maurer-Cartan forms of HI' as is already in the flat case. As in the flat case it is immediately verified that using the form (:n;) the equations in the theorem of § 4 can be written d(:n;) = (:n;) II (:n;) +(II),
(5.33)
where
(5.34)
and (n + 2) IIoo =
IIao =
- a~ , IIn2I = - i", IIn~l =
i 'Y, ) ! P,
(5.35)
II P= p _ _ l_ PY a. a. n+ 2 da y.. where the right-hand side members are exterior two-forms in w, w", wi, defined in § 4. For any such form
o == a"ji w" II w7i + terms quadratic in we or wU, mod w, we set
Tr 0
=
ga7i aaji.
(5.36) (5.37)
Then equations (4.37), (4.58), (4.70) can be expressed respectively by Tr III
=
0, Tr IIoo 0,) =
Tr II/ =Tr IIn~l =0,
(5.38)
Tr IIn~1 =0, and their totality can be summarized in the matrix equation Tr (II) =0.
Under the adjoint transformation of HI' (:n;) ..... ad (t)(:n;), (II) ..... ad (t) (II),
(5.39)
430 264
S. S. CHERN AND J . K. MOSER
the condition (5.39) remains invariant. We submit w, w", wfJ, to the linear transformation with the matrix (4.12) and denote the new quantities by the same symbols with asterisks. Since (:n;) is uniquely determined by (5.39) according to theorem 4.1 in § 4 and since these conditions are invariant under the adjoint transformation by HI' we have (:n;O)
=
ad (t)(:n;),
tEGI •
(5.40)
Therefore (:n;) satisfies the conditions of a connection form and we have the theorem: THEOREM
5.1. Given a non-degenerate integrable G-structure on a manifold M of
dimension 2n+1. Consider the principal bundle Y over E with the g'rOUp G1 cSU(p+l, q+l)/K. There is in Y a uniquely defined connection with the group SU(p+l, q+l), which is characterized by the vanishing of the torsion form and the condition (5 .39). In terms of Q-frames ZA which are meaningful under the group SU(p+l, q+l), the connection can be written (5.41) These equations are to be compared with (5.21) where the differential is taken in the ordinary sense. (d) Chains. Consider a curve A which is everywhere transversal to the complex tangent hyperplane. Its tangent line can be defined by
w"=O.
(5.42)
By (4.16) restricted to A, we get (5.43) The curve A is called a chain if b"=O. The chains are therefore defined by the differential system
w"=" =0.
(5.44)
They generalize the chains on the real hyperquadrics in Cn +! (cf. (1.33)) and are here defined intrinsically. It is easily seen that through a point of M and tangent to a vector transversal to the complex tangent hyperplane there passes exactly one chain. When restricted to a chain, equations (4.10), (4.26), (4.59), (4.72) give (5.45)
The forms w, ,
1p
being real, these are the equations of structure of the group of real
linear fractional transformations in one real variable. It follows that on a chain there is a preferred parameter defined up to a linear fractional transformation. In other words, on a chain the cross ratio of four points, a real value, is well defined.
431 REAL HYPERSURFACES m COMPLEX MANIFOLDS
265
6. Actual computation for real hypersurfaces Consider the real hypersurface M in en+l defined by the equation (4.6) . We wish to relate the invariants of the G-structure with the function r (z", z" , W , w), and thus also with the normal form of the equation of M established in § 2, 3. This amounts to solving the structure equations listed in the theorem of § 4, with the G-structure given by (4.7); the unique existence of the solution was the assertion of the theorem. We observe that it suffices to find a particular set of forms satisfying the structure equations, because the most general ones are then completely determined by applying the linear transformation with the matrix (4.12). In actual application it will be advantageous to allow g"p to be variable, which was the freedom permitted in § 4. Our method consists of first finding a set of solutions of the structure equations, without necessarily satisfying the trace conditions (4.37), (4.58), (4.70). By successive steps we will then modify the forms to fulfill these conditions. We set (6.1) ro = 0 = iOr, ro" = dz". Then (4.10) becomes (6.2) It is fulfilled if
g"p= -r"p+r;;;lr"rwp+r~lrprw"-(rwrw) - lrwwr,,rp } _ _ -l Ii + (Twrw) -1 Tww(r"dz " Ii , rw rw"dz, , _ Tw- l Tw{Jdz +rpdz)
(6.3)
where we use the convention (6.4) Exterior differentiation of (6.2) gives
i(dg"p+g"p«(1) = !cldz fi , 1p(I) =
(6.9)
flo·
Its most general solution, to be denoted by C/>l, c/>",
1p,
is related to the particular solution
(6.9), the "first approximation", by c/>l(l) = c/>«(I) = '1JI(I)
where
dfi~
c/>l=dlO, c/>a.+ dldz fi + e"O,
) _
(6.10)
=1p+gO+i(e"dz"-e pdz fi ),
satisfy
(6.11)
and g is real; cf. (4.35), (4.36). We will determine the coefficients in (6.10) by the conditions (4.37), (4.58), (4.70). In view of (4.53) we set
d.l.. y(l) _.I.. a(l) /\ 'f'fi· 'f'fi·
.I.. yell -
'f'a.
ig pa-di; /\
.l..y(l)
'f'
+ i.l..(I) /\ dzl' + ilJ Y (.1..(1) /\ dzy(2) + ic/>~) /\ dz1' + ilJl (c/>~) /\ dza,
'IjJ
so determined in successive steps satisfy now all the structure
equations, together with the trace conditions (4.37), (4.58), (4.70). Notice that our formulas allow the computation of the invariants from the function r. The determinations dfJ~'
ea , g involve respectively partial derivatives of r up to the fourth, fifth, and sixth
orders inclusive. The procedure described above can be applied when the equation of M is in the normal form of § 2, 3. Then we have (6.25) where N22 =
ba..a../J./J.za.·za·zP·zP'
NS2 =
N 23 = ka.a..a..p./J.za.·za·za·zP·zP.
N 42 =N-24 =l
(Xl • ••
--za.·za.·za.·za·zP,zP. a./hPt
(6.26)
- - - za·Za.·Z"'ZP·z/l·ZP. N 33 =m(tlazfXs{Jlp,(JS and N 22 and N 33 are real; the coefficients, which are functions of u, satisfy the usual symmetry relations and are completely determined by the polynomials. Moreover, we have the trace conditions 18 -742902 Acta malhematica 133. lmprime Ie 20 Fevrier 1975
434 268
S. S. CHERN AND J . K. MOSER
(6.27) (6.28) where the traces are formed with respect to ( , ) . The computation is lengthy and we will only state the following results: (1) Along the u-curve
r,
i.e., the curve defined by
z'" =V =0, we have cp"'=O. This means that
r
(6.29)
is a chain. In fact, this is true whenever the conditions
(6.27) are satisfied. (2) Along
r
we find (6.30) p _ 12 i p(i _ v",.y- - n+ 2 h k",ya,
_"'= _
qP.
48 h"';Y l-(n+l)(n+2) yp,
(6.31)
(6.32)
where the quantities are defined by (6.33) (6.34) (6.35) The situation is particularly simple for n = 1. Then conditions (6.27) and (6.28) imply (6.36) On the other hand, we have the remarks at the end of § 4; the invariant of lowest order is qll' Equation (6.32) identifies it with the coefficient in Nn-
Appendix. Bianchi Identities BY S. M. WEBSTER University of Oalifornia, Berkeley, Oalifornia, USA
In this appendix we will show that there are further symmetry relations on the curvature of the connection, which follow from the Bianchi identities and which simplify the structure equation.
435 REAL HYPER SURFACES IN COMPLEX MANIFOLDS
269
The Bianchi identities for the connection defined in section 5c are obtained by taking the exterior derivative of the structure equation (5.33). This yields 0= (TI) /\ (;71;)-(;71;) /\ (TI)+d(TI).
To write this more explicitly it is convenient to use the formulation given in the theorem of section 4. In the G1 bundle Y over E we have the independent linear differential forms
the relations with the gap constant, and the structure equations dw = igapw a /\ wP + w /\ dw a = w P/\
(A. I)
t/ + w /\ a
(A.2)
d = iwp /\ P + ip /\ w P + w /\ 'IjJ d{t =
/ /\ ua+ iwp /\ " d" = /\
ip /\ w" - io{t(u /\
W
U) -
(A. 3)
t Op"'IjJ /\ w + p"
" + p /\ p"- t 'IjJ /\ w" + "
(A . 4) (A . 5)
d'IjJ= /\ 'IjJ+2iP /\ p+ 'Y.
(A. 6)
The curvature forms are given by
p'" = Spe":awe /\ W U+ Vp~Qwe /\ w - V:p.w u /\ w
"= -
VQ~uwe
/\ w u +
V~~we 1\ W
-U
(A. 7) -
+ Pea we /\ w+Q,twU /\ w,
(A. 8)
where the coefficients satisfy the relations
P"p =P"p+P(J",
and
Vl e = gP-;'SPQ«. = g"P Pap = O.
Differentiating equations (A.I) through (A. 6) yields, respectively, a 0= (aP + Pa - gaP 0, w~
I
= - 2 d log 11 12 .
The affine metric is II = 2 Fdu dv
(2.25) where F
=
(1112)112
Then a standard computation implies that the Gaussian curvature of II, which is called the affine curvature is given by (2.26)
I o2iog F K= - - - - . F ouov
448
1I5
AFFINE GEOMETRY
By (2.20) j = dh lZ - h12W~ j hlljw = - 2hI2W~,
(2.27)
h Z2jWi
= -
+
hl2wl
hlZjw
hIZW~
=0
2hI2W~.
So we have hlZi ,.,2
(2.28)
""1 ,.,1 ""2
= 0, = - ~
2h 12
du
'
= _ h222 dv
2h 12
·
Therefore the Fubini-Pick form is (2.29)
Using the structure equation (2.6), we get
wI = ~u (log F) du (2.30)
w~ = ~v (log F) dv w~
w~
+ /~ dv = /~du + /dv, =
/du
where
/~ = _ (2.31)
1_av ~ (hill) 2F '
F3
/~ = _ ~ ~ (1z2?,J,)
au
2F '
=J-
K,
F3
(2.32)
L
=
F/
(2.33)
Since K, L are affine invariants, J is also an affine invariant. Next we develop a necessary and sufficient condition for a graph to be affine minimal. Let a surface be locally given by (2.34)
x 3 = f(x l , x 2 )
So x = (xl, x 2 ,f(XI , x 2») is the position vector. Then equations (2.4), (2.5) hold if we set
449 116
S. S. CHERN AND C. L. TERNG
_ (1,0, ()f) ()x! _ ( ()f) ez - 0, I, ()xz
(2.35)
el -
e3
= (0,0,
I)
with
,,()2f
3 _ Wi -
(2.36)
~ ~. j
Hence hij we let
ux'uX!
.
w!.
= ()ZfI()xi()x j and H = Hessian off. To find the affine normal, e1 = ei
(2.37)
ef
= e3 +
aIel
+
aZez
where ef is in the affine normal direction. Then a,.'s are determined by (2.38)
d loglHI
+ 4 L;
aihikdx k
= O.
i,k
Hence () I - L;j hij --. (log IH) 4 ()X!
(2.39) where (2.40) We compute
Wfl =
(def, ei,
en = da
(2.41)
l
+ a4
1
2
".*2
~3
dloglHI
a 2 = (e*1, de*3, e*) 3 = da + 4 d log IH I.
Therefore the affine mean curvature is
450
117
AFFINE GEOMETRY
We note that the equation for affine minimal surfaces is a fourth order equation in f If f is a non-degenerate quadratic polynomial, then H = constant. Hence the elliptic paraboloid x 3 = (x 1)Z + (x2)2 and the hyperbolic paraboloid x 3 = (X 1)2 - (x2)Z are affine minimal surfaces. Our next result is a formula for affine mean curvature in terms of Riemannian geometry. Let el> ez, e3 be a local orthonormal frame field on M such that el> ez are tangent to M, ()i is the dual coframe and ()ap are defined by (2.43)
de a
= I; ()apep, ()ap + ()Pa = O. P
Then we have equations (1.1) and (1.4) as in section I, and
(2.44)
H
= det(h ij) = K = Gaussian curvature of M .
To find the affine normal direction, we let
ei = ei
(2.45) where
ef
is in the affine normal direction, then ai's are determined by
(2.46) Hence (2.47) where (f)j denotes the covariant derivative ofJwith respect to ej. In fact
DJ = dJ = I;J..ui i
(2.48)
DJ.. = I; kw j = dJ.. - I; jjw{, j
j
We compute
en = w~ + Da Wf2 = (et, deL en = W5 + Da Wfl =
(2.49)
(def , e~,
1
l
+ 4a
Z
+ ~ d log IKI,
d log
IKI,
(2.50)
One immediate application of this formula is the following theorem
451 118
S. S. CHERN AND C. L. TERNG
THEOREM 3. Suppose M is a surface in R3 which is isometric to a piece of the elliptic paraboloid with its induced Riemannian metric. Then as an affine surface M is affine minimal. PROOF. Rewrite (2.50) as follows
21 K I- t L = hu{ - K -
i (log IKI)12 + 136 (log IKIMlog IKlh}
- 2h12{ -
(2.51)
! (log IKI)22 + ttl (log IKIH}
(In this formula the superscript 2 means square.) For the surface x 3
= (x1)2 + X(u, v)
(2.52)
(x2)2, we choose coordinates
= (v cos u, v sin u, v 2).
Then the coefficients of hjj in (2.51) vanish identically, and the theorem follows from the fact that these coefficients only depend on the first fundamental form of the surface. 3. Backlund theorem for affine surfaces. to prove our main theorem:
In this section we are going
THEOREM 4. Let M and M* be the focal surfaces of a W-congruence ill A3, with ihe correspondence denoted by 1': M --+ M* such that the affine normals at P and P* = I'(P) are parallel. Then both M and M* are affine minimal surfaces. PROOF. Choose an affine frame el
(3.1)
eb
e2, e3 such that
= PP*
e2 is tangent to M at P e3 is in the affine normal direction.
Suppose the position vector for M is given by X . Then the position vector for M* is given by (3.2)
X* = X
There exists a function k such that
(3.3)
+ el '
452
119
AFFINE GEOMETRY
is an affine frame on M*, with ef tangent to M* at X*. Let W*i be the dual frame of (3.3). Then
dX* (3.4)
= w*lef + w*zef = - w*lel + w*z (ez +
Le3)'
However, differentiating (3.2) we get (3.5)
+ del = (wi + WDel + (w Z + wi)ez + Wfe3'
dX* = dX
Comparing coefficients of (3.4) and (3.5), we get
+ wD
W*I = - (WI (3.6)
w*z
=
+ mf
wZ
Hence (3.7) Let
a = wi
+ wi +
kw~
= - w~ - dk + 2kw~ r = w 2 + WI - kWf = O.
~
(3 .8)
Then we have
W*I =
(3.9a) From (2.4) we
kw~
- a
w*z = kwf. h;:-'/ e
=( -el> ez + ke3, (J.9b)
= WI3
-
I
-del)
z
fWI
using (3.7) . Similarly, we have
1 - k Z (ka
+
m+ kI WI
453
120
S. S. CHERN AND C. L. TERNG
1 2 CV3*3 = CV33 - k CV3
(3.9c)
CVjl
= CV~
CVj2
= - CV§.
It follows from (2.18a), (3.9a), (3.9b) and (3.9c) that
= 21 IH*14l
L *cv*l /\ cv*2
1
(cv*l /\ cvj2
+ cvjl
l
= 21H*14 [( - kcv~ + a) =
/\ cv§
/\ CV*2)
+ kcv! /\
1 21H*14 [k(cv! /\ cv~ + cv§ /\ cv~) + a l
1
=~
IH*14l [k dcv~ + a
Since e3 is in the affine normal direction,
dcv~
cv~]
/\ cv§]
/\ cv§].
= 0 by (2.11). So we have
(3.10)
By hypothesis, e3 and ef are in the affine normal directions of M and M* respectively; we rewrite (2.11) as
cv~ (3.11) cvj3
= - !dlog IHI
= - ~dlOg
IH*I
Next we compute the following tensor by using (3.9a), (3.9b) and (3.9c) getting,
1: hjicv*' ® cv*i = ~j
(3.12)
1: cv*' ® cv1 3 i
=
(kcv~
=
cv~
=
L: hljcv
- a) ® (1cv 2)
® cv 2 + cv~ ® cv l i
® cvi -
+ kcv~ ® [ - b(ka + (3) + icvll -
1 k [a ® cv 2
+
cv~
® (ka + (3)]
k[a ® cv2 + cv~ ® (ka + (3)].
'. J
We note that the tensors 1:i.i h,jCV*i ® cv*i and 1: •. ihIjCV· ® cv i are symmetric and the same must be true of their difference. Because I is a Wcongruence (i.e., 11* is a multiple of II), these two tensors are proportional in the tensor space. Hence there exists a function b such that
454 121
AFFINE GEOMETRY
(3 .13)
a (is)
W
Z
+ w~ ® (ka + f3) = b 1.:: hijWi ® wi. i, j
This b oF k, for otherwise 11* = 0, contradicting the non-degeneracy of M*Suppose
ka
alw l = blw l
=
a
(3.14)
+ f3
+ azw2 + b2w 2.
Comparing the coefficients of W i (is) wi in (3.13), we get (3.15)
al
+ hllbz =
hlzb l
= hlzb
hllb l = hllb
(3.16) (3.17)
az
+ hl2 bz
= hZ2 b.
Since M is non-degenerate, hll and h lz cannot vanish simultaneously, and (3.15), (3.16) imply that bl = b. It also follows from (3.15) and (3 17) that (3.18) Using (3.9) and (3. 18), we get (3. 19)
W* I 1\ w*z
=
kH(b - k)w 1 1\
WZ
(3.20) However, (3.21) and (b - k) never vanishes, so we have (3 .22)
k4H*H
=
l.
Taking Ij4d log of (3.22) and using (3.11), we obtain dk - wj3 -
w~
=
o.
Then (3.9) implies that
f3 = o.
(3.23) Therefore by (3.14) (3.24)
b;
= ka;.
S.ubstituting (3.24) in (3.15) and (3.17), we obtain (3.25)
(l - khlz)al
-h22al
+ (I
+ hl\kaz = 0 + kh lZ )a2 = o.
455
122
S. S. CHERN AND C. L. TERNG
The determinant of (3.25) is
+
kZH If kZH
+
I¥-O then aj
I.
= 0, so a = O. And if kZH + I == 0, then
gZ - w5 = o.
(3.26)
But (3 = 0, so w~ = O. Therefore we have shown that either a or w~ is zero, so by (3.10) L * = O. Then by symmetry L = 0, i.e., both M and M* are affine minimal. We use the same notations as in the proof of the above theorem. We claim that if (j)~ = 0 then pp* is an asymptotic vector. Indeed using (3 .9),
= 0, L = 0, and L * = 0, we have wjl = /!lW*2 = k/!lW~ (3.27) = w! = /~z . Therefore hll == 0, i.e., el = PP* is an asymptotic vector. Suppose pp* is an asymptotic vector for all P, then hll == O. By using w~
the local theory for hyperbolic affine surfaces in section 2 and (3.7), we can conclude that wi = 0 and k = Ijh 1z . By (2.28), we have hlll = O. So J = O. But we have already shown that L = 0, hence K = O. Therefore we have proved the following two corollaries. COROLLARY I. Assumptions as in Theorem 4. totic direction for all P E M, then
a = 0, {3 = 0,
(3.28)
/fPP* is not in the asymp-
r = o.
COROLLARY 2. Assumptions as in Theorem 4. /f PP* is an asymptotic vector for all P E M, then both M and M* are affine minimal and affinely flat (i.e., the affine curvature is zero).
Now we wish to prove the integrability theorem. THEOREM 5. Suppose M is an affine minimal surface in A3. Given Vo E TpoCM) which is not an asymptotic vector, then there exist a surface M* and a W-congruence /: M -> M* with parallel affine normals at P E M
and P*
= /(P) E M* and PoPo* =
PROOF.
Taking the differential of the system (3.28), we have
da = (3.29)
Vo·
r
1\ w~ -
d{3 = - w! 1\ T dr
=a
1\
wi -
{3 1\ w~
+a
+ w~
a
1\
w~ 1\ {3.
+
1\ wI
2{3
1\
w~
-
21HI-t Lw l
1\ w 2
456 123
AFFINE GEOMETRY
That the system (3.28) is completely integrable follows from the fact that M is affine minimal. So there exist a function k and an affine frame el> ez, e3 with e3 in the affine normal direction and el(PO) = Vo such that a = 0, f3 = 0, T = 0. Let X be the position vector of Min A3, and X* = X + el' Using T = 0, we have
dX* = dX (3.30)
=
+ del
+ WDel + kwr(ez + ke3).
(WI
Since a = 0, (WI
(3.31)
+ wi)
A
kw¥ = kZwr
A w~
= kZHwl A wZ.
Since M is non-degenerate, WI + wi and kw¥ are linearly independent. Hence X* defines a surface M* having el> ez + l/ke3 as tangent at X* Therefore we can choose an affine frame on M* as follows
(3.32)
Then we have (3.9). Since a
= f3 = 0, (3.12) implies that
(3 .33) i.e.,
t' :
X ...... X ';' is a W-congruence.
Next we want to show that ej is in the direction of affine normal of M*. By (3 .9), (3.34) However, (3.35) =
-kZH*Hwl A
WZ
using (3.9),
so (3.36)'
k4H*H
=
Since e3 is in the affine normal direction, we have
1.
w3 = -
1/4 d log
IHI . By (3 .36),
457 124
(3.37)
S. S. CHERN AND C. L. TERNG
dk k
+ dlog4IH*I
-
3
QJ3
°
= .
Using (3.9), (3.37) and {3 = 0, we get
dk
k -
3 QJ3
i.e., ef is in the affine normal direction of M*. We note that if M is affinely flat and affine minimal in A3 with position vector X, then given any asymptotic vector field el on M such that X* = X + el defines a surface in A3, it follows from the local theory for hyperbolic surfaces in section 2 that E, such that the composition 7TOS is the identity. Since E is only locally a product, the differentiation of s needs an additional structure, usually called a connection. The resulting differentiation, called covariant differentiation, is generally not commutative. The notion of curvature is a measure of the noncommutativity of covariant differentiation. Suitable combinations of the curvature give rise to differential forms which represent characteristic cohomology classes in the sense of the de Rham theory, of which the Gauss-Bonnet formula (4a) is the simplest example [13]. I believe that the concepts of vector bundles, connections, and curvature are so fundamental and so simple that they should be included in any introductory course on multivariable calculus.
9. Elliptic differential equations. When M has a Riemannian metric, there is an operator • sending a k-form a to the (n-k)-form .a,n=dimM. It corresponds to the geometrical construction of taking the orthogonal complement of a linear subspace of the tangent space. With. and the differential d we introduce the codifferential (14)
and the Laplacian
il=d8+8d.
(15)
Then the operator 8 sends a k-form to a (k -I)-form and il sends a k-form to a k-form. A form a satisfying (16) ila=O is called harmonic. A harmonic form of degree 0 is a harmonic function in the usual sense. The equation (16) is an elliptic partial differential equation of the second order. If M is closed, all its solutions form a finite dimensional vector space. By a classical theorem of Hodge this dimension is exactly the kth Betti number bk • It follows by (12) that the Euler characteristic can be written
x(M)=d.-do,
(17)
where de (respectiveiy, do) is the dimension of the space of harmonic forms of even (respectively odd) degree. The exterior derivative d is itself an elliptic operator and (17) can be regarded as expressing X(M) as the index of an elliptic operator. The latter is, for any linear elliptic operator, equal to the dimension of the space of solutions minus the dimension of the space of solutions of the adjoint operator. The expression of the index of an elliptic operator as the integral of a local invariant culminates in the Atiyah-Singer index theorem. It includes as special cases many famous theorems, such as the Hodge signature theorem, the Hirzebruch signature theorem, and the Riemann-Roch theorem for complex manifolds. An important by-product of this study is the recognition of the need to consider pseudo-differential operators on manifolds, which are more general than differential operators. Elliptic differential equations and systems are closely enmeshed with geometry. The CauchyRiemann differential equations, in one or more complex variables, are at the foundation of complex geometry. Minimal varieties are solutions of the Euler-Lagrange equations of the variational problem minimizing the area. These equations are quasi-linear. Perhaps the "most" non-linear equations are the Monge-Ampere equations, which are of importance in several geometrical problems. Great progress has been made in these areas in recent years (14). With this heavy intrusion of analysis George Birkhoffs remark quoted above sounds even more disturbing. However, while analysis maps a whole mine, geometry looks out for the beautiful
467 348
[May
SHIING-SHEN CHERN
stones. Geometry is based on the principle that not all structures are equal and not all equations are equal. 10. Euler characteristic as a source of global invariants. To summarize, the Euler characteristic is the source and common cause of a large number of geometrical disciplines. I will illustrate this relationship by a diagram. (See Fig. 8.) Combinatorial Topology
Elliptic Topology
Total Curvature
Homology and Sheaf Cohomology
Characteristic Gasses
FIG. 8
11. Gauge field theory. At the beginning of this century differential geometry got the spotlight through Einstein's theory of relativity. Einstein's idea was to interpret physical phenomena as geometrical phenomena and to construct a space which would fit the physical world. It was a gigantic task and it is not clear whether he said the last word on a unified field theory of gravitational and electromagnetic fields. The introduction of vector bundles described above, and particularly the connections in them with their characteristic classes and their relations to curvature, widened the horizon of geometry. The case of a line bundle (i.e., when the fiber is a complex line) furnishes the mathematical basis of Weyl's gauge theory of an electromagnetic field. The Yang-Mills theory, based on an understanding of the isotopic spin, is the first example of a nonabelian gauge theory. Its geometrical foundation is a complex plane bundle with a unitary connection. Attempts to unify all field theories, including strong and weak interactions, have recently focused on a gauge theory, i.e., a geometrical model based on bundles and connections. It is with great satisfaction to see geometry and physics united again. Bundles, connections, cohomology, characteristic classes are sophisticated concepts which crystallized after long years of search and experimentation in geometry. The physicist C. N. Yang wrote [15]: "That nonabelian gauge fields are conceptually identical to ideas in the beautiful theory of fiber bundles, developed by mathematicians without reference to the physical world, was a great marvel to me." In 1975 he mentioned to me: "This is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere." This puzzling is mutual. In fact, referring to the role of mathematics in physics, Eugene Wigner spoke about the unreasonable effectiveness of mathematics [16]. If one has to find a reason, it might be expressed in the vague term "unity of science." Fundamental concepts are always rare. 12. Concluding remarks. Modern differential geometry is a young subject. Not counting the strong impetus it received from relativity and topology, its developments have been continuous.
468 19791
FROM TRIANGLES TO MANIFOLDS
349
I am glad that we do not know what it is and, unlike many other mathematical disciplines, I hope it will not be axiomatized. With its contact with other domains in and outside of mathematics and with its spirit of relating the local and the global, it will remain a fertile area for years to come. It may be interesting to characterize a period of mathematics by the number of variables in the functions or the dimension of the spaces it deals with. In this sense nineteenth century mathematics is one-dimensional and twentieth century mathematics is n-dimensional. It is because of the multi-variables that algebra acquires paramount importance. So far most of the global results on manifolds are concerned with even-dimensional ones. In particular, all complex algebraic varieties are of even real dimension. Odd-dimensional manifolds are still very mysterious. I venture to hope that they will receive more attention and substantial clarification in the twenty-first century. Recent works on hyperbolic 3-manifolds by W. Thurston [17] and on closed minimal surfaces in a 3-manifold by S. T. Yau, W. Meeks, and R. Schoen have thrown considerable light on 3-manifolds and their geometry. Perhaps the problem of problems in geometry is still the so-called Poincare conjecture which says that a closed simply connected 3-dimensional manifold is homeomorphic to the 3-sphere. Topological and algebraic methods have so far not led to a clarification of this problem. It is conceivable that tools in geometry and analysis will be found useful. This paper, written with partial support from NSF Grant MCS77-23579, was delivered as a Faculty Research Lecture at Berkeley, Ca1ifornia, on April 27, 1978. Refereaces
1. O. Veblen and 1. H. C. Whitehead, Foundations of Differential Geometry, Cambridge, England, 1932, p. 17. 2. Elie Carlan, Le role de Ia theorie des groupes de Lie dans I'evolution de la geometrie moderne, Congres Inter. Math., Oslo, 1936, Tome I, p. 96. 3. George D. Birkhoff, Fifty years of American mathematics, Semic.:ntennial Addresses of Amer. Math. Soc., 1938, p. 307. 4. A Wei!, S. S. Chern as friend and geometer, Chern, Selected Papers, Springer Verlag, New York, 1978, p. xii. S. H. Whitney, On regular closed curves in the plane, Comp. Math. 4 (1937) 276-284. 6. William F. Pohl and George W. Roberts, Topological considerations in the theory of replication of DNA, 10urnal of Mathematical Biology, 6 (1978) 383-386, 402. 7. lames H. White, Self-linking and the Gauss integral in higher dimensions, American 1. of Math ...91 (1969), 693-728; B. Fuller, The writhing number of a space curve, Proc. Nat. Acad Sci., 68 (1971) 815-819; F. Crick, Linking numbers and nucleosomes, Proc. Nat. Acad. Sci., 73 (1976) 2639-2643. 8. S. Smale, A classification of immersions of the two-sphere, Transactions AMS, 90 (1959) 281-290; cf. also A. Phillips, Turning a surface inside out, Scientific American, 214 (May 1966) 112-120. A film of the process, by N. L. Max, is distributed by International Film Bureau, Chicago, III. 9. H. Weyl, Philosophy of Mathematics and Science, 1949, p. 90. 10. A. Einstein, Library of Living Philosophers, vol. I, p. 67. 11. 1. Hadamard, Psychology of Invention in the Mathematical Field, Princeton, 1945, p. 115. Il. R. Godement, Topologie algebrique et theorie des faisceaux, Hermann, Paris, 1958. 13. S. Chern, Geometry of characteristic classes, Proc. 13th Biennial Sem. Canadian Math. Congress, 1-40 (1972). 14. S. T . Yau, The role of partial differential equations in differential geometry, Int. Congress of Math., Helsinki, 1978. 15. C. N . Yang, Magnetic monopoles, fiber bundles, and gauge fields, Annals of the New York Academy of Sciences, 294 (1977) 86-97. 16. E. Wigner, The unreasonable effectiveness of mathematics in the natural sciences, Communications on Pure and Applied Math., 13 (1960) 1-14. 17. W. Thurston, Geometry and topology in dimension three, Int. Congress of Math., Helsinki, 1978. DEPARTMENT OF MATHEMATICS,
UNIVERSITY
OF CALIFORNIA, BERlCEl.BY, CA 94720.
469
manuscripta mathematica
manuscripta math. 28, 207 - 217 (1979)
© by Springer-Verlag 1979
LIE GROUPS AND KdV EQUATIONS Shiing-shen Chern * and Chia-kuei Peng Dedicated to Hans Lewy and Charles B. Morrey, Jr. I.
In troduct ion In recent years there have been extensive studies
of evolution equations with soliton solutions, among which the most important ones are the Korteweg-deVries and sine-Gordon equations.
We will show that the
alge-
braic basis of these mathematical phenomena lies in Lie groups and their structure equations; their explicit solutions with special properties often give the evolution equations. The process is thus similar to the introduction of a "potential".
In fact,
from SL(2; R),
the special linear group of all (2 X 2)-real unimodular matrices, one is led naturally to the KdV and MKdV (= modified Korteweg-deVries) equations of higher order. A Miura transformation exists between them. H.H. Chen,
Following
(2) this leads to the Backlund transforma-
tions of the KdV equation.
2.
KdV equations Let
(1)
SL(2; R)
* Work
done under partial support of NSF grant MCS77-23579. 207
470 be the group of all (2 X 2)-real unimodular matrices. Its right-invariant Maurer-Cartan form is Wl 1
(2)
w=dXX
1 1
=
(
w 2
where
1
w
(3)
l
2 + w = O. 2
The structure equation of SL(2;R), or the Maurer-Cartan equation, is
(4)
dW=W"w,
or, written explicitly, =w (4a)
U
1
2 l
2w
dwl 2
2W
dW
Let
2
l 1 1
2
2 " Wl 1 1
" W
be a neighborhood in the (x,t)-plane and
consider the smooth mapping (5)
f : U
-7
SL(2; R)
.
The pull-backs of the Maurer-Cartan forms can be written 1
W l
T)dx + Adt qdx + Bdt
(6 )
w
1
2
= rdx + Cdt ,
where the coefficients are functions of
x,t.
The forms in (6) satisfy the equations (4a). gives
208
This
We c o n s i d e r t h e s p e c i a l c a s e t h a t
i s a p a r a m e t e r i n d e p e n d e n t of
x,t.
we g e t from (7)
and
7
Writing
,
q = u(x,t)
(8)
r = +1
,
S u b s t i t u t i o n i n t o t h e second e q u a t i o n of (7) g i v e s (10)
u
,
= K(u)
t
where K(u) = u C X
+
2uC
X
+
2 27 C - x 2
cxxx
A s a n example we t a k e
Then (10) becomes
which is t h e well-known KdV (= Korteweg d e V r i e s ) equa-
t ion. I t is t h e r e f o r e n a t u r a l t o take
7.
C
t o be an a r b i -
S i n c e t h e e x p r e s s i o n i n (11) 2 i n v o l v e s e x p l i c i t l y o n l y 7 , we s h a l l s u p p o s e C t o 2 be a polynomial i n 7 , i . e . , p r e c i s e l y . t r a r y polynomial i n
472
where
C.(x,t)
are functions of
x,t.
Substi tuting
J
(14) into (11) and equuting to zero the powers of
Tj
2
,
we get (15)
Co = const.,
(16)
C
1
j+l,x
2
+ !. C u C. - uC 4 j,xxx j,x x J
o
~
n-l
Rj + l , we can The rightO.
given by (33) , will be de-
R
Then we have
n
(35)
j
j
x
M (v)
n
=
-v
-1
R
n+l, x
(v)
From (34) we immediately observe that
R . are even, J
i. e. ,
R. (-v) J
(36)
It follows from (35) that M (-v) n
(37)
=
R . (v) J M (v) n
are odd, i.e.,
-M (v) .
n
There is no loss of generality in supposing
(38)
are found to be
R.' s
and the first
J
RO
=1
Rl
=- 2
1
V
2
213
RO = 1,
476
~
4.
3
= 8"
1
4
v
2
1
vv
- "4
+ 8 Vx
xx
Miura transformation The connection between the KdV and MKdV equations
is furnished by the Miura transformation.
To define it
we observe that (17) can be written formally as (39)
K 1 (u) = TK (u) n+ n
where (40)
1 u'
T
2
d D = dx'
-1
D,
I'
u
u
x
Similarly, we write (35) as
(41)
M (v) = SM (v) n+l n
,
where 1
2
- v
S -- -4 D
(42)
2
- v
,-1
D
v.
By an easy computation the following commutativity relations can be verified: ,
(2v + D)S(v)
T(v
(2v - D)S(v)
T(-v
2 + v ) (2v + D)
(43) I
,
2 + v ) (2v - D)
.
It follows that K (v n (44) K (-v n In fact, and (32), Ml (v)
for
n = 1,
+ v 2)
I
,
2 + v )
(2v - D)M (v) n
this follows directly from (13)
their right-hand sides being
respectively.
,
(2v + D)M (v) n
Kl (u)
and
The general case follows from in-
duction by applying (43)
to
M l(v). nThese results were derived by a different method
214
477
in [1], in which they, or at least a part of them, were attributed to P. Olver.
Formula (44) gives a funda-
mental relation between the KdV and MKdV equations and is at the basis of the Miura transformation.
Of impor-
tance is the relation (45) If
= vx
u u
and
v
+ v
2
are so related and if
R . (v)
satisfy
J
then, by a straightforward computation, we find
(34),
that 1 = - - M.
C . (u)
J
2
J-l
(v)
+ R . (v) J
satisfy the recurrent relation (16).
-!.2
(46a)
It follows that 2 C. (v + v ) J x
M (v) + R . (v) j-l J
.
Similarly, we have 1
= C . (-v
-2 M. l(v) + R.(v) JJ
(46 b)
J
x
2 + v )
.
These relations can also be written 2 2 -C. (v + v ) + C . (-v + v ), x J J x
M. 1 (v) J(47)
= CJ. (v x
2R . (v) J
In particular, we draw from (47) M. (v), R.(v) J
are polynomials of
J
tives.
Moreover, if
v
2 2 + v ) + C . (-v + v ) x J the conclusion that v
and its x-deriva-
and the subscript
given the weight I,
then
isobaric of weight
2j.
M. 1 (v)
J-
and
x
R . (v) J
are each are
From (44) we get
~~
(48)
where
u
- Kn (u)
=
(D + 2v)
is given by (45).
tion of MKdV,
then
u(x,t),
tion of KdV.
But then
-v
(~~
Thus, if
- Mn v
(V)) is a solu-
given by (45), is a soluis also a solution of MKdV, 215
478 so that the KdV has a new solution given by (49)
u
=
-v
to
x u
formation, and that from
u
This passage from
v
+ v
2
is called a Miura transto
u
a Backlund trans-
10rmation, following an approach of To pass from
u to
u
H.H. Chen [3).
we set
(50)
u = w x
Then u - u
=
(w - ';)
2v
x
x
and we can suppose 2v
(51)
w - w
I t follows that
(w +
';)
x
~ 2 1 (w - w) 2
(!
(52) ~
(w - w) t Wi th
w
given, such that
(w 2M n 2 u = w
x
;))
is a solution of (18),
the system (52) is completely integrable. w
of (52)
gives a new solution
is a Backlund transform of
u = w x
A solution of (18);
u
u.
From the MKdV equation one can pass to a twice modified KdV equation by a similar procedure.
This and
other results will be reported later. References 1.
M. Adler and J. Moser, On a class of polynomials connected with the Korteweg-deVries equation, Communications in Math Physics 61, 1-30 (1978)
2.
HSing-Hen Chen, Relation between Backlund transformations and inverse scattering problems, Lecture Notes in Math, no. 515, 241-252, Springer 1976
216
479 3.
M. Crampin, F.A.E. Pirani, D.C. Robinson, The soliton connection, Lett. Math. Phys. 2, 15-19 (1977)
4.
C. S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Korteweg-deVries equation and generalizations VI, Methods for exact solution, Comm. Pure and Appl. Math. 27, 97-133 (1974)
University of California Berkeley, California 94720, USA and University of Science and Technology of China Hofei, Anhwei People's Republic of China
(Received February 23, 1979)
2 17
Reprinted from Diferential Geometry and Complex Analysis, Springer Verlag, 1984.
Deformation of Surfaces Preserving Principal Curvatures By Shiing-shen Chern '
1. Introduction and Statement of Results The isometric deformation of surfaces preserving the principal curvatures was first studied by 0.Bonnet in 1867. Bonnet restricted himself to the complex case, so that his surfaces are analytic, and the results are different from the real case. After the works of a number of mathematicians, W. C. Graustein took up the real case in 1924 -, without completely settling the problem. An authoritative study of this problem was carried out by Elie Cartan in [2], using moving frames. Based on this work, we wish to prove the following: Theorem: The non-trivial families of isometric surfaces having the same principal curvatures are the following: 1) a family of surfaces of constant mean curvature; 2) a family of surfaces of non-constant mean curvature. Such surfaces depend on six arbitrary constants, and have the properties: a) they are W-surfaces; b) the metric d i 2 = (grad H)2 ds2/(H2- K), where ds2 is the metric of the surface and H and K are its mean curvature and Gaussian curvature respectively, has Gaussian curvature equal to - 1. By a non-trivial family of surfaces we mean surfaces which do not differ by rigid motions. The theorem is a local one and deals only with pieces of surfaces. We suppose that they do not contain umbilics and that they are C5. The analytic formulation of the problem leads to an over-determined system of partial differential equations. It must be the simple geometrical nature of the problem that the integrability conditions give the clear-cut conclusion stated in the theorem. The surfaces in class 2) are clearly of interest. An analogous problem is concerned with non-trivial families of isometric surfaces with lines of curvature preserved. They also have a simple description and are given by the molding surfaces; cf. [I, pp. 269-2841. I wish to thank Konrad Voss for calling my attention to this problem.
' Work done under partial support of NSF grant MCS 77-23579
481 Defonnation of Surfaces Preserving Principal Curvatures
156
2. Formulation of Problem We consider in the euclidean space E3 a piece of oriented surface M, of sufficient smoothness and containing no umbilics. Over M there is then a well-defined field of orthonormal frames xel e 2e3' such that x E M, e3 is the unit normal vector at x, and el>e 2 are along the principal directions. We have then dx = wle l + W2e2 del = W12e2 + W13e3, de 2 = - wlt"e l + W23 e 3, de3 = - W13 e l - w23e~,
(1)
the w's are one-forms on M. Our choice of the frames allows us to set W I2
WI3
= hWI + kW2 = aWl, W23 = CW2,
a> c.
(2)
Then a and c are the two principal curvatures at x. As usual we dellote the mean curvature and the Gaussian curvatures by H
= -Ha + c), K = ac.
(3)
The functions and forms satisfy the structure equations obtained by exterior differentiation of (1). They give dW I dW 12 dW I3
= W12 /\ W2 , dW 2 = WI /\ W12 , = - KWI /\ W2, = W I2 /\ W23, dW23 = W13 /\ W12 ·
(4)
The equation in the second line of (4) is called the Gauss equation and the equations in the last line of (4) are called the Codazzi equations. Using (2), the Codazzi equations give {da - (a - c) hw 2} /\ WI = 0, {dc - (a - c) kwd /\ W2 = 0.
(5)
We introduce the functions u, v by 2dH
= d(a + c) = (a
- c) (UWI
+ vw 2 ).
(6)
Then we have 1 - - da a-c 1 - - dc a-c
=
(u - k)wI
+ hW2, (7)
= kWI + (v - h)W2'
and d log(a - c)
= (u - 2k)wI - (v - 2h)W2 '
(8)
We note also the relation 4(grad H)2 = (a - C)2 (u 2
+ v 2 ).
(9)
482 Defonnation of Surfaces Preserving Principal Curvatures
157
For our treatment we introduce the forms
= - VWI + UW2, exl UWI - VW2' ex 2 = VW I + UW 2 • = 0 is tangent to the level curves H = const and (Jl
= =
UWI
+ VW2'
(J2
(10) (11)
Thus (JI exl = 0 is its symmetry with respect to the principal directions. If H =l= const, the quadratic differential form d§2
=
(Ji
+ (J~ = exi + ex~ = (u 2 + v2 )(wi + w~) (grad H)2
= H2_K ds
2
(12)
defines a conformal metric on M . We find it convenient to make use of the Hodge *-operator, such that *W2= -WI'
(13)
on one-forms. Then we have
* (JI = (J2, * (J2 = -
(14)
(Jt>
(15)
Using these notations Eq. (6) and (8) can be written
+ c) = (a + 2 * W12.
2dH = d(a d log(a - c)
=
exl
C)(JI'
(6 a) (8 a)
Suppose M* is a surface which is isometric to M with preservation of the principal curvatures. We shall denote the quantities pertaining to M* by the same symbols with asterisks, so that a* = a,
c*
= c.
(16)
As M and M* are isometric, we have wT = cos rWI - sin rw 2 , w! = sin rWI + cos rw 2 •
(17)
Exterior differentiation gives dwT = (- dr + W 12 ) 1\ wt = wT 1\ ( - d r + W 12),
d w!
so that (18)
By (8 a) we get Applying the *-operator to this equation, we find WT2 -
W I2
=
i(ex! - ex 2)·
483 Defonnation of Surfaces Preserving Principal Curvatures
158
This gives dr=Hx2 - a
n
(19)
We wish to simplify the last expression. From (6a) we have
Or = 01> i.e. which gives, in view of (17), u* v*
= cos r u - sin r v = sin r u + cos r v.
(20)
It follows that a!
= sin 2 r . a I + cos 2 r . a2'
Putting t
= cot r,
(21)
we get from (19), dt
=
tal - a2'
(22)
This is the total differential equation satisfied by the angle r of rotation of the principal directions during the isometric deformation. In order that the deformation be non-trivial it is necessary and sufficient that the Eq. (22) be completely integrable. This is expressed by the conditions dal
= 0,
(23)
When the mean curvature H is constant, we have tt=v=O
and t = const. This gives the theorem of Bonnet (cf. [3]) : Theorem (Bonnet) : A surface of constant mean curvature can be isometrically deformed preserving the principal curvatures. During the deformation the principal directions rotate by a fixed angle.
3. Connection Form Associated to a Coframe Given the linearly independent one-forms WI, W 2 , the first two equations in (4) uniquely determine the form W 1 2' We call WI, W2 the (orthonormal) coframe of the metric (24)
484 Deformation of Surfaces Preserving Principal Curvatures
159
and W I2 the connection form associated to it. The discussions leading to (18) give the following lemma: Lemma 1. When the coframe undergoes the transformation (17), the associated connection forms are related by (18).
We now consider a conformal transformation of the metric d§2 = A2 ds 2 = A2(wf + w~),
(25)
where A > 0 is a function on M. Let (26) Then we have: Lemma 2. Under the changes of coframe (26) the associated connection forms are
related by Wf2 = W 12 -
i(iJ - 0) log A.
(27)
Here iJ,"O are the differentiation operators relative to the complex structure = WI + iW2 of M. The proof is by straightforward calculation and will be omitted. We note, however, the useful formula
W
*(iJ - 0)/= - id/
(28)
where / is a function on M.
4. Surfaces of Non-Constant Mean Curvature Suppose H =l= const. Then A = + (u 2 + V2)1/2 > 0,
(29)
and we write u
+ iv = A
exp(it/!).
(30)
Let
= () =
W
WI
()I
rx = rxl
+ iW2' + i()2, + irx2.
(31)
Then
() = A exp( - it/!)w, rx = A exp(it/!)w,
(32)
rx = exp(2 i t/!)().
(33)
so that The forms w, (), rx define the same complex structure on M and the operators *, can be used without ambiguity.
a, "0
485 Defonnation of Surfaces Preserving Principal Curvatures
160
Let (0\2,0 12 ,0(\2 be the connection forms associated to the co frames (01' (02; 01>0 2 ; 0(1,0(2 respectively. By Lemmas 1 and 2, Sect. 3, we have the fundamental relation 0\2
= (0\2 + d", - i(a - 0) log A = 2d", + 0(12.
(34)
In addition, from (23) we have (35) The second equation of (23) then implies that the metric d§2 on M has Gaussian curvature equal to - 1. Moreover, the Eq. (35) shows that the curves 0(2 = 0 are geodesics and the curves 0(1 = 0 have geodesic curvatures equal to 1, i.e., are horocycles relative to the metric d§2. From (8 a) and (23) we get d*(O\2
= O.
(36)
* to (34), we get, by using (28), *0\2 = *(012 + *d", - d log A = 2*d",
Applying
-
0(1.
(37)
Exterior differentiation of the last equation gives, in view of (23), (36), d*d", = 0,
(38)
which says that", is a harmonic function. Differentiation of (37) then gives (39)
d*012=O.
By differentiating (6a) and using (8 a), we get dOl
+ (0(1 + 2*(Od
/\ 0 1 =
o.
But (40)
From (37) we find
- * 0 12 + 0(1 + 2 * (012 =
2d log A.
It follows that
d log A /\ 0 1 = 0,
(41)
and we set
d log A = B0 1 •
(42)
This is a differential equation in log A . But a"O log A is related to the Gaussian curvature K of M . We wish to combine these facts to draw the remarkable conclusion that M is a W-surface. This involves further computation of the integrability conditions. The simplest way is to make use of the coframe 0( 1,0(2' because their exterior derivatives satisfy the simple Eq. (23). For a function f on M we define df=flO(I +f20(2·
(43)
486 Defonnation of Surfaces Preserving Principal Curvatures
161
Its cross covariant derivatives satisfy the commutation formula
121 - 112
+ 12 = o.
(44)
Moreover, the condition for 1/1 to be a harmonic function is
+ 1/1 22 + 1/1 1 = o.
1/1 11
(45)
Note also that, by (37),
*0 12 = - (21/12 + 1)0: 1 + 21/110: 2.
(46)
By (6 a) and (8 a), the condition for M to be a W-surface is (0: 1 + 2 * W 12)
1\
01 =
o.
Using (37) and (42), this can be written
21/11 cos 21/1
+ (21/12 + 1) sin 21/1 = O.
(47)
From (42) we have (log A)l = B cos 21/1,
(log A)z = B sin 21/1,
(48)
whose differentiations give (log A)\i = Bi cos 21/1 - 2Bl/li sin 21/1, (log A)zi = Bi sin 21/1 + 2Bl/li cos 21/1,
i
= 1,2.
(49)
The commutation formula (44) applied to log A gives
B1 sin 21/1 - B2 cos 21/1
+ B{21/11
cos 21/1
+ (21/12 + 1) sin 21/1} = 0. (50)
But there is another equation between B 1 , B 2 , to be derived from the Gauss equation (51)
as follows: From (34) we have W 12
= d 1/1 + 0:2 + (log A)2 0:1 - (log A)I 0:2·
(52)
Substituting into the above equation, we get - (log A)11 - (log A)22
+ {- (log A)I +
I}
+ ac A - 2 = 0,
or, by (49),
- B I cos 21/1 - B 2 sin 21/1 + B {2 1/1 I sin 21/1 - (21/12
+ 1) cos 21/1} + 1 + a c A - 2 = 0.
(53)
Solving for B I , B2 from (50), (53),
BI
+ B(2"'2 + 1) -
(1 B2 - 2BI/II - (1
+ acA -2) cos 21/1 = 0, + acA -2) sin 21/1 = 0.
(54)
Differentiating the first equation with respect to the second index, the second equation with respect to the first index, subtracting, and using the Eq. (45) that 1/1
487 162
Deformation of Surfaces Preserving Principal Curvatures
is a harmonic function, we get
- 2(1
+ ae A -2) {21/11 cos 21/1 + (21/12 + 1) sin 21/1} + A -2 {- (ae)1 sin 21/1 + (ae)2 cos 21/1} = o.
(55)
The expression in the last braces is the coefficient of IXI /\ IX2 in
- * d(ae)
/\ ()2.
Now 4ae
= (a + e)2 - (a - e)2,
and its differential can be calculated, using (6 a) and (8a). We get 2d(ae) -- =
a-e
(a
+ e)()1 -
(a - e)(IXI
+ 2*wd
and
-
2 (a - c)
2 (* d(ae)) /\ ()2 = (IX2 - 7.wd /\ ()2 = - {21/11 cos 21/1
+ (21/12 + 1) sin 21/1} IXI
/\ IX2·
Hence (55) becomes (1
+ H2 A -2) {21/11
cos 21/1
+ (21/12 + 1) sin 21/1} = O.
Since the first factor is non-zero, the second factor must vanish, which is the condition (47) for M to be a W-surface. On M with the metric d §2 of Gaussian curvature - 1 we search for a harmonic function 1/1 satisfying (47). We shall show that such a function depends on two constants. In fact, Eq. (47) allows us to put 21/11 = C sin 21/1,
21/12
+ 1 = - C cos 21/1.
(56)
Differentiation gives Ci sin 21/1 + 2Cl/li cos 21/1, 21/1li = - Ci cos 21/1 + 2Cl/li sin 21/1, i = 1,2 .
2 1/1 Ii
=
(57)
The commutation formula for 1/1 and Eq. (45) give - C I cos 21/1 - C 2 sin 21/1 + 2CI/I I sin 21/1 - C(21/12 + 1) cos 21/1 -1 = 0, C I sin 21/1 - C 2 cos 21/1 + 2CI/I I cos 21/1 + C(21/12 + 1) sin 21/1 = o. (58)
Solving for C I, C 2, we get C I + C(21/12+ 1) + cos 21/1 =0, C 2 - 2 C 1/1 I + sin 21/1 = O.
(59)
It can be verified by differentiating (59) that the commutation relation for C is satisfied. Hence there exist harmonic functions 1/1 satisfying (47). The solution depends on two arbitrary constants, the values of 1/1 and C at an initial point. From our discussion the differentials of the functions log A, B, a, e are all determined. Hence our surfaces, e.g., the surfaces of non-constant mean curvature which can be isometrically deformed in a non-trivial way preserving the principal
488 Defonnation of Surfaces Preserving Principal Curvatures
163
curvatures, depend on 6 arbitrary constants. This proves the main statement of our theorem in Sect. 1, the other statements being proved before. Our derivation makes use of the 5th order jet of the surface M, which is therefore supposed to be of class 5.
References [1]
Bryant, R.; Chern, S.; Griffiths, P. A.: Exterior differential systems. Proceedings of 1980 Beijing DD-Symposium. Science Press, Beijing, China and Gordon and Breach, New York, 1982, vol. 1, pp. 219-338 [2] Cartan, E.: Sur les couples de surfaces applicables avec conservation des courbures principales. Bull. Sc. Math. 66 (1942), 1- 30, or Oeuvres Completes, Partie III, vol. 2, 1591-1620 [3] Darboux, G .: Theorie des surfaces, Partie 3. Paris 1894, p. 384
491 Reprinted from Differential Geometry and Topology. Lecture Notes in Math. 1369. Springer Verlag. 1989.
DUPIN SUBMANIPOLDS IN LIE SPHERE GEOMETRY Thomas E. Cecil and Shiing-Shen Chern
1. Introduction .
Consider a piece of surface immersed in three- dimensional Euclidean space E3 .
Its normal lines are the common tangent lines of two surfaces. the focal
surfaces .
These
focal
surfaces
may have singularities.
and a classical
theorell says that if the focal surfaces both degenerate to curves. then the curves are conics. and the surface is a cyclide of Dupin . [CR . pp. 151-166].)
(See. for example.
Equivalently. the cyclides can be characterized as those
surfaces in E3 whose two distinct principal curvature s are both constant along their corresponding lines of curvature . The
cyclides
have
been
generalized
hypersurfaces in En. the Dupin hypersurfaces .
to
an
interesting
class
of
Initially. a hypersurface M in
En was said to be Dupin if the nUlllber of distinct principal curvatures (or focal points) is constant on M and i f each principal curvature is constant along the leaves of its corresponding principal foliation. [GU] . )
(See [CR] . [Th].
More recently. this has been generalized to include cases where the
number of distinct principal curvatures is not constant.
(See [P3]. [CC] . )
The study of Dupin hypersurfaces in En is naturally situated in the context of Lie sphere geometry. developed by Lie [LS] as part of his work on The projectivized cotangent bundle PT * En of En has a n In fact. if xl .. . .• x are the coordinates in En. the
contact transformations . contact structure .
contact structure is by the linear differential defined form n 1 n-l dx -p dx - . . . -Pn-ldx . Lie proved that the pseudo- group of all contact transformations carrying
(oriented)
hyperspheres
in
the
generalized
sense
(i.e .. including points and oriented hyperplanes) into hyperspheres is a Lie group. called the Lie sphere group . isomorphic to 0(n+l.2)/±I. whe re 0(n+l . 2) n 3 is the orthogonal group for an indefinite inner product on R + with signature (n+l . 2) .
The first
The Lie sphere group contains as a subgroup the Moebius group of
author was
s upported by NSJi Grant No . DMS 87-06015.
author by NSF Grant No . OMS 87-01609 .
the second
492 2
conformal transformations of En and , of course , the Euclidean group ,
Lie
exhibited a bijective correspondence between the set of oriented hyperspheres in En and the points on the quadric hypersurface Qn+1 in real projective space n+2 n+3 P given by the equation <x , x> 0, where is the inner product on R n+1 mentioned above . The manifold Q contains projective lines but no linear n+2 subspaces of P of higher dimension . The 1-parameter family of oriented spheres
corresponding
to
the
points
of
a
projective
consists of all oriented hyperspheres which are
line
ly i ng on Qn+1
in oriented contact at a
certain contact element on En. Thus, Lie constructed a local diffeomorphism .. n 2n-1 n+1 between PT E and the manifold A of projective lines which lie on Q . An imme rsed submanifold f : 14k -+ En naturally induces a Legendre sub2n 1 n 1 n 1 manifold A: B - -+ A - , where B - is the bundle of unit normal vectors to f n n 1 1 (take B - = M - in the case k n-1). This Legendre map A has similarities with the familiar Gauss map, and like the Gauss map, it can be a powerful tool in the study of submanifolds of Euclidean space .
In particular , the Dupin
property for hypersurfaces in En is easily formulated in terms of the Legendre map,
and
it
is
immediately
seen
to
be
invariant
under
Lie
sphere
transformations . The
study
of
Dupin
submanifolds
has both local and global aspects .
Thorbergsson [Th) showed that a Dupin hypersurface 14 with g distinct principal curvatures at each point must be taut. i . e .. every nondegenerate Euclidean 2 distance function Lp (x) = 1p- x 1 , pEEn, must have the minilRulR nUllber of critical
points
on M.
Tautness
was
shown
transformations in our earlier paper [CC) .
to
be
invariant
under
Lie
Using tautness and the work of
Munzner [Mu] , Thorbergsson was then able to conclude that the number g must be 1 . 2 , 3.4 or 6, as with an isoparametric hypersurface in the sphere Sn .
The
case g = 1 is . of course, handled by the well-known classification of umbilic hypersurfaces.
Compact Dupin hypersurfaces with g=2 and g=3 were classified
by Cecil and Ryan (see [CR. p . 168]) and Miyaoka [M1] respectively . recent
preprints,
classification Meanwhile,
of
Grove
Miyaoka
(142),
compact
Dupin
and
Halperin
(143)
has
aade
hypersurfaces [GH]
have
further
progress
In two on
the
in the cases g=4 and g=6 .
deterllined
several
important
topological invariants of compact Dupin hypersurfaces in the cases g=4 and g=6. In this paper. we study Dupin hypersurfaces in the setting of Lie sphere geoaetry using local techniques .
In Section 2 , we give a brief introduction
493 3
to Lie sphere geometry .
In Section 3, we introduce the basic differential
geometric notions: the Legendre map and the Dupin property .
The case of E3 is
handled in Section 4, where we handle the case of g=2 distinct focal points for En .
This was first done for n > 3 by Pinkall [P3) .
Our main contribution
lies in Section 5, where we treat the case E4 by the method of moving frames . This case was also studied by Pinkall [P2), but our treatment seems to be more direct and differs from his in several essential points .
It is our hope that
this method will provide a framework and give some direction for the study of Dupin hypersurfaces in En for n > 4 .
2 . Lie Sphere Geoaetry. We first present a brief outline of the main ideas in Lie ' ~ geometry of n This is given in more detail in Lie ' s original treatment [LS) ,
spheres in IR
in the book of Blaschke [B), and in our paper [eel . The basic construction in Lie sphere geometry associates each oriented n U {~} sn with a point on the
sphere, oriented plane and point sphere in IR quadric
Qn+1
in
projective
space
pn+2
given
in
homogeneous
coordinates
(xl " .. ,x + ) by the equation
n 3
(2.1 )
<X,X>
We will denote real
O.
+ . .• +
(n+3)-space endowed with the metric (2.1) of signature
(n+1,2) by 1R~+3 We can deSignate the orientation of a sphere in IR minus sign to its radi us .
n
by assigning a plus or
Positive radius corresponds
to the orientation
determined by the field of inward normals to the sphere, while a negative radius corresponds to the orientation determined by the outward normal . (See n A plane in IR is a sphere which goes through the point 00 .
Rellark 2 . 1 below) . The orientation of
t~
plane can be associated with a choice of unit norllal N.
The specific correspondence between the points of Qn+1 and the set of oriented n spheres, oriented planes and points in IR u {Go} is then given as follows :
EuclidRan Points: u E R~
2
P'P+r
Spheres: Center p, signed radius r
2
w: u-N
h, unit normal N
, p.r)
I
[(h.-h.N.l)I
+ ~ by Here the square brackets denote the point in projective space P ~ given the homogeneous coordinates in the round brackets, and u - u is the standard Euclidean dot product in R". From (2.2). we see that the point spheres correspond to the points in the n+2 with the hyperplane in P given by the equation
intersection of Q"+' 0.
Xn+3 A
The manifold of point spheres is called Moebius space.
fundamental notion in Lie sphere geometry is that of oriented contact
of spheres.
Two oriented spheres S1 and S2 are in d e n t e d contact if they If pl and
are tangent and their orientations agree at the point of tangency.
p2 are the respective centers of S1 and S2, and rl and r2 are the respective signed 'radii, then the condition of oriented contact can be
expressed
analytically by
If S1 and S2 are represented by [k 1] and [k2] as in (2.2). then (2.3) is equivalent to the condition
In the case where S1 and/or S2 is a plane or a point in Rn, oriented contact has the logical meaning.
That is, a sphere S and plane m are in
oriented contact if n is tangent to S and their orientations agree at the point of contact.
Two oriented planes are in oriented contact if their unit
normals are the same. They are in oriented contact at the point
m.
A point
sphere is in oriented contact with a sphere or plane S if it lies on S , andoo is in oriented contact with each plane.
In each case, the analytic condition
for oriented contact is equivalent to (2.4) when the two "spheres" in question are represented in Lie coordinates as in (2.2).
495 5
Remark 2 . 1 : In the case of a sphere equation (2 . 4) is equivalent to p·N
[k1l
and a
h+r .
plane
[k2l
as
in (2.2),
In order to make this correspond
to the geometric definition of oriented contact, one must adopt the convention that the inward normal orientation of a sphere corresponds to positive signed radius . radius,
To get one
[(h,-h,N,I)].
the outward normal
should
represent
orientation to correspond to posi ti ve
the
plane
Then (2.4) becomes p'N
by
[( -h, h, -N, 1)]
instead
of
h-r, which is the geometric formula
for oriented contact wi th the outward normal
orientation corresponding to
positive signed radius . Because of the signature of the metric (2.1) , the quadric Qn+l contains lines in pn+2 but no linear subspaces of higher dimension .
A line on Qn+l is
determined by two points [xl, [y] in Qn+l satisfying <x,y> O. The lines on Qn+l form a manifold of dimension 2n-l, to be denoted by A2n - 1 In R n , a line on Qn+l corresponds to a I-parameter family of oriented spheres such that any two of the spheres are in oriented contact, tangent
to an oriented plane at
i.e., all the oriented spheres
given point, i.e. ,
a
an
oriented
contact
element. Of course, a contact element can also be represented by an element of T Sn , the bundle of unit tangent vectors to the Euclidean sphere Sn in En +1 1 with its usual metric .
This
is
the
starting
point
for
Pinkall's
[P3]
considerations of Lie geometry . A
~ ~ transformation is a projective transformation of pn+2 which
takes Qn+l to itself.
Since a projective transformation takes lines to lines,
a Lie sphere transformation preserves oriented contact of spheres.
The group
G of Lie sphere transformations is isomorphic to O(n+l,2)/{±I}, where 0(n+l,2)
is
the
group
of
orthogonal
transformations
for
the
inner product
(2.1).
Moebius transformations are those Lie transformations which take point spheres to point spheres .
The group of Moebius
transforlllations
is
isomorphic to
O(n+l,1 )/(±I}. 3. Legendre Subaanifolds . Here we recall the concept of a Legendre submanifold of the contact manifold A2n - 1 (= A) using the notation of [ee]. In this section, the ranges of the indices are as follows : (3.1 )
1
3
~
A,B,e
~
n +
3,
i,j , k
~
n
1.
+
496 6
Instead of using an orthonormal frame for the metric defined by (2 . 1), it is useful to consider a
.L.iJ:~ ,
that is, an ordered set of vectors Y in A
1R~+3 satisfying
(3 . 2)
(3.15)
R
o
is
equivalent
to h
f·t,
while
Note that the condition the Legendre condition
is the same as the Euclidean condition
( ' df
o .
Thus, ( is a field of unit normals to the ia..ersion f on U.
Since f is an
imlllersion, we can choose the Lie frallle vectors Y3 ' .. . . 'Y +1 to satisfy n (3 . 16)
499 9
for tangent vector fields X ' . . . ,X + on U. 3 n 1
Then, we have
(3 . 17)
Now using (3 . 14) and (3 . 16), we compute (3 . 18)
!
4 Cfol 1
3
~7
2
-a-p ,
is a multiple of
5
Similarly, differentiation
~1'
pw~, yields the following analogue of (5.14),
(5 . 16)
and differentiation of ~~
(5 . 17)
=
PW~ yields
(c+p+u
(-dp
In each of the equations (5 . 14), must vanish .
(5 . 16),
(5 . 17)
both sides of the equation
From the vanishing of the left-hand sides of the equations, we
get the fundamental relationship, -a-p
(5 . 18)
a+r-s
c+p+u .
Furthermore, from the vanishing of the right-hand sides of the three equations (5 . 14),
(5.19)
(5 . 15) and (5 . 17), we can determine after some algebra that
dp + p
w~
q
w~
-
t
~~
The last equation shows the importance of
the
function p .
notation introduced in (5.8), we write (5 . 19) as (5.20)
where
(5 . 21)
D
. 1
b ,
are the "covariant derivatives" of p. Using the Maurer-Cartan equations , we can compute
Following the
Using (5.8) and (5.21). this can be rewritten as
The trick now is to express everything in terms of p and its successive covariant derivatives. We first derive a general form for these covariant derivatives.
Suppose
that a is a smooth function which satisfies a relation of the form
for some integer m .
(Note that (5.19) is such a relationship for p with m=l.)
By taking the exterior derivative of (5.23) and using (5.13) and (5.22) to express both sides in terms of the standard basis of two forms 9 A 9 1 2' 02A0
3
and9
A 01, one finds that the functions
01,02,03 satisfy
equations
of the form
where the coefficient functions o
aP
In particular, from relations on p
p : 1' 2' 3 p
equation
satisfy the commutation relations
(5.20).
we have the following commutation
519 29 We next take the exterior derivatives of the equations
(5 . 10)-(5 . 12) .
We
first differentiate the equation (5.27)
Col
7
4 7
On the one hand, frail the Maurer-Cartan equation (3 . 7) for dIooI , we have (by 4 not writing those terms which have already been shown to vanish), (5 . 28)
On the other hand, differentiation of the right-hand side of (5 . 27) yields
~
!
da
(5 . 29)
A
da A l- db
~
Iol
+ a
Col~
A
+
~~
a(Col~
+ db A
Iol~ + b(lol~
A
Col~
Iol
+ b
P Col~
Iol~
P
+ (al-u-c)p
Cool~)
A
~
A
Iol~
~~
Col~)
A
Iol~)
Equating (5.28) and (5 . 29) yields (da + 2a ColI 1-
(5 . 30)
Since b
(5 . 31)
+(db + 2b
P3'
Cool
1 l
Cool~
it follows from (5 . 19) and (5 . 24) that
db + 2b
Cool
1
1
5 3 By exallining the coefficient of ColI A Col
7
9
2
A 9
(5 . 31) , we get that (5 . 32)
A
P
33
p(c-a-·u).
Furthermore, the remaining terlls in (5 . 30) are
3
in equation (5.30) and using
520 30
(5 . 33)
(da
1
7
3
2a Col 1 - Col 2
+
5
4
2pb Col 7 - (pt + P31)Col I ) A ColI
+ terms involving
5
~l
and
3
~7
only . 4
Thus, the coefficient in parentheses must be a multiple of ColI' call it - 4
We can write this using (5 . 8) and (5.21) as
ac.> 1 .
(5.34)
In a
si~ilar
manner, if we differentiate
we obtain,
(5 . 35)
Thus, from the two equations in (5.10), we have obtained (5 . 32), (5.35).
In
completely
analogous
fashion,
we
can
(5.34) and
differentiate
the
two
equations in (5 . 11) to obtain (5 . 36)
Pll
(5.37)
dp + 2p ColI
7 Col 2
(5 . 38)
1 dr + 2r ColI
Col 2
J
p(s+r-p)
=
+ 2pp1 9 1 +
7
(-P
13
-PP )9 + 2 2
pe3
2pp 9 + r9 2 + (-P12 + PP )9 , 3 3 1 1
while differentiation of (5 . 12) yields (5 . 39)
p(p-r-s) ,
P22 + P33
(5 . 40)
ds
+
2s Col
1 1
s9
(5 . 41)
du
+
2u
1 1
(-P
In these
w
equations,
1
+ (P
23
the
-PP
31 1
)9
+PP
1
2
)9
+ (P
2
13
+ ( -P -PP
2
)9
21 2
+ PP
+ U9
coefficients a,c , p,r , s,u
3
)9
3
3
remain
undetermined .
However , by differentiating (5 . 18) and using the appropriate equations among those involving these quantities above, one can show that
521 31
(5.42)
a
-6PP1
c
6PP2
p
-6PP3
r
6PP2
s
-12pp1
u
I2pp3
From equations (5.32). (5.36). (5.39) and (5.18). we easily compute that (5 . 43)
Using (5.42). equations (5.40) and (5.41) can be rewritten as (5 . 44)
ds
(5 . 45)
+
du
1
2s 1,)1 +
-12Pp 9 + (P + PP )9 + (-P + PP )9 1 1 31 2 2 3 3 21 1
2u 1,)1
(-P 23 -PPl)9
1
+ (P
I3
-PP2)9
2
+ 12PP393
By taking the exterior derivatives of these two equations and making use of (5.43)
and
derivatives.
of
the
commutation
relations
(5 . 25)
one ultimately can show after a
for
P and its various
lengthy calculation that
the
following fundamental equations hold:
PP 12
.+
2
P P + P P 1 2 3
2
PP21 + P 1P 2 - P P 3
2
PP23 + P~3 + P PI
(5 . 46)
PPn + P~3
2 P PI
2
PP 3I + Pf>1 + P P 2
2
PPI3 + Pf>1 - P P 2
0 0 0 0 0 0
We now briefly outline the details of this calculation . have (5.47)
The commutation relation (5 . 25) for s with m=2 gives (5.48)
By (5 . 44). we
522 32 On the other hand, we can directly compute by taking covariant derivatives of (5 . 47) that (5 . 49)
The main problem now is to get P311
into a usable form.
By taking the
covariant derivative of the third equation in (5 . 26), we find (5.50)
Then using the commutation relation
we get from (5 . 50) (5.51)
Taking
the
covariant
derivative
of Pll
=
p(s+r-p)
and
substituting
the
expression obtained for P113 into (5.51), we get
(5.52)
If we substitute (5 . 52)
for P
311
in (5.49) and then equate the right-hand
sides of (5.48) and (5.49), we obtain the first equation in cyclic permutations are obtained in a similar way from s23 Our
frame
attached
to
the
line
[VI' V ) 7
is
still
(5.46) .
S32 ' etc . not
completely
determined , viz . , the following change is allowable : (5.53)
The Vi's ,
The
3,4,5 being completely determined, we have under this change,
523 33
4· 5· 4 3· 5 3 1.>1 = <JI.)l' 1.>1 <JI.)l' 1.>7 = <JI.)7 , 7· -1 7 + ~4 ~ 0: Co) 4 Co) 4 1 1· -1 1 3 I.> = a 1.>3 IN 7 3
which implies that a*
• p We choose 1.1 to make a •
p*.
a a
-2
-2
a + a
a
p
-1
-1
1.1 1.1
After dropping the asterisks, we have from
(5 . 18) that (5 . 54)
a
-P
= p
Now using the fact that a
2
, r
+ S
t
C
u.
p, we can subtract (5 . 37) from (5 . 34) and get that
(5 . 55)
We
are
finally
in
position
to
proceed
toward
the
main
results .
Ul timately , we show that the frame can be chosen so that the function P is constant, and the classification naturally splits into the two cases P = 0 and p
~
o.
The case
P"'O .
We now assume that the function p lemma is the key in this case . his
function
c
is
the
is never zero on B.
The following
This is Pinkall's Lemma [P2, p . 108). where
negative
of
our
function p .
Since p
fc.
0,
the
fundamental equations (5.46) allow one to express all of the second covariant derivatives Pap in terms of p and its first derivatives Pa '
This enables us
to give a somewhat simpler proof than Pinkall gave for the lemma . Le••a 5,1 : Suppose that
p never vanishes on B.
Then P1 - P
2
- P3 -
0 at
every point of B.
.fI:.!!2.L. First note that
i f the function
and the assumption that p
~
P3 vanishes identically, then (5 . 46)
0 imply that P1 and P2 also vanish identically .
524 34 We
now
cOllplete
everywhere .
the
proof
of
the
lemma by showing that P3 lIust vanish
This is accomplished by considering the expression s12
the commutation relations (5 . 25) , we have
By (5.46) and (5.47) , we see that
and so (5 . 56)
On the other hand , we can compute s12 directly from the equation sl Then using the expression for P12 obtained from (5 . 46), we get
(5.57)
Next we have from (5 . 47) ,
Using (5.46), we can write
and thus (5 . 58)
Then, we compute
Using (5 . 36) for Pll and (5 . 46) to get P
31
, this becomes
525 35
(5 . 59)
Now
equate
the
expression
(5 . 56)
for
s21
with
that
obtained
by
subtracting (5 . 59) from (5 . 57) to get
This can be rewritten as O - P (12p 2 3
(5 . 60)
+ 3 s+r-p
2p2 -2) 1P
Using the expressions in (5 . 54) for rand p, we see that 3s+r - p
45 + 4p2,
and so (5.60) can be written as (5 . 61)
Suppose that P3 " 0 at some point b neighborhood U of b . (5.62)
on U.
E
B.
Then P
3
does not yanish on some
By (5 . 61), we have
o
16p2 + 4s
We now take the 8 - covariant derivative of (5 . 62) and obtain 2
o
(5. 63)
We now substitute the expre ss ion (5 . 58) for s2 and the formula
obtained from (5 . 46) into (5.63).
Since p " 0 , this implies that P2
After some algebra, (5.63) reduces to
o on U.
But then the left side of the
526 36 equation (5 . 46)
must vanish on U.
Since p
to our assumption .
~
o
0 , we conclude that P3
on U, a contradiction
Hence , P3 must vanish identically on B and the lemma is
proven . We now continue with the case P '" covariant
derivatives
of p
are
zero ,
o.
According to Lemma 5 . I, all the
and our
formulas
simplify greatly.
Equations (5 . 32) and (5 . 36) give c-a-u
o ,
s+r-p
o .
These combined with (5 . 54) give c = r
(5 . 64) 7
By (5.55) we have w 2
O.
-s
So the differentials of t he frame vectors can now
be written
5
w1Y 1 1 wI Y 1 7 + 1 YS
From this we see that the following 4-dimensional subs paces , Span(Y l 'Y4 'YS ,Z4'ZS} (5 . 107 )
Span(Y 7 'Y3 'YS ,Z3,Z5} Span{Y1+Y7'Y3'Y4,Z3,Z4}
are invariant under exterior differentiation and are thus constant.
Thus,
each of
in
the
three
focal
point maps Y , Y and Y +Y is 1 7 1 7
contained
a
4-dimensional subspace of p6, and our Dupin submanifold is reducible in three different ways . the
space of
Each of the three focal point maps is thus an immersion of
leaves
of
its
principal
cyclide of Dupin in a space x 3
foliation onto an open subset of a
p4 n QS .
We state this result due to Pinkall [P2] as follows : Theorem 5 . 3: Every Dupin subllanifold with p obtained from a cyclide
in R3 by one of
0 is reducible . the
four
Thus,
it is
standard constructions
(5 .9 7) . Pinkall [P2, p. 111) then proceeds to classify Dupin submanifolds with p
0 up to Lie equivalence.
ca n follow his
proof
using
We will not prove his result her e. the
fact
that his
constant s a
The reader
and fJ are our
co ns tants sand -u, respectively.
REFERENCES [B]
W. Bl asc hke , Vorlesungen uber Differentlalgeometrie , Vol . 3, Springer,
[eC]
T. Cecil and S . S . Chern , Tautness and Lie sphere geometry, Math. Ann.
Berlin, 1929. 278 (1987), 381-399.
538 48 [CR]
T. Cecil and P
Ryan. Tight and taut imMersions of manifolds. Res. Notes
Math. 107. Pitman. London. 1985. [E)
L. Eisenhart. A treatise on the differential geometry of curves and
[GH]
K. Grove and S. Halperin. Dupin b¥persurfaces
surfaces. Ginn. Boston. 1909. group actions and the
double mappings cylinder. J . Differential Geometry 26 (1987). 429-459. [L]
E.P . Lane. A treatise on projective differential geometry. U. Chicago
[LS]
S. Lie and G. Scheffers. Geometrie der Beruhrungstransformationen.
[M1]
R. Miyaoka. Compact Dupin hypersurfaces with three principal curvatures.
Press. Chicago. 1942 . Teubner. Leipzig. 1896 . Math. Z. 187 (1984). 433-452. [M2]
____ . Dupin hypersurfaces with four principal curvatures. Preprint. Tokyo Institute of Technology.
[M3]
____ . Dupin hypersurfaces with six principal curvatures. Preprint. Tokyo Institute of Technology.
[Muj
H.F. MUnzner . Isoparametrische
Hyperflachen in Spharen
and II. Math .
Ann . 251 (1980). 57-71 and 256 (1981). 215-232 . [N]
K. Nomizu. Characteristic roots and vectors of a differentiable family of symmetric matrices. Lin. and Multilin. Alg. 2 (1973). 159-162 .
[P1]
U. Pinkall. DUDin'sche Hyperfl&chen. Dissertation. Univ . Freiburg. 1981.
[P2j
Dupin'sche Hyperfl&chen in E4. Manuscr .
[P3j
Dupin hypersurfaces. Math. Ann . 270 (1985). 427-440 .
[5]
Math 51 (1985). 89-119.
D. Singley. Smoothness theorems for the principal curvatures and principal vectors of a hypersurface. Rocky Mountain J . Math .. 5 (1975). 135-144.
[Thj
G. Thorbergsson. Dupin hypersurfaces. Bull . Lond. Math. Soc. 15 (1983). 493-498.
Thomas E. Cecil Department of Mathematics College of the Holy Cross Worcester. MA 01610
Shiing-Shen Chern Department of Mathematics University of California Berkeley. CA 94720 and Mathematical Sciences Research Institute 1000 Centennial Drive Berkeley . CA 94720
539
Historical Remarks on Gauss-Bonnet Shing-Shen Chern * Mathematics Department University of California Berkeley, California
Dedicated to Jiirgen Moser Let At be a two-dimensional oriented Riemannian manifold and D a compact domain on 111 bounded by a sectionally smooth curve C. The Gauss-Bonnet formula says
2::(11" - a) +
1
kgds
+
JL
I M be the bundle of frames of the complex vector bundle. Then the pull-back 7T'·C a becomes a derived form, i.e., ( 16) where Tc a , a form of degree 20' - 1 in P, is uniquely determined by certain properties. This operation is called transgression and TC a have been called the Chern-Simons forms (5). These forms have played a role in three-dimensional topology and in recent works of E. Witten on quantum field theory (20) . This theory can be developed for any fiber bundle; see (3). The above provides the geometrical basis of gauge field theory in physics. Here M is a four-dimensional Lorentzian manifold, so that the Hodge '" -operator is defined, and we define the codifferential
( 17) There is a discrepancy of terminology and notation, as given by the following table: mathematics connection w cUlvature n
physics gauge potential A strength F
554 1990)
WHAT IS GEOMETRY?
685
Maxwell's theory is based on a U (I)-bundle over 'M, and his field equations can be written
dA =F,
of=J,
(18)
where J is the current vector. Actually, Maxwell wrote the first equation as
dF
=
0,
(19)
which is a consequence. For most applications (19) is sufficient. But a critical study of an experiment proposed by Boehm and Aharanov and performed by Chambers shows that (18) are the correct equations [21). A generalization of (18) to an SU(2) bundle over M gives the Yang-Mills equations
DA =F,
of=J.
(20)
It is indeed remarkable that developments in geometry have been consistently parallel to those in physics.
7. An application to biology. So far the most far-reaching applications of geometry are to physics, from which it is indeed inseparable. I wish to mention an application to biology, namely, to the structures of DNA molecules. This is known to be a "double helix", which geometrically means a pair of closed curves. Their geometrical invariants will clearly be of significance in biology. The following three are most important: I) The linking number introduced by Gauss; 2) the total twist, which is essentially the integral of the torsion; 3) the writhing number. James White proved that between these invariants there is the relation [18)
Lk
=
Tw
+ Wr.
(21)
This formula is of fundamental importance in molecular biology. 8. Conclusion. Contemporary geometry is thus a far cry from Euclid. To summarize, I would like to consider the following as the major developments in the history of geometry: I) 2) 3) 4) 5) 6)
Axioms (Euclid); Coordinates (Descartes, Fermat); Calculus (Newton, Leibniz); Groups (Klein, Lie); Manifolds (Riemann); Fiber bundles (Elie Cartan, Whitney).
A property is geometric, if it does not deal directly with numbl!rs or if it happens on a manifold, where the coordinates themselves have no meaning. Going to several variables, algebra and analysis have a tendency to be involved with geometry. This story is clearly one-sided and incomplete, representing only my personal viewpoint, and my limitations. It is clear that the story will not end here. Recent developments in theoretical physics, such as geometric quantum field theory, string theory, etc, are pushing for a much more general definition of geometry [19). It is sa tisfying to note that so far almost all the sophisticated notions introduced in geometry have been found useful. Finally, I wish to call attention to an early paper of mine [2), which could be read as a companion to this one.
555
686
[October
SHIING-SHEN CHERN
*)01 Jj} -y-j~ ~)I. ,~ff.ft ~f'?o- ~
:t
1* tt
~ -r t, /.7 ~ JJ.&..ll.. ~
MI~ HE .$..~~ ,.!1l ~
~~3t~~AJF1
:J
-t6
->+ _
1 t-
......... '''':1-.
.!HE.--§-
*- l::iK. ""
0
REFERENCES I.
2. 3. 4. 5.
6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
16. 17 18. 19. 20. 21.
G. Birkhoff and M. K. Bennett, Felix Klein and His "Erlanger Programm" , History and Philosophy of Modern Mathematics (W. Aspray and P. Kitcher, editors), Univ. of Minn . Press, 1988, 145-176. S. Chern, From triangles to manifolds, this Monthly, 86 (1979) 339-349. S. Chern, Complex Manifolds without Potential Theory, 2nd edition, Springer 1979. S. Chern, Vector bundles with a connection, Studies in Global Differential Geometry, Math. Asso. Amer. Studies no. 27, (1989) 1-26. S. Chern and J. Simons, Some cohomology classes in principal fiber bundles and their application to Riemannian geometry, Proc. Nat. Acad. Sci. USA, 68 (1971),791-794; or, characteristic forms and geometrical invariants, Annals of Math , 99 (1974) 48-69. Rene Descartes, Discours de la methode pour bien conduire sa raison et chercher la verite dans les sciences, 1637. Pierre de Fermat, Oeuvres, edited by Paul Tannery and Charles Henry, Gauthier-Villars, Paris, 1891-1912. C. F. Gauss, Disquisitiones generales circa superficies curvas, 1827; Ges. Werke, 4. F. Hausdorff, Grundziige der Mengenlehre ,. Leipzig 1914; dritte Auflage, Dover, N.Y. 1944; English translation , Chelsea, N.Y. 1957. D. Hilbert, Uber die Grundlagen der Geometrie, Gottinger Nachrichten , 1902, 233-241. F. Klein, Vergleichende Betrachtungen iiber neuere geometrische Forschungen, Math. Annalen 43 (1893), 63- 100 or Ges. Abh I (1921), 460-497. ___ , Vorlesungen iiber nicht-euklidische Geometrie, Springer. 1928. ___ , Hohere Geometrie, Springer, 1926, p. 314. B. Riemann, Uber die Hypothesen welche der Geometrie zu Grunde liegen , Habilitationschrift 1854; Gott Abh 13, 1868; Ges. Werke 1892. Clifford H. Taubes, Morse theory and monopoles; topology in longe range forces, Progress in Gauge Field Theory, Cargese 1983,563-587, NATO Adv. Sci. Inst, Ser B, physics 115, Plenum New York-London, 1984. H. Weyl, Die Idee der Riemannschen Flache, Leipzig, 1913; 3 te Auflage, verandert , Leipzig, 1955. ___ , Riemanns geometrische Ideen , ihre Auswirkung und ihre Verkniipfung mit der Gruppentheorie, Springer, 1988. James H. White, Self-linking and the Gauss integral in higher dimensions, Amer. J. Math., 91 (1969) 693-728. E. Witten, Physics and geometry, Proc. Int. Congo of Math . Berkeley 1986, Amer. Math . Soc., 1987, Vol. 1,267-303. ___ , Quantum field theory and the Jones polynomial, Braid Group, Knot Group, and Statistical Mechanics, (c. N. Yang and M. L. Ke editors), World Scientific, 1989, 239-329. C. N. Yang, Magnetic monopoles, fiber bundles, and gauge fields. Annals of New York Academy of Sciences, 294 (1977), 86-97.
556 Reprinted from Differential Geometry, Longman, 1991.
AN INTRODUCTION TO DUPIN SUBMANIFOLDS Shiing-Shen Chern * To Manfredo on his 60th Birthday
1. Dupin submanifolds and Lie sphere geometry Consider a piece of surface immersed in R3:
(1) Its normal lines are the common tangent lines of two surfaces, the focal surfaces . We have the theorem:
If the focal surfaces degenerate to curves, they are conics.
The surfaces in question are called the Dupin cyclides. The simplest example is given by a torus obtained by rotating a circle about a line not meeting it . Bllt there are other interesting examples, some with singularities. The cyclides were studied by Dupin in his famous book published in 1822. He defined them as follows: Consider in R3 three fixed spheres and the family of spheres which are tangent to all of them. Their envelope is a cyclide. The equivalence of the two definitions follows from the Lie line-sphere contact transformation . The corresponding problem in R4 was studied by U. Pinkall in 1985. An immersed hypersurface
(2) is called a Dupin hypersurface, if the three focal hypersurfaces all degenerate to two-dimensional surfaces. A remarkable situation arises: Besides the general case which generalizes the classical result in R3 there is an exceptional case where the three focal surfaces merge into the same Veronese surface. The classification of Dupin hypersurfaces in Rn with n 2: 5 remains an open problem. When the hypersurfaces is analytically given as a graph, the lowering • Work done under partial support of National Science Foundation grant DMS-87-01609 .
557 96
in dimension of the n - 1 focal hypersurfaces is given by n - 1 nonlinear partial differential equations, so that we are involved with an over-determined system. To understand the problem we make the key observation that the Dupin property is invariant under the Lie sphere group, which contains as a subgroup the group of euclidean motions. Consider the projectivized cotangent bundle
(3) where the left-hand side stands for the non-zero one-forms
0,
with
>. i- 0, identified. As local coordinates in PT" R n we take (Xl, ... ,xn , Xi,
1:< o. We can call " points" of R"
Let A be a point of pn+2 satisfying the hyperspheres X satisfying
(7)
< A, X > =
o.
The resulting geometry is Mobius geometry whose group, the Mobius group, is the subgroup of the Lie sphere-group O(n
+ 1, 2)/ ±
I, leaving the point A fixed .
Similarly, let C be a fixed point on Qn+l .
The hyperspheres X satisfying
< C, X > = 0 can be regarded as the hyperplanes. The subgroup of the Lie sphere-group leaving the point C fixed is called the Laguerre group. In Laguerre sphere-geometry there are hyperplanes , but no points. The euclidean group is the intersection of the Mobius group and the Laguerre group. Consider now the diagram
B"-l (= B)
(8)
11"
1
Mk -------t
R" ,
x
where Mk is immersed in R" and B"-l is its unit normal bundle . By definition, a point b E B is a unit normal vector at x
= 1I"(b) .
Let b.L be the oriented tangent
hyperplane perpendicular to it . The oriented hyperspheres tangent to b.L at x form a one-parameter family having the property that any two of them are tangent to each other. This defines a line on Qn+l, which we denote by >.(b) . We have therefore the mapping
(9)
>.:
B
--+
A2n-1 ,
where A2"-I, of dimension 2n - 1, is the manifold of all lines of Q"+l. We will call
A the Legendre map. Let b(t) be a curve on B. Then A(b(t)) is a ruled surface formed by lines of Q"+l. In order that it be a developable surface, i.e. , a ruled surface formed by
the tangent lines of a curve, the tangents of b(t) must be in principal directions . In analogy to surface theory we will call such a curve a line of curvature . The
SS9 98 submanifold Mk is called a Dupin submamfold if the lines ),(b(t)) pass through a point for any line of curvature b(t). This definition reduces to the classical one for a hypersurface and is clearly invariant under Lie sphere-transformations. An important class of examples of Dupin submanifolds is given by the extrinsic symmetric submanifolds Mk in Rn, which satisfy the equation
(10)
D II
= 0,
where II is the second fundamental form, a quadratic differential form with value in the normal bundle, and D is the covariant differential. These were completely determined by D. Ferus; cf. [6]. A local Euclidean characterization of the Dupin property seems to be complicated.
2. Taut imbedding Dupin submanifolds are closely related to the global problem of taut imbedding of manifolds in Rn. Given a compact manifold Mk, the question is how can it be best immersed in Rn ? The immersion is called tight if every height function has the minimum number of critical points. In 1957 Lashof and I proved that the k-sphere Sk is tightly immersed if and only if it is imbedded as a convex hypersurface in Rk+l
In 1970, T. Banchoff considered immersion of a compact manifold Mk in Rn, where every distance function from a fixed point has the minimum number of critical points. Such an immersion has since been called taut. It is a stronger property than tightnes: a taut immersion is necessarily tight. Carter and West proved that it is always an imbedding. Banchoff proved that a taut surface in R3 must be a round-sphere or a Dupin cyclide. Tautness can be defined for general subsets of R" by a homological condition. It has been proved that a taut submanifold remains taut under a Lie sphere-
transformation. All indications are that the taut submanifolds in Lie sphere geometry are characterized locally by the Dupin property. We refer to [4], [5], [7] for information on taut imbeddings.
560 99
3. A generalization of Lie sphere geometry and an equivalence problem The Lie sphere geometry can be generalized to the study of
oo"+!
hypersurfaces in
R" under contact transformations. We will formulate it a.s an equivalence problem: Cartan's equivalence problem is the following: Given two sets of linear differential forms 0', Xk,
0'; in the coordinates
l
x· respectively, 1 :S i,j,k,l :S n, both linearly independent, and
given a Lie group G
c
G L( n, R). To find the conditions that there are
functions
such that 0'; , after the substitution of these functions, differ from Oi by a transformation of G .
The problem generally involves local invariants, and Cartan gave a procedure to generate such invariants. In R" consider
oon+!
hypersurfaces defined by
(12)
Xn
= F( X 1 , ••• ,x"-1 ,
POt
= 8x Ot (x,a),
We have
(13)
8F
These equations can be solved for aA,
(14) Geometrically we have in
l:Sa:Sn-l.
1:S A :S
n
+ 1,
with a parameter t:
1 :S i :S n. R2n
two sets of coordinates, x Ot , aA and
tively, related by the above equations. The submanifolds aA
=
Xi,
POt, t respec-
const
define a
leaved structure, of dimension n - 1, in R 2n, which is also defined by a completely integrable system
561 100
dx" - EPadxa
(15)
=0
i
dpa - EPa(3 ( x , P"1't)dx dt - Et a (x 'P"1,t)dx a i
where Pa(3, ta are functions of
Xi,
(3
=0
1
~
0:,/3,[
~
n - 1
= 0,
Pa, t. For a function in these 2n variables we
define
(16) Then the complete integrability is expressed by the conditions
(17)
Pa(3
!!:!La
= P(3a,
dx
To an element of contact (xi, p) there are the parameter t. Thus a point of R
2
n
00 1
leaves tangent to it, depending on
is a leaf of the foliation (15) together with
a point on the leaf or a point of PT'Rn and a leaf tangent to the coresponding element of contact. As an example consider the hyperspheres of Rn:
(18) We have
(19)
We can set
The completely integrable differential system defining the hyperspheres is
562 101
(20)
dp'"
1 + -E(6"'i3 + p",pp ) dx p_ - 0, t
In the general case let (21) tw being the transpose of the one-columned matrix w. It is seen that w is defined
up to the transformation
(22)
1 :; a., {3 :; n - 1.
Thus the determination of the local invariants of the family (12) under contact transformations of R n is reduced the solution of the equivalence problem in R 2n where the group G is the group of all the (2n x 2n) matrices in (22). This, together with a new proof of Lie's theorem, will be presented in a forthcoming paper.
References [1
I Steven G.
Buyske, "Lie sphere transformations and the focal sets of hyper-
surfaces", thesis, Brown University, 1988. [2
I T.
Cecil and S. Chern, "Tautness and Lie sphere geometry" , Math. Annalen
278 (1987) 381-399. [3
IT.
Cecil and S. Chern, "Dupin submanifolds in Lie sphere geometry", Differ-
ential Geometry and Topology, Lecture Notes in Math . 1369, Springer-Verlag,
1989, 1-48. [4
I T.
Cecil and P.
Ryan, Tight and Taut Immersions of Manifolds (Pitman,
London, 1985) . [5
I Eugene
Curtin,
"Intermediate tautness and relative tautness for submani-
folds" , thesis, Brown University, 1988.
563 102 [6
I D.
Ferus, "Symmetric submanifolds of Euclidean space", Math. Annalen 247
(1980),81-93 . [7
I U.
[8
I G.
Pinkall, "Dupin hypersurfces", Math. Annalen 270 (1985), 427-440 . Thorbergsson,
"Dupin hypersurfaces", Bull. London Math.
(1983), 493-498.
Department of Mathematics University of California and Mathematical Sciences Research Institute Berkeley, California 94720
Soc . 15
S64 Reprinted from International Symposium in Memory of Hua Loo Keng, Springer Verlag, 1991.
Families of Hypersurfaces Under Contact Transformations in Rn IN MEMORY OF Loo-KENG HUA
SHIING-SHEN CHERN*
1. Introduction.
We consider the cotangent bundle T* Rn of Rn, which consists of the linear differential form 8 of Rn . By identifying the non-zero one-forms differing from each other by a factor, we get the projectivized cotangent bundle PT' Rn. If xi, 1 :S i :S n, are coordinates of Rn, we write
(1)
8=dxn-
2: p",dx'" ,
1
:S a :S n -1,
Then (xi, p",) can be taken as local coordinates in PT* Rn . A local diffeomorphism of PT" Rn, which preserves the equation
(2) is called a contact tran3formation. A submanifold of PT" Rn satisfying (2) is called a Legendre submanifold. Consider a family of hypersurfaces depending (smoothly) on n + 1/ parameters:
(3)
1/
2 o.
A hypersurface lifts to a Legendre submanifold in PT" Rn in an obvious way. The problem of the local invariants of such a family under contact transformations is a complicated one. The simplest case is of course when 1/ = O. In fact, we have shown in [2] that in this case there are no local invariants, i.e., the family is equivalent to the family of all hyperplanes in Rn. This will be given a new proof in this paper, and we will give in §3 the structure equations of the subgroup of the pseudo-group of contact transformations leaving invariant the hyperplanes. The case 1/ = 1 is also of importance, because the family (3) has as d. special case the family of all the hyperspheres of Rn, which is the geometrical structure leading to the Lie sphere geometry. In §4 we shall exhibit the first local invariant. "Work done under partial support of NSF Grant No. DMS87-01609.
565 50
Shiing·shen Chern
2. Formulation as a problem of equivalence. From (3) we get
po = -8f 8 (;,; 1 , ... ,;,; n-l ,al,·· · ,a n+ v ) · ;,;0
(4)
Equations (3) and (4) can be solved for
at, .•• , Un+v
in terms of v parameters giving
The hypersurfaces (3) in Rn can be considered as the integral manifolds of the differential system
(6)
en:
= d;,;"
11"0:
= dpo - LPO/l(;,;i,p_pt:..)d;,;/l = 0,
T:..:
- LPod;,;o
= dt:.. -
= 0,
Lt:..o(;,;i,p-y,tl')d;,;O
= o.
From here on in this section we will agree on the following ranges of indices:
(7)
1
:S a,{3,"(:S n -1, 1 :S i,j,k:S n, 1 :S >',I1-:S v.
The system (6) is completely integrable. Let E be the space of the variables ;,;i,Po,t:.., which is of dimension 2n - 1 + v. The differential system (6) defines a foliation in E, with leaves of dimension n - 1 given by aA = const, corresponding to the hypersurfaces of the family (3). Fixing;,; i, Po, we fix an element of contact in Rn and there are OOv hypersurfaces of (3) tangent to it, parametrized by t:... Let F be a function in E. We define
(8) Then the complete integrability of (6) is expressed by the conditions
(9)
Poll = P/lo,
The forms in (6), together with the forms d;,;o, give rise to a G-structure, where G is the group of the non-singular matrices
o (10)
up o o
~ ).
g~
We take the elements of M as auxiliary variables. In fact, in the space of ;,;i ,po, t:.. and the variables in M we introduce the one-forms
566 Families of Hypersunaces Under Contact Transformations in R"
51
(11) where the t's denote the transposes of the row-matrices. Suppose there be a second family of hypersurfaces of the same kind in R". We denote the corresponding quantities with the same notations with asterisks. Then the two families are equivalent under a contact transformation if and only if there is a diffeomorphism in the space of all the variables under which
(12) i.e., the corresponding forms are equal. Thus a condition among these forms is an invariant condition. We find, for example, mod w". Comparing with (11), it follows that we can have mod w",
(13) if and only if
(14)
These are thus invariant conditions on the auxiliary variables, and it is natural to impose them.
3. A notable pseudo-group of contact transformations. We consider the case
1/
= O. An important special case is the family of hyperplanes
(15) We shall show that this is no restriction vis-a.-vis the pseudo-group of contact transformations. In fact, taking the exterior derivative of (11), we see that we can write
(16)
dw" =
7r /\
d"P", =
L 7r~
w"
+ /\
LW'" /\ "P""
"P",
+ 7r", /\ w",
567 52
Shiing- shen Chern
where (17) are one-forms, which are not completely determined_ Because of (14) there are linear relations between them_ The easiest way to get these relations is to take the exterior derivative of the first equation of (16), obtaining (18) It follows that
where
On the other hand, the forms in (17) are subject to the change
7rp -+ 7rp + L P;"'cp.., + p$wn, P;'" = PJ"', p'{J
-+
p'{J
+ L R'{J..,w'" + LRp"'cp.., + R'{Jwn,R'{J.., =
R~p _
We can therefore utilize this indeterminancy to arrive at (19) Equation (18) then becomes
from which it follows that
where
d aP ..,
+ d Pa -r =
d aP
+ dPa
0,
= 0,
and, on substituting,
0l,{3,"( mutually distinct_ Keeping the condition (19), the allowable change of the forms in (17) is now given by
568 S3
Families of Hypersurfaces Under Contact Transformations in R"
11"fJo
+ Pw", p0"'/,n + POw" -+ ~o ufJ + ~ ~ fJ T-r fJ '
Pp
-+
Pp - L
pofJ
-+
pofJ _ LP;fJw-r + LQofJ-r'P-r + QOfJ w "
11"
-t
11"
(20)
pp-r'P-r
PfJ°-r
=.
PfJ-r o ,
+ (-P; + DpP)W n
where
(21) We can choose pofJ to make (22) The change (20) keeping the conditions (22) satisfies the further conditions
(23) It is not difficult to prove, using the Cartan-Kahler theory of exterior differential systems, that the system
Wn
= W*n ,
'Po
* = 'Po'
w
Q
=w
*0'
is in involution. From this it follows that any family of hyper surfaces (3) with v = 0 is equivalent under contact transformations to the family of hyperplanes (15) . A different, and more elementary, proof of this theorem was given in [2]. This proof has, however, another consequence. It leads to the subgroup of the contact transformations leaving the hyperplanes invariant . In fact, equations (16) are the structure equations of this pseudo-group in the sense of Elie Cartanj cf. [1] .
4. An invariant for oon+1 hypersurfaces. We consider the case v = 1, so that in our notations the indices A, /-L will be dropped . We write the equation (11) explicitly as
Wo = uO(dx" - LPpdxfJ) (24)
+L
updxfJ
+L
rofJ(dpp - LPlhdx-r),
fJ 'Po = fo(dx" - LPpdx ) + Lf!(dPfJ - LPp"dx"), 1/J = s(dxn - LPodx O) + L
where the subscripts have the ranges
hO(dpo - LPafJdxfJ)
+ g(dt -
L
todxO),
569 54
Shiing- shen Chern
(25)
1::; 0.,(3,"(::; n -1,
and Pat], ta are functions of
xi ,Pa, t_
1::; i,j::; n,
The other coefficients are auxiliary variables subject
to the conditions (14)_ As a consequence we can write, as in §3, dw" =
(26) where
7r
7r
1\ w"
+L
w a 1\ 'Pa,
is defined up to an additive term in w" _ Our objective is to impose further
invariant conditions, if they are available_ In fact, we have mod wn,'PP_
It is therefore natural to make the genericity assumption
(27) and impose the invariant condition p 'L~ " 8p -y fPf-Y b a p =ug ap,
(28) or
(28a) As a result we can write
(29)
The difference of this equation from the second equation of (16) is the presence of its first term_ It turns out that this does not affect the discussion in §3_ As a result we have the same conclusion: We have
(30)
dw
a
=
LP~ I\w P + Lpa P I\'PP +pa I\w n ,
and P~
(31)
+ 7rp - bp7r =
0,
pap = pP,
where the forms are defined up to the change in (20), with the conditions (23). Since the system w
n
= 0, 'Pa = 0, 'I/J = 0
is completely integrable, we can write
570 Families of Hypersurfuces Under Contact Transformations in RD
(32)
dl/J =
u "w" + L U
O
"'Po
+U
55
"l/J.
Taking the exterior derivative of (29), we get mod wn,'P..,. It follows that
(33)
mod wn,'P.."l/J,
where p 0.., -
(34)
C
c'"op·
Comparing with (20) and (32), we see that c~.., are invariant under the change (20) and a possible change of u. Hence
c~.., = 0
(35)
is an invariant condition, which we will examine closely. The issue is whether it can be fulfilled by imposing conditions on the auxiliary variables or it represents a property of the 00"+1 hypersurfaces under contact transformations. This investigation needs the explicit expression of c~.." and hence those of the left members of (33). We take the exterior derivatives of the expressions in (24) and compare them with (26), (29), (30), (32). This is a lengthy calculation, but is greatly simplified by noticing that it suffices to calculate mod w",'P""l/J. The result is:
7r =: u-1du 7rp
=: u- 1
u- 1
L
f..,w\
L u~dfJ + u- fpw'" 1
u-
1
L
op..,u~h"w'"
Of"f!l0P,,!' 'I ~u;.. P ..,-,,--w + U -2 ~ uP;..
(36)
P~ =: u- 1
L du~fJ -
u- 1 fpw'" - opu- 1
+ u- 1 ~ o~ u"'h"w'" ~ ~.., " u = g-ldg
-
(37)
~ uOf"f!lOP,,!' W.., ~ ;.. p .., oP;"
+ u- 1 ~ fP otp w.., + u- 1 ~ uPo hOw'" ~ .., at ~ .., "'P •
These relations are taken mod w ", 'Pc" l/J. We express the condition (35) by
or using (31) , by
u- 2
L f'lw'"
S71 56
Shiing· shen Chern
We differentiate the equation (28a), obtaining, by (28a), (38) Substituting (36) into (37) and using (38), we can write (35) as
8",8U- (L u~hP - f-y 1
(39)
+ u- 1 {-(l,8
-
+L
a: f~)
8t
L u~h(7)8~ -
- u- 2 "'(+u"f17fl' L.J A,8 -y
(I" -
L u~h(7)8~}
p 17 + u,8f17fl')8 1' >. " -y 8PA
-
-
0
.
We consider this equation as a system of linear equations in f", h,8. For n = 2 there is only one equation. But for n 2: 3 there will be compatibility conditions. Since (35) is an invariant condition , these compatibility relations will be the first invariants of the family of hypersurfaces under contact transformations. These invariants deserve to be further investigated, with geometrical interpretations. Our simplest, and most important, example is the family of hyperspheres in R n , defined by
(40) We find
(xn _ an)p" = _(XU - a,,), (xn - an )p",8 =
-8",8 -
P"P,8 .
Setting
t
(41)
= x" - an,
the hyperspheres are the integral manifolds of the differential system
dx" - L
(42)
dp"
+c
p"dx 1
u
= 0,
L(8",8
+ Pop,8)dx,8
= 0,
dt - LP"dx'" = O.
REFERENCES L E. Cartan, Sur la .t",cture de. groupe. infini. de tranoforTnation., Annales Ecole Norm . Sup. 21(1904), 153-206; Oeuvres Completes, Partie II, Vol. 2, 571-624. 2. S. S. Chern, Projective geometry, contact tran.forTnation., and C R-.t",ctive., Archiv der Mathematik 38 (1982), 1-5.
572 Reprinted from Miscellanea Mathematica, Springer, 1991.
Shiing-Shen Chern
Surface Theory with Darboux and Bianchi
The treatises of Darboux (1842-1917) and Bianchi (1856-1928) on surface theory are among the great works in the mathematical literature. They are: G. Darboux, Theorie generale des surfaces, Tome 1 (1887), 2 (1888),3 (1894), 4 (1896), and later editions and reprints. L. Bianchi, Lezioni di Geometria DifJerenziale, Pisa 1894; German translation by Lukat, Lehrbuch der DifJerentialgeometrie, 1899. The subject is basically local surface theory. There are beautiful spots and I wish to guide you through some of them. Needless to say, the corresponding global questions deserve study. They are interesting and are usually difficult.
1.
ISOMETRY
Classically this is known as the form problem : Given (1)
ds 2 =Edu 2 +2Fdudv+Gdv 2 , d s' 2 = E' d u' 2 + 2 F' d u' d v' + G' d v' 2,
both positive definite, to decide whether there is a transformation u'=u'(u, v),
(2)
v' = v' (u, v),
such that after substitution (3)
The fundamental invariant is the Gaussian curvature K(u, v). They have to be equal at corresponding points : (4)
K(u , v)=K' (u', v').
If one is a constant, the other must be the same constant. The surface then admits a three-parameter group of isometries. It is the euclidean plane, the hyperbolic plane, or the elliptic plane, according as K = 0, < 0, or > O.
573
s.-s. Chern
60
In the general case the main tool consists of the Beltrami differential parameters. For a function F(u, v) on the surface the first Beltrami differential parameter 17 F is the square of the norm of its gradient. The second Beltrami differential parameter ,1 F is its Laplacian. If G(u, v) is another function, we have also the polarization V (F, G), so that V (F, F) = V F. Given two invariant functions (5)
ep(u, v) = ep'(u', v') I/I(u, v)=I/I'(u', v'),
it follows that (6)
Vep=V'ep',
VI/I=V'I/I',
V(ep, I/I)=V'(ep', 1/1').
If the functions ep, 1/1 are independent, so that it determines the transformation (2), equations (6) are necessary conditions for the isometry (3). They are also sufficient. For, by the definition of the differential parameters, we have, by taking ep, 1/1 as parameters, ds 2 = Vl/ldep2 -2V(ep, 1/1) depdl/l + V(ep) dl/l 2 (7) VepVI/I-(I7(ep,I/I»2 K being an invariant function, not a constant, we search the second invariant function from V K, ,1 K. Hence the problem is solved, when there are two independent functions among K, V K, ,1K. The remaining case is when V K, ,1K are functions of K, say
(8)
VK=f(K),
,1K=g(K).
For (3) to hold we must have (9)
V' K' = f(K'),
,1'
K' = g(K'),
with the same functions f, g. This condition is sufficient. For with K and another parameter 1/1 we can write 2 2 2 (10) f(K)ds =dK +eXP (2S ~~~~ dK)dl/l Such a ds 2 is isometric to that of a surface of revolution. We shall call it rotation-like. It admits a one-parameter group of isometries. We notice the gap phenomenon: A ds 2 is generally rigid. It may admit a one-parameter group of isometries (rotation-like surfaces) or a three-parameter group of isometries (K = const), but
574 Surface Theory with Darboux and Bianchi
61
not a two-parameter group. Such a property persists in high dimensions. It should be interesting to study the global problem of complete rotation-like surfaces. Is it always a surface of revolution?
2.
ApPLICABLE SURFACES
Classically two surfaces with the same ds 2 are called applicable; in fact, one is also called a deformation of the other. Applicable surfaces may not be congruent. Their investigation is clearly an interesting and important problem. Let the surface S be (11)
x(u, v)=(x (u, v), y(u, v), z(u, v)),
with a ds 2 given by (1). Then the metric d x 2 + dy2 = ds 2 -dz 2
has Gaussian curvature zero. Expressing this fact, we get a long partial differential equation in the unknown function z, whose leading term is (12)
Given a surface, to find another surface applicable to it thus becomes analytically the study of a Monge-Ampere equation. Its characteristics are the asymptotic curves of S. This fact is the basis of the following theorem: Let C be a curve on S. If C is not an asymptotic curve, a surface keeping C fixed and applicable to S must be S itself. If C is an asymptotic curve, there is an infinite number of surfaces through C and applicable to S. Mon>. generally, given a curve C on S and a curve C in space, one asks the question whether there is a surface S' through C and applicable to S such that C goes into C . For this to be true it is necessary that C and C have the same geodesic curvature at corresponding points. For C this is equal to P sin (), where p is the curvature of C (which we suppose to be ~ 0) and () is the angle between the principal normal of C and the surface normal of S; the same notation, with dashes, will be used for C. It follows that C must satisfy the condition p' ~ Ip sin ()I. It can be proved that if p' > Ip sin ()I there are exactly two applicable surfaces S'
575 s.-s.
62
Chern
through C . On the other hand, if P' = Ip sin 81, C will be an asymptotic curve of S'. By a theorem of Beltrami-Enneper, its torsion is equal to ± K, which is another condition to be fulfilled by C. When C is given on S, the curve C is then determined up to a rigid motion and there is an infinite number of surfaces S' applicable to S with C going to C . Bonnet, and others, studied applicable surfaces with further conditions imposed. Analytically such a problem leads to an overdetermined system of partial differential equations. Bonnet proved the theorem : An isometry between two non-ruled surfaces which maps a family of asymptotic curves of one surface into the asymptotic curves of another is a rigid motion.
V-
Two other problems of this nature are: (X) isometries preserving the lines of curvature ; fJ) isometries preserving the principal curvatures or the mean curva-
ture, as the Gaussian curvature is always preserved. The study of such problems leads to long calculations. A more interesting question is the study of a family of applicable surfaces with the above properties (cf. [1 , 2]). A family of 00 1 surfaces is called non-trivial if it is not the orbit of one of them by a one-parameter group of rigid motions. We have the theorem : A non-trivial family of applicable surfaces preserving the lines of curvature is a family of cylindrical molding surfaces. We recall that a cylindrical molding surface is constructed as follows : Take a cylinder and a tangent plane 1t to it. On 1t take a curve C. A cylindrical molding surface is the locus of C as 1t rolls over the cylinder. When the cylinder is a line, the molding surface becomes a surface of revolution. Concerning the property fJ) Bonnet observed that a surface of constant mean curvature can be deformed continuously in a nontrivial way. In general we have the theorem : There exist non-trivial families of applicable surfaces of non-constant mean curvature, depending on six constants, such that the mean curvature is preserved during the deformation. It should be remarked that the proofs of this theorem and the theorem in the last section involve the studies of the respective over-determined systems and their integrability conditions, which look complica~ed but lead to unexpected simple conclusions.
576 Surface Theory with Darboux and Bianchi
3.
63
W-SURFACES
A Weingarten surface or a W-surface S is one which satisfies a relation between the principal curvatures: (13) Such surfaces include the minimal surfaces, the surfaces of constant mean curvature, the spherical and pseudospherical surfaces, etc. The first properties come from the congruence of normals. The latter consists of the common tangent lines of the evolute or focal surfaces of S. In fact, let x(u, v) be a point on S and let v(u, v) be a unit normal vector at that point. Then the focal surface F;, i = 1, 2, is the locus of the point (14)
where r; = 11k; is a principal radius of curvature. Weingarten proved the remarkable theorem : If S is a W-surface, F; is applicable to a surface of revolution, whose form depends only on the relation (13). The converse of this is also true. The normals establish a map between the two focal surfaces by mapping one focal point to the other. A congruence is called a W-congruence if this map preserves the asymptotic curves on the two focal surfaces. We have the following theorem of Ribaucour : A surface is a W-surface if and only if its normal congruence is a W-congruence. Let K; be the Gaussian curvature of F; at y;, i= 1, 2. For a W-surface we have the formula of Halphen : (15)
Sophus Lie proved another characterization of the W-surfaces: A surface is a W-surface if and only if the quadratic differential form (16)
P =(v, dx, dv)
has Gaussian curvature zero. Since the equation P =0 defines the lines of curvature, it follows that on a W-surface these can be determined by quadratures. The same is true of the asymptotic curves. Such properties are of importance, but they have been neglected in modern works on differential geometry.
S77
s.-s. Chern
64
An example of a minimal surface is the catenoid. One of its focal surfaces is a surface of revolution obtained by rotating the evolute of the catenary. By Weingarten's theorem one of the focal surfaces of any minimal surface is applicable to it. Similarly, consider Beltrami's pseudosphere obtained by the rotation of a tractrix. Since the catenary is the evolute of the tractrix, it follows that one of the focal surfaces of a pseudospherical surface is applicable to a catenoid. It should be remarked that the relation (13) is essentially a partial differential equation, generally non-linear, in two independent variables. For instance, if the surface is given as a graph z = z(x, y), the condition for a minimal surface is (17)
and the condition K = - 1 becomes (18)
The latter is thus the equation of a pseudospherical surface. It can be put in a different form: It 1/1 denotes the angle between the asymptotic curves, there are asymptotic parameters u, v, such that (19)
I/Iuv=sin 1/1.
This is called the sine-Gordon equation. Thus the study of pseudospherical surfaces is equivalent to that of the sine-Gordon equation. The above are remarks on some of the important local properties of W-surfaces. Their global properties, particularly those of minimal surfaces, have recently been exhaustively studied. For the global study of general W-surfaces I wish to refer to the works of H. Hopf; cf. [4].
4.
W-CONGR UENCES
An important feature of euclidean geometry is the role played by the straight lines. Thus the study of a surface is intimately tied to that of its normal congruence. On the other hand, it is justified to study line congruences, i.e., a two-parameter family of lines, on their own right. The first fundamental paper on line congruences was written
578 Surface Theory with Darboux and Bianchi
65
in 1860 by E. Kummer, the great algebraic number theorist. Let the lines be given by a point x(u, v) and a direction ~ (u, v), the latter being a unit vector. Kummer based his study on the two quadratic differential forms (20)
I'=(d~, d~),
II'=(dx, d~).
(We use the dashes to distinguish them from the forms in surface theory. Actually ~(u, v) defines an analogue of the Gauss map and I' is a generalization of the third fundamental form in surface theory.) The line congruence is called isotropic, if the forms I' and I I' are proportional. The notion is a generalization of the sphere (or plane) in surface theory. It has the following geometric interpretation: If the corresponding points of two applicable surfaces have a constant distance, the lines joining them form an isotropic congruence. The line ..1. with the parameters u, v and a neighboring line (u + d u, v + d v) have a common perpendicular. Its foot as d u -+ 0, d v -+ 0 gives a point on ..1.. All such points lie on a segment of ..1., whose endpoints L 1 , L2 are called the limit points on ..1.. On the other hand, the equations u=u(t), v=v(t) define a ruled surface consisting of lines of the congruence. There are in general two directions when it becomes a developable, whose lines are the tangent lines of a curve. The points of contact give two points Fl, i = 1, 2, on ..1.. They are called the foci and their loci j at F;, i = 1,2. For a W-congruence we have (21) where d is the distance between the limit points. A W-congruence for which both distances Fl F2 and Ll L2 are constants is called pseudospherical. In this case iJ>;, i = 1, 2, is a pseudospherical surface of curvature - d - 2 The correspondence between iJ>j is called a Backlund transformation. An important family of W-congruences was constructed by Darboux as follows : Consider a surface of translation (22) The lines of intersection of the osculating planes of the generating curves form a W-congruence such that the generating curves correspond to the asymptotic curves of the focal surfaces. Moreover, if the generating curves have the constant torsions + wand - w respectively, the W-congruence is a normal congruence of a Wsurface satisfying the relation (23) Its focal surfaces are applicable to a paraboloid of revolution. These Weingarten-Darboux surfaces have many interesting properties. A W-congruence is a projective property. Its study is an important chapter in projective differential geometry.
5. TRANSFORMATION OF SURFACES As remarked above, most properties of surfaces are described by partial differential equations. It is interesting, and mysterious, that the same property could be defined by equations which are very different in appearance. For example, the pseudospherical surfaces with K = -1 can be characterized either by Monge-Ampere equation (18) or the sine-Gordon equation (19). It is thus of clear interest to study the transformations of surfaces which preserve certain geometrical properties. We shall give some examples: IX) BONNET'S TRANSFORMATION. Let S be a surface with the principal curvatures k j and the radii of principal curvatures rj = l lk j ,
580 Surface Theory with Darboux and Bianchi
67
i = 1, 2. The principal curvatures of its parallel surface Sh at a distance h are given by
(24)
k I-hk /
k ~ = --'I
i= 1,2.
From this Bonnet made the following observation : If S has a constant Gaussian curvature l/a 2 , its parallel surface at a distance ± a has a constant mean curvature 1/2a. If S has a constant mean curvature I / a, its parallel surfaces at the dista nces a and a/2 have respectively the constant mean curvature - l/a and the constant Gaussian curvature l/a 2 (Note : mean curvature =(k 1 +k 2)/2.) Thus surfaces of constant mean curvature 0 and surfaces of constant positive Gaussian curvature are in a sense equivalent problems.
+
*'
(3) a -TRANSFORM . When the surface S is oriented, it has a complex structure defined by the rotation of a tangent vector by 90° (multiplication by i !). This leads to the definition of the operators and 0, which are respectively the exterior differentiations with respect to the holomorphic and anti-holomorphic coordinates. In particular, ax, where x(u, v) is the position vector, is a vectorial form of bidegree (1 , 0). The ratios of its components define a new complex surface, called the a-transform of S. A theorem on minimal surfaces says that they can be characterized by the condition
a
(25)
aox=o,
i.e., the coordinate functions are harmonic. This is equivalent to saying that the a- transform is a holomorphic curve. The latter property is the main reason for the Weierstrass formulas of a minimal surface. The notion of a a-transform is playing an important role in the study of minimal surfaces in other spaces; cf. [3]. y) BACKLUND TRANSFORMATIONS . In 1883 Backlund proved the
remarkable theorem : Let Sand S' be the focal surfaces of a pseudospherical congruence (for which the distances between the foci and the limit points are both constant). Then Sand S' have the same Gaussian
581
s.-s. Chern
68 2
curvature _d- , where d is the distance between the limit points. The transformation so defined between Sand S' is called a Backlund transformation. Given a pseudospherical surface S, to construct S' it suffices to construct a vector field on S such that the tangent lines to S along the vector field form a pseudo spherical congruence. This leads to a completely integrable total differential equation whose sorution depends on the solution of a Riccati equation. Since a pseudospherical surface corresponds to a solution of the sine-Gordon equation (19), a Backlund transformation can be interpreted as transforming one solution of (19) into another. In this way new solutions of (19) are produced. The method plays an important role in the theory of solitons in mathematical physics. 0) LAPLACE TRANSFORM (=!= LAPLACE TRANSFORM IN HARMONIC ANALYSIS). A net of curves N on a surface is called conjugate, if at every point the tangent directions to the curves of the net separate harmonically the asymptotic directions. Taking the net to be the parametric net with parameters u, v, we have, by one of the Gauss equations (26)
A conjugate net has the following geometrical interpretation: Take a v-curve Cv • The tangent lines of the u-curves at the points of C v form a developable surface. On such a tangent line there is thus a point Xl where it is tangent to the edge of regression. Reversing the role of u, v, we get a point X _ 1 on the tangent line of the v-curve Cv. As u, v vary, Xl (u, v) and X -1 (u, v) generally describe surfaces, which are called the Laplace transforms of the net N . The remarkable fact is that the u- and v-curves also form conjugate nets on x_du,v) and x 1(u, v); we will denote them by N- 1 and N1 respectively. Moreover, the positive (resp. negative) Laplace transform of N-1 (resp. N 1) is N itself. Continuing this process, we get a Laplace sequence of conjugate nets (27)
such that each one is the Laplace transform of the one to the left and is the negative Laplace transform of the one to the right. A conjugate net is a projective property. For its treatment it is advantageous to use homogeneous coordinates in the three-
582 Surface Theory with Darboux and Bianchi
69
dimensional ambient space. The homogeneous coordinates of a surface x(u, v) satisfy an equation of the form (28)
xuu+ ax u+ bxu+ ex =0,
if and only if the parametric net is a conjugate net. Equation (28) is called a Laplace equation. To every conjugate net is associated a Laplace equation, and vice versa. If x(u, v) defines N, its Laplace transforms are given by (29)
To the Laplace sequence (27) corresponds a sequence of Laplace equations, and the solution of a Laplace equation reduces to the solution of one of the equations in the sequence. In particular, the last problem could become a simple one, when the corresponding surface degenerates to a curve.
REFERENCES
1. Bryant, R., Chern, S., Griffiths, P.A. (1990) : Exterior differential systems. Proc. of 1980 Beijing DD-Symposium (1980), 219-338 or Bryant-ChernGardner-Goldschmidt-Griffiths, Exterior Differential Systems, Springer 2. Chern, S. (1989) : Deformation of surfaces preserving principal curvatures. Differential geometry and complex analysis, volume in memory of H. Rauch. Springer, 1984, pp. 155- 163, or Chern, selected papers, vol. 4, pp. 95-103, Springer 3. Chern, S., Wolfson, 1. (1989) : Harmonic maps of the two-sphere into a complex Grassmann manifold, II. Annals of Mathematics 125 (1987) 301 - 335 or S. Chern, selected papers, vol. 4, p. 189- 223, Springer 4. Hopf, H. (1983) : Differential geometry in the large, part. II. Lecture notes in mathematics, vol. 1000. Springer
Work done under partial support of NSF Grant DMS-87-01609. Research at MSRI supported in part by NSF Grant DMS-8505550.
583
TRANSGRESSION IN ASSOCIATED BUNDLES In memory of Guang-Lei Wu
SHIING-SHEN CHERN Mathematical Sciences Research Institute Berkeley CA 94720 USA Received 5 August 1990
1.
Introduction and Review
One of the fundamental notions in the differential geometry of fiber bundles is that of transgression. It can be fonnulated as follows: (cf., for example [3]): Let (1)
n:P-+M
be a principal bundle over a manifold M with structure group G a Lie group. Denote by L(G) the Lie algebra of G. A connection in the bundle is given by an L(G)-valued one-form in P satisfying certain conditions. Its curvature form is an L( G)-valued 2-form defined by (2)
In tenns of a chart n- l (U) ::::: U x G, with the local coordinates (x, su), x has the expression
E
U, Su E G,
(3)
where Su is an L(G)-valued two-fonn in U, and ad(su) is the adjoint representation of L(G). The property (3) prompts us to consider (real or complex-valued) functions F(X I , . · . , X h ), Xl' . .. , X h E L(G), which satisfy thefollowing conditions: (1) It is linear in each of the arguments and remains unchanged when any two arguments are interchanged; (2) It is "invariant", i.e., aE
Research at MSRI supported in part by NSF Grant DMS-87-01609.
383
Internationat Journal of Mathematics Vol 2 No 4 (1991) 383- 393 tF.\ World Scientific Puhlishin2 Comnanv
G.
(4)
584 384
S. S. CHERN
We can then substitute for each Xi the curvature form, getting F($) :
= F($, . .. ,$) = F(0 u , . . ,0 u ),
(5)
which is then a form of degree 2h in M. This form F($) has remarkable properties. In fact, we deform qJ to the zeroconnection by setting qJ(t) = tqJ ,
(6)
Its curvature form is (7)
We have the fundamental formula 1d
h dt F($(t» = dF(qJ, $(t), .. . , $(t)) . Define the transgression operator TF(qJ)
=h
r
F(qJ, $(t), ... , $(t)) dt.
(8)
(9)
Then we have d(TF(qJ))
= F($) = n*F(0 u ).
(10)
In other words, the "characteristic form" F(0 u ) is pulled back by n* into a coboundary in the principal bundle P. An important special case is when G = GL(q ; C). Then L(G) can be identified with the space of all (q x q)-matrices and the curvature form $ can be considered as a matrix-valued two form. We define the invariant forms c j ($), 1 ~ i ~ q, by (11)
so that cj ($) is a 2i-form in M. By the above general result it can be transgressed in the principal bundle P. We wish to show in this paper that it can be transgressed in a "smaller" associated bundle. For this purpose we identify P with the space of all frames, i.e., ordered sets oflinearly independent vectors e I , ... , eq , e I /\ ... /\ eq #- 0, with the same origin x E M. There is a natural projection
585 385
TRANSGRESSION IN ASSOCIATED BUNDLES
A..: P -+ p., which sends the vectors {e I ' ... , eq } to {e., .. . , eq }. Its image is an associated bundle having as fiber the Stiefel manifold Yq.q-.+1 of ordered sets of q - s + 1 independent vectors of the q-dimensional vector space, so that P = Pl. We have the fiberings
n = n.
0
(12)
A.•.
In Sec. 2 we will prove the theorem: Theorem 1. Consider an hermitian complex vector bundle and an admissible connection. The form n:c. is a coboundary in p•. An analogus result holds for hermitian holomorphic bundles. In this case we have the operators a, and their combinations
a,
d
= a+
a,
d C = i(a - a),
(13)
with (14)
Suppose H(t)(e, e l ), when e, e l are vectors with the same origin x hermitian metrics. The formula
E
M, be a family of
(15)
defines an endomorphism in the fiber ofthe vector bundle. For an invariant polynomial F, Bott and I proved in [2] the formula
-
d
haaF(L(t), (t), .. . , (t» = dt F((t» .
(16)
This allows a "double transgression". For defining
fF(f{J)
=
h
f
F(L(t), (t), ... , (t» dt,
(17)
we will have
aafF(f{J)
= F((t), . .. , cI>(t»
"---v----' h-I
= F(d", cI>(t), ... , cI>(t» - (h - l)F(",q>(t)
1\
cI>(t) - cI>(t)
1\
q>(t), cI>(t), ... , cI>(t».
By (37) the last tenn is equal to (h - 1)F(",q>(t)
= F(q>(t)
1\
1\
cI>(t) - cI>(t)
"+,,
1\
1\
11' (t), cI>(t), ... , cI>(t»
q>(t),cI>(t), ... ,cI>(t».
Hence
dF(", cI>(t), ... , cI>(t» = F(d" - q>(t)
1\ " -
" 1\
q>(t), cI>(t), ... , cI>(t»
(38)
By (33), (34), and (21) we have d" - q>(t)
1\ " -
" 1\
q>(t) = d" - 11'(0)
1\ " -
-2tq>~
1\
" 1\
11'(0) - 2t"
q>t
= ( cI>f + (1 - 2t)q>i
1\
q>!
1\ "
cI>! + (1 + (1 -
{
2t)q>~ 2t)q>/t
1\ 1\
q>i) 11'1 . (39)
On the other hand, from (30) and (31) we have
By (39) and (40) we observe that the matrix in (39) is equal to dcI>(t)/dt. Hence if we put TF(q» = h
r
F('1,cI>" ... ,cI>,)dt,
(41)
we will have
dTF(q» = F(cI>(1» - F(cI>(O».
(42)
TRANSGRESSION IN ASSOCIATED BUNDLES
389
By construction the form TF(cp)is in Q. It is generated by daerentiais in M and by the forms cpt, cp; and their complex conjugates. By (29) they satisfy the relations
and can be expressed as linear combinations of differentialson P,. We claim that TF(cp) itself is in Ps. For this it sufficesto show that it is invariant under a change of the vectors em.But this follows from the invariant property of F. Theorem 1 follows by taking F = c,, in which case we have
We wish to rewrite our formula as follows:
where
Notice that we have used the fact that cp is an admissible connection of an hermitian metric on the bundle, as expressed by (29). 3. Holomorphic Bundles
Now suppose M be a complex manifold and our complex vector bundle be a holomorphic bundle. The notations in Sec. 2 will be used. In particular, an hermitian metric will be given by (24). There is a uniquely determined type (1,O)-connection given by
Its curvature is
whose exterior differentiation gives
the so-called Bianchi identity. For completeness of our treatment we wish to give a proof of (16). We now have a
590 390
S. S. CHERN
family of hennitian metrices h"j(t) and the formulas (47), (48) are valid with the parameter t. By definition (15) we have (50) By exterior differentiation we have, using (49) aF(L(t), (t), ... , (t»
= F(aL(t), (t), ... , (t».
The same argument used in the proof of (38) gives aaF(L(t), (t), ... , (t»
= F(aaL(t) -
. 'so Therefore, by Euler's theorem, we have (1.5a) Throughout this paper, Flli denotes ::. (higher order partial derivatives are denoted similarly) and, unless specified otherwise, is evaluated at the point (Xl, .. . , Xn; yl, ... , yn). Successive differentiation ::If (1.5a) with respect to y (as above, we abbreviate partial derivatives by subscripts) yields: (1.5b)
. . -- 0 YiF,lI'y'
,
(1.5c)
(l.5d) and so forth. Another consequence of (1.2) is that the length of a Curve is independent of its parametrization.
615 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
137
Let T M (resp. T· M) be the tangent (resp. cotangent) bundle over M, and T:r:M (resp. T; M) its fiber at x EM. T;r:M and T; M have dual bases {a~.} and {dxk}, so that k
(1.6)
0
k
(dx ) ( ~) = 6 i vX'
.
A vector of T;r:M can be written as y = yi a~. and we can use (Xi , yi ) as local coordinates in T M . Two such local coordinates (Xi, Yi) and (x· k , y. k) are related by the transformation
(1.7)
x .k
= X .k (X,1 ... , x n) ,
y .k
8
.k
= -x8 . yi x'
,
from which we have
(1.8) It follows that the I-form
(1.9) and the symmetric covariant 2-tensor . . (1.10) 9 = gijdx'®dx J :=
1
('2 F
2
..
.
.
)lhjdx'®dxJ = (FFlI' lIi+FlI,Fyj )dx'®dxJ
are intrinsically defined on TM. From (1.10) and (1.5), one finds that
(1. 11 a)
Yig ij = FFyi,
yiyjg"J'
= F2
and
(1.11b)
i Ogij 0 y--k= , oy
kogij y oyk
0
= .
Of importance to our treatment is the projectivized tangent bundle PTM obtained from TM by identifying its non-zero vectors which differ
616 138
DAVID BAO AND 5.5. CHERN
from each other by a multiplicative factor. Geometrically PT M is the manifold of the line elements of M. As its local coordinates we can take (xi, yi) where (yl, .. ., y") # (0, ... ,0) are now homogeneous coordinates. Thus, local calculations on PT M can be performed conveniently on T M using all the yi 's (i = 1, ... , n), provided that the result is invariant under the scaling y -+ AY, A # O. Since Fyi is homogeneous of degree zero in y, the form w in (1.9) lives in PT M. In classical calculus of variations it is called Hilbert's invariant integral; we will call it the Hilbert form. Similarly, the metric tensor 9 in (1.10) also lives in PTM . In fact, the same is true of other geometrical quantities such as the special connection and its curvature that we will subsequently derive from F, even though the function F itself is defined on TM but not on PTM. Note that we are led to PTM, as opposed to the sphere bundle (equivalently, T M without its zero section), by the stipulation that (1.2) be valid for all A, rather than for positive A 's only. We would like to briefly justify this stipulation. Let us define the distance d(P, Q) between two points P and Q on M to be the infimum of the lengths of curves which connect these points. If (1.2) were to fail for A :5 0, then d(P, P) would in general not be zero, and d(P, Q) would likely differ with d(Q, P). FUrthermore, we will in section 4 need the fact that the reverse of a geodesic a(t) with velocity T(t) is again a geodesic. As shown there, the proof reduces essentially to the following question: Is the value of the connection at the point (a, T) in T M the same as that at the point (a, -T) ? Of course, such is automatically true if the connection lives on PTM. A more detailed argument is given at the beginning of section 4. Let 11": PT M -+ M be the canonical projection map 11" (xi ,yi) = (xi) . Using 11", the tangent bundle T M pulls back to a n-dimensional vector bundle 1I"·TM over the (2n-l)-dimensional manifold PTM. We would like to describe the geometry of this vector bundle, using the method of moving frames. Over each point (x, y) of PT M, the fibre of 1I"·T M has a basis {8~'} and a metric 9 given by (1.10). The regularity assumption (which we previously alluded to) on F is then: (1.12)
9(:z:,y) (P,p)
:=
that is, the metric 9
(9ij)(:z:,I/)
pipi >
°
unless (PI, ... ,p") = (0, ...,0) ;
= 9(:z:,1/) is positive-definite.
We shall list below some
617 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
139
immediate consequences of (1.12); complete proofs can be found in Rund [20]. We begin with the Cauchy-Schwarz inequality (which follows from (1.12)) (1.13) where equality holds iff p and q are collinear. Double contract (1.10) with p; to the quantity (Fyi pi)2 we apply the first part of (LIla), (1.13) , then the second part of (LIla) , and find that (1.14)
(Fy ' y ,
kr,y)
pirJ ~ 0 , and
= 0 only if p and y are collinear ;
in other words, the matrix (Fy ' y ') is positive semi-definite with rank n - 1, and its I-dimensional null-space is spanned by y. Expanding F( x, p) in terms of F(x, y) up to second order, namely
o < € < 1, and applying (1.14), we see that for
each fixed x, the graph of
F(x, y) is a convex function of the variable y : (1.15a) Simplifying the left hand side with Euler's theorem (1.5) gives the following fundamental inequality (1.15b) where equality holds iff p and y are collinear. Such an inequality will later play an important role in our proof that geodesics in a Finsler space minimize distance locally. In the meantime, we would like to point out that in the Riemannian case (1.3), where the metric 9 does not depend on y, (1.15b) reduces to 9(z)(P, y) :5 J9(z)(P,P)J9(z)(Y, y), which is 'half' of the Cauchy-Schwarz inequality (1.13). Finally, from (1.15b) one Can deduce the triangle inequality -
(1.16)
F(x,p+ q) :5 F(x,p) + F(x,q) .
618 140
DAVID BAO AND 5.5. CHERN
Again, equality holds iff p and q are collinear. Let {ei} be a local orthonormal (with respect to g) frame field for the vector bundle 7r'TM such that
=
(1.17)
yi 0 F(x, y) ox i
.
Let {wi} be its dual co-frame field . The wi's are local sections of the dual bundle 7r' T' M. One readily finds that (1.18) which is the Hilbert form given in (1.9). We observe that along a curve Xi = xi(t) , with yi = dtt', Euler's theorem (1.5) allows us to rewrite the
J:
n integral (1.1) as w . Starting with a given F, our geometrical setup has the following number of essential variables: 2n-1 from the local coordinates (xi, yi) of PTM, and ~(n-l)(n-2) from the freedom to specify the remaining vectors e1, ... , en -1 in our orthonormal frame. So far, the number of linearly independent Pfaffian forms depending on these variables (and F) is n, namely the wi'S . It follows that we need to find (n-1)+~(n-1)(n-2) more, for then by a theorem of E. Cartan's [6], one can decide in a finite number of steps whether the two sets of Pfaffian forms corresponding to F(x, y) and F' (x· , y') are equivalent under a transformation of the essential variables. This information in turn decides whether F and F· will transform into each other under a diffeomorphism between x and x·; see Chern ([9], [10]) . In the next section, we shall produce these (n-1)+~(n-1)(n-2) extra Pfaffian forms as a special choice of connection I-forms for our co-frame field wi .
2. A special connection and its structural equations. In this section, we shall recall (from Chern [9]) a torsion-free connection which is not entirely compatible with the metric 9 defined in (1.10) . However, we shall see from this section on that it has retained, in some sense, just the right amount of metric-compatibility to let us carry out certain basic Riemannian geometry calculations in Finsler spaces. The curvature of this connection is also quite remarkable. It has only a P part and
619 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
141
a R part; we will see that in all the specific computations where this curvature is called upon, the P part eventually drops out, while the R part (when contracted with the 'appropriate' vectors) exhibits all the symmetries and skew-symmetries of the classical Riemann-Christoffel curvature tensor. Let us express our frame and co-frame fields as (2.1a) equivalently, (2.1b) This shows that wi and ej may be regarded, respectively, as I-forms and vector fields on PT M even though they are strictly speaking local sections of certain n-dimensional co-vector and vector bundles over PT M. Equations (1.17) and (1.18) give u nl
(2.2) Duality says that
hence
(2.3)
i Uk vk).
= Uj. J:i
Orthonormality amounts to (2.4a) which, when given (2.3), is equivalent to (2.4b)
v
k ~ i ukl
v
I
j
=
gij
,
and (2.4c)
Uj
I
=
i kl hji V k 9
Also, (2.2) and (2.3) imply that
(2.5)
v\ = .
b
i)"
I Uj glk.
620 DAVID BAO AND S.S. CHERN
142
while (1.10), (2.2), and (2.4) imply that (2.6)
We now compute. Differentiating (1.18) and repeatedly using Euler's theorem (1.5) on both F and Fzi, we find that (2.7)
where the I-forms
w/
can be chosen as
(2.8)
wOrn (2.9)
The arbitrary parameters )..Or{3 are symmetric in the two lower indices and will be fixed later. Next, differentiate wOr as expressed in (2.1). Using (2.3) and (2.5), one finds that
(2.10)
The parameters J-lOr{3"1 are arbitrary and symmetric in the two lower indices, and will be determined later along with the )..Or{3 's. The ~i 's are also arbitrary but we shall fix them now by demanding that the I-forms wnOr and wOrn be negatives of each other, that is,
(2.lla) put another way,
(2.llb)
w Or n
= _6 Or {3wt:/n "
• '
621 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
143
One can check that this is achieved by the choices (2.12)
(2.13) The
wrt in (2.10) are then determined as W{30 --
(2.14)
VO k du {3k
-
vCOCT(U CTiu (3kFyk;z;J
+ /lCT{3 \ )w n +
J-L0{3~,"'" " . '/ .
The connection I-forms w/ satisfy the torsion-free condition because (2.7) and (2.10) collectively say that (2 .15) We emphasize that d here means exterior differentiation on PT M . So far, the matrix of connection I-forms, namely (w/), has skew-symmetric last row and last column, as evident from (2.8) and (2.11). We shall now choose the parameters AO{3 and J-L0{3/ such that the remaining (n-l) x (n-1) block of the matrix (w/) is 'almost' skew-symmetric; specifically, we want wpCT+wcrp , that is, wpoOocr + wcrooop, to be a linear combination of the wn o 's. To this end, let us apply d (on PTM) to the first equation in (2 .6), contract the resulting equation twice with u, into this we substitute what one gets by differentiating (2.3); the result is (2.16) Using (2.16) and (2.14), one finds that
wpcr
+ wCTP = - ujupi[d(FFyiyi) + (Fyi;z;i + Fyi;z;i )w n] -
(2.17) Let us write (2.18)
+ (oocrJ-L°p/ + OopJ-L°cr/)w/
.
2ApcrWn
622 144
DAVID BAO AND S.S. CHERN
where both Kijo and Giil are symmetric in i, j; their explicit formulas will be given later in this section. One can now check (for details, see Chern [9]) that with the choices (2 .19)
(2.20) equation (2.17) can be reduced to (2.21) with
(2.22a) which is totally symmetric. We shall call Hebe the Minkowski potential. If we now observe that, on account of (2.2) and (l.l1b), (2.22b)
Hebe
is zero whenever any of its three indices has the value n ,
then (2.11) and (2.21) combine to give
-H"1 k"w J ni
(2.23)
.
Having specified>. in (2.19), we rewrite the formula for won, namely (2 .9), as
won = -u o FI/oy.dyk $
(2.24a)
+ u; [~~FI/I:(Gr.!n + Fyrzo
-~(G.kn + FI/oz. -
- FI/ozr)
k Fyl:zo)]dx .
623 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
145
As we shall see below, this can be more compactly re-expressed as (2.24b)
wQ
n
d k F d k] = -u s [9F t aGt ayk X + y'yk Y s
Q
,
where
G .- ~F2 2 '
(2.25a)
(2.25b)
GI
(2.25c) Formula (2.24b) follows from (2.24a) because of (2.5) and the identity
We shall obtain (2.26) as a by-product of the formulas for the coefficients Ki j Q and Gijl in (2.18). To derive the latter, substitute (2.24a) into (2.18) and contract with ~ and el respectively; then, using (2.1) - (2.6), it is not difficult to find that: (2.27a) that is, (2 . 27b)
y;) Q( 'F F 2 F. i kg kl) K ij Q= -v QI 9 Ik Fo(FFyi oyk = -v I y 11'11; + lI ll;lI ;
624 146
DAVID BAO AND 5.5. CHERN
and Gijl
=
(P:z:kFy' y; + PFy.y;:z:k)U/
+ (yS PY ' y; + p2 Pyiy;yrgrs) [~~ bnl(Gksn -
Fy.:z:k
+ Pyk:z:.)
(2.28a)
where (2.28b) Using (2 .28b) and (2.25), it is straightforward to verify (2.26). As remarked before, (2.24b) then follows. Now that we have gotten (2.24b), we can use it (instead of (2.24a» to re-calculate the coefficients G ijl . Specifically, substitute (2.24b) into (2.18) and contract with el, the result is: (2 .28c) in particular, by (2.2) , (2.28d) Before we compute the curvature 2-forms, let us briefly discuss the significance of formula (2.23) . Let V denote covariant differentiation (of sections of tensor products of 7r*T M and 7r*T* M), on PTM, relative to the connection Then
w/
(2.29) and, along any tangent vector
tI,
one finds (with the help of (2.23» that
625 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
147
In other words, g(e n , ek) and g(ei, en) are covariantly constant in all directions, whereas g(e p , e a ) is covariantly constant only along those directions v which satisfy wna(v) = 0, that is, when en is parallely displaced. Thus, our torsion-free connection does exhibit a great deal of metric-compatibility; we will see later that it has just the right amount of such to effect the generalization of a good number of tools from Riemannian geometry to Finsler spaces. Furthermore, from (2.23) we see that the linearly independent entries in the matrix (w/) are w! and w! (0' < f3), which number exactly (n-1)+~(n-1)(n-2); since by construction these are also linearly independent (though not functionally independent) from the wi's, therefore together they can be used to solve the equivalence problem described at the end of section 1. Finally, from (1.10) and (2.22a), we see that the metric 9 is Riemannian (that is, independent of y) if and only if the Minkowski potential Hikj is zerG, and in that case the connection reduces to the usual Levi-Civita connection (pulled back from M onto PT M). In sum, the connection here, as characterized by (2.15) and (2.23), may be viewed as a natural analogue, in the Finsler setting, of the Levi-Civita connection of Riemannian geometry. Our work in sections 3-5 will further corroborate this belief. Differentiating (2.15) yields
w/
w/
o·,
(2.31) this says that the curvatures r'I i ._ Hk'-
(2.32)
dw
k
i
which are 2-forms on PT M, do not involve w na Aw! type terms, and hence can be expressed as (2.33)
r'lki H =
'12 R ki jlW j
Aw
I
+
Pk i jaW j
A 1\
wna .
Let us comment on the conventions and normalizations used in these structural equations:
(i) The minus sign in (2.32) indicates that the group action on the bundle 7r·T M is implicitly taken as a left action.
626 148
DAVID BAO AND 5.5. CHERN
(ii) The R in (2.33) is -2 times the R in Chern [9]; the change is made here in order to agree with the conventions of Spivak [21] and, Cheeger and Ebin [8]. In this paper, wi 1\ wi := wi ® wi - wi ® wi. (iii) The P in (2.33) is, except for the a index written as a subscript, numerically the same as that in Chern [9]. In (2.33), we will suppose without loss of generality that (2.34)
also, (2.31) and (2.33) imply a Bianchi identity (2.35)
and the following symmetry on the first and third indices of P, (2.36)
More can be deduced about R and P. For such purpose, let us apply d to (2.23), use (2.32) and (2.8), and get (2.37)
where C\7H)kio:
(2.38)
= dHkio -
s s Hsiowk - Hksowi - HkisWos
= : Lkio/3 w!
+ Qkios W
S
is the I-form-valued (k, i, a)th component of the covariant derivative of the Minkowski potential Hiki. From (2.38) it is immediate that (2.39) (2.40)
Qnios =
o.
Substituting (2.33) and (2.38) into (2.37) shows that: L is symmetric in the indices a and f3; in addition, (2.41 )
627 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
149
In particular, when k is fixed as n, (2.42) Rnisl
+
Rinsl : = R! slDji
+
R/s1Djn
=
0 (hence Rnnsl
0)
and, (2.34), (2.35), (2.42) then give (2.43)
Rknnl = Rlnnk;
equivalently, Rnknl
Rnlnk.
Also, one finds that (2.44) this, together with (2.36), suggests that the quantity P kiso is algebraically similar to the following Christoffel symbols of the first kind in Riemannian geometry, namely {iks} := (9ik,s - 9ks,i+9si,k)' Imitating the usual derivation of the formula for {iks}, that is, applying (2.44) to the combination (Pkiso + Pikso) - (Pskio + P ksio ) + (Pisko + P siko ), leads to an intennediate formula for Pkiso which, in conjunction with (2.39) and (2.40), imply that
!
(2.45a)
P nino
= 0,
and then (2.45b) consequently one finds that 1
Pkiso =
(2.45c) where
Hpqr
·
4[Hi/Qjson -
. Hks'Qjion
1 - 2"[Qikos - Qksoi
.
+ Hs/Qikon]
+ Qsiok]
,
is given by (2.22) [of course, (2.45c) embodies (2.45a,b)]. Hence
PkisCi is expressed in terms of the Minkowski potential Hikj (which, as we
recall, vanishes if and only if the Finsler space is actually Riemannian) and the Q part of its covariant derivative.
628 150
DAVID BAO AND 5.S. CHERN
For the remainder of this section, we shall re-express the covariant differentiation formula (2.29) in terms of the natural homogeneous coordinates (x, y), (yl, ... , yn) =I (0, ... ,0), on PTM. Specifically, we seek I-forms r i k on PT M such that (2.46)
V'
a axk
=
-
a
r' k ® axi
.
rik
and w k i describe the same connection, the former concerns the 's while the latter concerns the ek 'so We write r i k instead of r ki in order to match certain conventions regarding Christoffel symbols_ Recall that V' denotes covariant differentiation of sections of tensor products of rr*T M and rr*T* M, hence quantities such as V' are not considered because they do not make sense. On the other hand, the r i k 's, being I-forms on PTM, should a priori be expected to have both dx and dy components. However, as we shall see, the torsion freeness (see (2 .15)) of V' implies that
Both
-/xr
a?-
(2.47a) and (2.47b) In particular, the dy components are absent afterall. Furthermore, if we define r jkl by (2.48a) we will (as demonstrated below) find that (2.48b) where (2.48c)
MJ-kl
'-
,- -
8g'k 8Gt J Byt
Byl '
629 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
151
and G t is defined in (2.25). The M tenns measure the lack of metriccompatibility of V' . We would like to emphasize that in (2.48b), the gij 's depend on both x and y. However, if the Finsler function F does not depend on the argument y, then (gij) depends only on x and is a Riemannian metric. Also, from (2.25), we see that when F depends only on x, then Mjkl = 0 and the f jkl 's are literally the Christoffel symbols of the first kind in Riemannian geometry. Thus, in that case, V' is none other than the Levi-Civita connection, a fact we already know (see thf' paragraph before (2.31) ). Let us outline the steps leading to (2.47) and (2.48) . Applying the Cart an exterior differential to the statement ek (8) w k = ~ (8) dx k , we get (V'ek) 0 w k + ek 0 dw k = (V'~) 0 dxk. Into this we substitute (2.29), (2.46), and find that (2.49)
ei
"
k
.
0 (dw' - w 1\ wk')
0 " k =~ (8) (f\ 1\ dx ) . vx'
By torsion freeness (see (2.15)), the left hand side of (2.49) vanishes. So we must have (2.50) which then implies (2.47a) and allows us to specify, without any loss of generality, that fi k1 obeys (2.47b). Next we apply V' to the statement gij = g( a~., a~J ), getting ~~i' dx k +
~~i' dyk = (V' g)( a~.' a~)) + g(V' a~., a~)) + g( a~.' V' a~J)' For the first term on the right, use (2.1b) , followed by (2.30). For the other two tenns on the right, use (2.46), (2.47a), and (2.48a). The right hand side then becomes vPivqjHpqswnS + (fjik + fijk)dx k . Consequently, (2.51 )
(f jik
d k + f ijk )dX k = Ogij ox k X -
h d p q t e x part of v iV jHpq3WnS
and (2.52)
ogij dYk = t he d y part 0 f v piV)" q H --k pqsWnIJ .
oy
630 DAVID BAO AND 5.5. CHERN
152
By (2.22b) and (2.11),
Thus, upon substituting (2.24b) for w~ n, equations (2.51) and (2.52) become (2.53)
rijk
+ rjik =
Ogij
act (C-'~ U{5gst )
OXk - Oyk u
a
{3
V iV
H a{3-,
j-Y- ,
(2.54) It can be checked that (2.54) is equivalent to (2.22a). Next, note that 6-'~u~Sgst = 6-,c u / gst which, by (2.4b), is equal to V-'t . Using this, then
(2.22b), (2.22a), (1.10), and (2.48c), the right hand side of (2.53) can be re-written and we get (2.55) As in the derivation of the formula for the Christoffel symbols of the first kind in Riema.nnian geometry, we now apply (2.55) to the combination (r jk/ + rkjI) - (rklj + rlkj) + (rljk + rjlk), use (2.47b), and obtain (2.48b). This finishes our derivation of an explicit formula for the Christoffel symbols corresponding to the connection characterized by (2.15) and (2.23). The formula of the curvature tensor in terms of natural coordinates will be reported elsewhere.
3. The first and second variation of arc length, and the flag curvature. In this section, we shall derive the formulas for the first and second variation of arc length using two methods: one uses differential forms and produces an elegant calculation, while the other uses covariant differentiation along curves and is somewhat more geometric. We shall see that the special connection introduced in section 2, when used to express the second variation, gives a formula which is formally identical to that in Riema.nnian
631 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
153
geometry and, as a by-product, shows immediately how the sectional curvature of Riemannian manifolds can be generalized to Finsler spaces. We call this generalization the flag curvature. Finally, we identify the index form. We should mention that the second variation of arc length in Finsler geometry also appeared in two other authors' works, namely, that of L. Auslander ([2], [3]), and P. Foulon ([14], [15]) . The connection used by Auslander is that of Cartan's [7], which is metric-compatible but has torsion. On the other hand, Foulon approached the subject from a dynamical systems point of view; though he has a formula for the second variation in [14], it is not clear whether a specific connection is being used (or needed) in his work; also, he informs us that our flag curvature is analogous to the Jacobi endomorphism that he defined in [15]. Let us begin our derivations. Let b. := {(t, u) : to ~ t ~ t1 , -1 ~ u ~ + I} be a rectangle. We map it into M using a : b. -+ M, such that the t-curves (u = constant) are smooth. One obtains two vector fields defined over the square :
(3.1) T gives the velocity vectors to the t-curves. Strictly speaking, T and U are maps from b. into T M; but we will occasionally find it convenient to regard them, by a slight abuse of notation, as maps from b. into 7r·TM. The map a admits a lift a : b. -+ PTM, defined by
(3.2)
O'(t, u) := (a(t, u), T(t, u)) .
Correspondingly, one gets the following vector fields over the lifted square: (3.3) At each point O'(t, u) in PTM, we use the metric 9 given by (1.10) to calculate
(3.4)
IIT(t,u}II:= Jg(T,T).
There is also a preferred en satisfying (see (1.17))
(3.5)
632 DAVID BAO AND 5.5. CHERN
154
which is then completed to an orthonormal basis by eo. With respect to this frame field, we obtain the dual co-frame field wi and the connection I-forms These I-forms can be pulled back to A by a·; we write
w/.
(3.6) (3.7)
£1*w/ = a/dt + b/du .
Since wi has no dy terms, one has wi(T) = wi(T), w(U) = Wi(U), consequently
(3.8) Also,
(3.9) and, from (2.11),
(3.10) Using £1· to pull back the torsion-free condition (2.15), then substituting in (3.6) and (3.7), one gets (3.11)
aa i
abi
-au - + -at
.. '
.. ,
= a'b ·' - b'a ·'
in particular,
(3.12) n
(3.13)
aa n ab b = + au at
O
a0
n
Doing the same to the structural equations (2.32) and (2.33), namely those involving the curvature, gives (after some simplification and the use of (2.34))
633 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
in particular, specializing k to one finds that Po nnf3 = 0 and
0',
155
i to n, and using (2.39), (2.40), (2.45c),
(3.15) We now compute the first and second variation of arc length. A priori, the length L(u) of a t-curve is given by It:l F(a, T)dt; but from (1.10), Euler's theorem (1.5), and (3.4), one finds that F(a, T) = IITII. Hence by (3.8), one has (3.16) The formula for the first variation of arc length now follows from (3.13): (3.17) Since bi = U i , the variational vector field, it is clear that the boundary term in (3.17) vanishes when all the t-curves have the same end-points. By specializing to delta-like variational vector fields, we see that the condition for the base curve (u = 0) to be critical-that is, a geodesic-is a on = 0, which by (3.10) is equivalent to (3.18) In other words, a geodesic in Finsler geometry is a curve (! in M whose lift a := (a, T) into PTM is a solution of the following differential system: (3.19a) more explicitly, (3.19b) As for the second variation of arc length, we differentiate (3.16) with respect to tt, which entails the calculation of Applying to (3.13),
a;:lft.
:u
634 DAVID BAO AND 5.5. CHERN
156
The terms bOb on and -bnobon can be re-expressed using (3.9) and (3.10); also, one re-writes the R term using (2.42). Finally, we restrict (3.20) to u = 0 and assume that the base curve is a geodesic, then by (3.10) and (3.18) the terms proportional to aon drop out. The formula for the second variation of arc length along a geodesic in Finsler geometry is therefore: n
(3.21)
L"(O)
8b = [-8 - 6ij bibj] t=t + 1tl an [6;jb;bj + Rninjbibildt 1
U
t=to
.
to
We pause to note that the absence of any P term in (3.21) is traceable to the fact that Po nnl3 = 0, which in turn follows from the basic formula (2.45c) . In (3.21), the term Rninjbibi, that is, Rn;njUiU j (where U is the variation vector field, see (3.8)), is proportional to the negative of the following quantity (3.22)
-R . ·UiU j K(U, en) := 6ijUi~;~ (U n )2
which depends on the flag {x,e n , U /\ en} (in the context of (3.21), x = o(t)), and will be called a flag curvature. More generally, given any flag {x, Y, U /\ Y : Y, U E TzM} in M, the flag curvature K(U, Y) is defined as (3.23) where both R and 9 are evaluated at the point (x, Y) in PTM. As a result, the flag curvature depends not just on U /\ Y, but also on Y . In the
635 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
157
Riemannian case, the flag curvature reduces to the sectional curvature, and the minus sign in (3.22) and (3.23) reflects our adherence to the convention that the round Euclidean 2-sphere has (constant) positive curvature . We would like to point out that, even though the calculations in this section so far have been done with respect to a particular moving frame and its dual co-frame, the key formulas obtained (especially (3.21) , the second variation of arc length) are tensorial-that is, independent of the moving frame used. For the remainder of this section, we shall recast most of the formulas of this section in the language of covariant differentiation along curves. Thif:j has the advantage of making more explicit the geometric content and sets the stage for generalizing the theory to infinite-dimensional spaces. As a bonus, we shall readily identify the index form and see that it is, up to a multiple that will be made precise later, the integral in (3.21) plus a n )]2a n dt. 11011 ...L(aan)2dt an a u ' that is , 10 [.2..-(log au
111
Let a(t) be a curve in M with velocity T(t) := a.ft . It admits (however, see remark (i) below) a lift into PTM, namely o(t) := (a(t), T(t)), whose velocity is T(t) := o.ft = ft(a(t), T(t)). At each image point o(t) in PT M, take an orthonormal frame {ei} for the vector bundle 7r·T M, with en pointing in the direction of T (see (3.5)); correspondingly, one obtains the connection I-forms Wki. Given any vector field W(t) in M defined along a, we expand it as W = Wiei and define its covariant derivative along a as i
(3.25)
dW DT W := dt
ej
+ W'. V t
ej
=
[dW
j
- ] dt + W k wk'· (T) ej
•
Two remarks are in order: (i) In case a is a constant curve, the velocity T is identically zero, and the aforementioned lift 0 will not make sense. The solution is simple because all the vectors W(t) are now based at a common point of M, hence a meaningful derivative can be taken without first having to perform any parallel transport. In this case, one defines DTW to be ftW(t). (ii) As indicated at the beginning of this section, when dealing with a square a(t, u) in M, in other words a one parameter (namely u) family of t-curves in M, one always lifts both the t-curves and the u-curves into PT Musing the velocities of the t-curves, through 0 := (a, T). Using o· to pull back the torsion-free condition (2.15) will yield the statement
(3.26)
DTU = Du T ,
636 DAVID BAO AND 5.5. CHERN
158
where the left hand side is defined by (3.25); but what about the right hand side? Well, even though it is given by a formula which is structurally identical to (3.25), namely
Du T
(3.27)
:= [
8T
i
· - ] au + T k wk'(U) ei ,
we hasten to add, however, that the quantities wk i and ei in (3.27) are evaluated at the point a = (o,T) [not (o,U)]; also, the vector U, as defined in (3.3), is (0, T) as opposed to (0, U). The point being that in computing the above covariant derivative-namely DuT-of T along au-curve, we did not lift the u-curve into PTM using its velocity U, as the instructions preceding (3.25) would have directed us to do; instead, we lifted it by the transversal directions T. Nevertheless, the elegance of (3.26) indicates that the tool of covariant differentiation along curves is worth retaining. At the same time, the fact that (3.27) is structurally the same as (3.25) suggests that they can be unified by a minor generalization of the instructions leading to (3.25). This we shall do below. Let us choose symbols that will not bias us into thinking about tcurves or u-curves. Let /,(s) be a curve in M with velocity 8(s); let W(s) be a vector field in M defined along /'. Corresponding to any lift ..:y( s) := (/'(s), y(s)) of /' from Minto PTM, we can define a covariant derivative of W along " as follows:
tu
tu
i
(3.28a)
DsW:=
dW . [dWi k · -] ds ei + W' 'f:J s ei = ds + W wk'(8) ei
where
(3.28b)
d d 8 := ..:y. ds = ds (/'(s), y(s))
and
(3.28c)
Wki and
ei
are evaluated at (/'(s), y(s)) .
The dependence of Ds W on the choice of the lift will be kept implicit. In each application, the specific lift involved should be clear from the context;
637 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
159
quite often it is given by the tangent to the curve in question, but the previous paragraph explains why such is not always the case. From (2.23), one derives readily the following: (3.29a)
d d)g(V, W)] = g(DsV, W)
. k + g(V, DsW) + VJW Hjk/3
(3 Wn
-
(S) ,
where (3.29b)
g, Hjk/3, and
w! are evaluated at (-y(s), y(s))
.
The third term on the right hand side of (3.29a) may be abbreviated as (Dsg)(V, W). We now mention two special cases under which this term drops out: (3.30a)
if V or W is proportional to y (and hence to en) ,
for example when 'Y is a u-curve, but y
= T, and either V
or W is T; or
(3.30b) for example when 'Y is at-curve, y (3.30c)
= T, and 'Y is a geodesic; then
d ds[g(V, W)] = g(DsV, W)
+ g(V, DsW)
,
a formula that would be true without the qualifications (3.30a), (3.30b) if the connection were metric-compatible. It turns out that even though the connection here is, strictly speaking, not metric-compatible, the contexts in which we shall need to calculate a quantity of the form g(V, W) always satisfy either (3.30a) or (3.30b). In this sense, one can say that the special connection introduced in section 2 has just the right amount of metriccompatibility. Return now to the rectangle o(t, u), 0 : ~ -+ M. As agreed, we shall lift it into PTM using the velocity T of the t-curves, namely through U := (0, T). Using u· to pull back the structural equations (2.15), (2.32) and (2.33), while keeping in mind that
is
(3.31)
638 DAVID BAO AND 5.5. CHERN
160
one finds that DTU = DuT, which is (3.26), and (3.32a)
k
. -
-
DUDTZ = DTDUZ + Z Ok'(U,T)e;
for any vector field Z(t, u) in M defined over the square. The 0 term can be computed using (2.33), (3.31), and (2.34); the result is k
.
.
DUDTZ = DTDUZ + Z [Rk' jnUJIlTIl (3.32b) Formulas (3.26) and (3.32) are equivalent in content to (3.11) and (3.14), respectively. We can now give the variational formulas in terms of covariant derivatives. The length of a t-curve is L(u) := Jt~ Jg(T, T)dt, where 9 is evaluated at aCt, u) := (a(t, u), T(t, u)) in PTM. Differentiating with respect to u, using (3.30a,c), then (3.26), and then (3.30a,c) again, we find that tuJg(T,T) = lIing(DTU,T) = g(DTU, II~II) = %t[g(U, II~II)J g(U, DT( II~II )). Therefore the analogue of (3.17) is:
(3.33)
L'(u) =
1:
1
%t [g( u, II~II)] - g( U, DT( II~II))
dt .
The equation for a geodesic is thus (3.34) which is equivalent to (3.18) or (3.19). If the geodesic is given a constant speed parametrization, then (3.34) reduces to a form familiar from most Riemannian geometry texts, (3.35)
DTT=O.
Next we apply tu to (3.33). For the first term on the right, we use (3.30a,c) to get %t[g(DuU, frrr)] (which we retain) plus %t[g(U,Du(II~II))] (to which we apply (3.29)). This gives
%t [g( DuU, II~II)]
+ g(DTU,Du (II~II))
+ 9 (V, DTDU (II~II)) + H jk /3Vi [Du (II~II)] kw!(,i).
639 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
161
As for the second term on the right hand side of (3 .33), we apply (3.29) and the result is equal to
Now restrict to u = 0 and assume that the corresponding base t-curve is a geodesic, that is, w!(T) = 0 and D T ( II~II) = O. The two H terms, together with one other, then drop out. Thus the second variation L"(O) consists of a boundary term [g(DuU, II~II )]~~~~ plus the integral g(DTU, Du( II~II)) + g(U, [DTDU - DUDT]( II~II)) dt . By (3.32b), (3.31), (2.34), (2.45a), and
It:'
the fact that wnQ(T) = 0 at the base geodesic, we find that g(U, [DTDU DUDT]( frrr)) = liTIiRninjUiUj. The product rule and (3.26) show that
9(DTU,Du(II~II))
=
IIT"2 a~:lIg(DuT,T)
(3.30a,c), = [liT"' a~:I1][~ together gives
+
IIhg(DuT,DuT) which, by
;u IITII2] + nhg(DuT, DuT).
Putting the above
the second variation of arc length along the base (u = 0) geodesic. Let us compare (3.36) with its equivalent, (3.21). From (3.28), (3.9) and (3.8), one sees that (3 .37)
9 ( DuU,
T)
IITII
=
8b n
.
t
.
8u - 6ij b b,? .
Using (3.27), (3.5), (3.8), (3.9), we find that (3.38)
_1_ [g(DuT,Du T ) _
liT II
(8~TII)2] = ancijb;b,! uU
.
640 162
DAVID BAO AND 5.5. CHERN
Finally, it's clear that (3.39) In (3.36), let us use (3.26) to rewrite DuT as DTU. To highlight the formal similiarity with the Riemannian case, we write the R term as (3.40) Formula (3.36) now reads L"(O)
(3.41)
= [g(Du U, II~II)]:::: +
tl IITII1 [g(DTU, DTU) + g(Rt,T , U)T, U)Jdt
1 to
_1tl _1 (oIlTII)2 liT I ou dt . to
The first integral on the right of (3.41) is the quadratic form of the so-called index form along a geodesic aCt), a :$ t :$ b, with velocity T(t): (3.42)
leV, W):=
lb II~II
[g(DTV,DTW)
+ g(R(T, V)T, W)Jdt .
Here, 9 is evaluated at the point (a(t), T(t)) of PTM and, from property (2.43) of R, we see that leV, W) is indeed symmetric, just like in the Riemannian case. We would like to mention that in some texts, geodesics are defined by (3.35) and hence automatically has a constant speed parametrizais a constant and is tion imposed on them; in that case, the factor typically omitted from the definition of leV, W). We will enumerate two fundamental properties of the index form in the next section. The second integral on the right hand side of (3.41) is, as remarked before, the term that must be added to the integral in (3.21) in order to get the index form. Our derivation of the first variation shows that 8~~1I = g(DTU, hence the integral in question may be re-expressed
nh
frrr),
641 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
163
It:'
as IIh [g(DTU, II~II )Fdt, and now the formula for the second variation of arc length looks formally identical to the one in Riemannian geometry; see, for example, Cheeger-Ebin [8]. We regard this as further evidence that the connection described in section 2 is the correct generalization of the Levi-Civita connection to Finsler geometry. Finally, the flag curvature defined in (3.23) now reads (3.43)
-g(R(Y, U)Y, U) K(U, Y) = g(U, U)g(Y, Y) - [g(U, Y)]2 ,
where Rand 9 are evaluated at the point (x,Y) in PTM. The object R(U, V)W encountered in both (3.40) and (3.43) is defined as (3.44)
R(U, V)W := [R/ kl
ei](x,y)
wj UkVI
.
Though R(U, V) behaves algebraically like the curvature operator V' V V' v V'vV'u - V'[U,V) in Riemannian geomt::try, we must emphasize that here, it
is only a piece of the Finsler curvature. We will utilize (3.44) again when we write down the Jacobi equation in the next section.
4. Geodesics, Jacobi fields, and the index form. In this section, we write down the geodesic equation on the Finsler space M in terms of the natural coordinates on PTM, summarize some analytical properties of geodesics, derive the Jacobi equation, demonstrate that the calculus of Jacobi fields generalizes verbatim to Finsler geometry, and review two fundamental properties of the index form . From (3.34), we recall that on M, a curve a(t) with velocity T(t) is a geodesic if DT( II~II) = o. In order to be able to refer to such Riemannian geometry texts as [8], [16], [19], and [21], we shall from this point on require that all geodesics be given a constant speed parametrization. From (3.34) and (3.30a,c), we see that a geodesic has constant speed if and only if its defining equation is DTT = O.
(4.1)
a:'
Let us write T as Ti = d~i 8:" where the ai(t) 's are the coordinate functions of the curve a(t). Using (2.1b) and then (3.25), it is straightfordT i 8 k 8 dlr ward to show that DTT = (ita;; + T V't a;r; . Here, T(t) := Tt, where
642 164
DAVID BAO AND 5.5. CHERN
a(t) := (a(t), T(t)) is the lift of a into PTM. By (2.46) and (2.47a), we a . a . a . la have 'Vi' a;k = [(f\)Ii1(T)] ax' = [(f\)Ii1(T)]Fx' = (f\I)Ii1 T Fx'. Therefore (4 .1) , in natural coordinates, gives the ODE (4.2a) for the coordinate functions of the geodesic a(t). Using (2.48a,b ,c) and (l.llb) , we see that the ~ part of f does not contribute to the double contraction . So (4.2a) reduces to 2
(4.2b)
d a dt 2
k
i
+
l
da da [gij (8 g jk _ 8g kl dt dt 2 8x l 8x j
8 91j )] _ 0 ,
+ 8x k
which is formally identical to the equation for geodesics in Riemannian geometry. However, we hasten to caution that, unlike the Riemannian case, the metric coefficients grs here depend on both x = a(t) and y = T(t) = ~~. Hence, in the second term on the left hand side of (4.2b) , the non-linear dependence on the velocity is more than quadratic. We also remark that, since our connection here is derived solely from the Finsler function F, it is not surprising that our geodesic equation agrees with that in Rund [20], even though he takes a classical calculus of variations approach. Let us now show that, despite the above caution, the left hand side of (4.2b) [and, for that matter, that of (4.2a) as well] still scales in the usual way under an affine change of parameter t 1--+ ot + (3. More precisely, let ,,(s) := a(os + (3), 5 := 1(S) := (1'(s),5(s)), then
¥S'
We shall outline the elementary argument [which produces (4.3a)] because it explains clearly why we choose our base space to be PTM rather than, say, the sphere bundle or the full tangent bundle. Let us do so in two remarks: (i) By the chain rule, ~ = o(~~)lt=Q.s+fl. To avoid clutter, for the rest of this discussion it will be implicit that in any statement involving sand t, the t is set to the value os + (3. Hence ~i = 0 2 d;t~i
d:.
.
643 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
165
(ii) (rikl)Vy = (rikl),a because 1'(s) = (')'(s), ~~) = (I'(s),a~~), which is the same point in PTM as (')'(s), ~~) = (a(t), ~~) = a(t) . Note that the key identity 1'(s) = a(t) would fail completely if our base space were TM, and would only hold partially (namely for the case a > 0) if our base space were the sphere bundle. For later use, let us generalize the discussion here. Let a, T, a, 1', S, l' be as defined above. Let V(t) be a vector field along a and define W(s) := V(as + (3). Then (DsW)
(4.3b)
with lift
'Y
= a(DTV) with lift a
where, as before, the argument t on the right hand side is implicitly set dW'(s) to the value as + f3 . To see this, recall that (DsW )with lift 'Y = [-d-s- + i) ' Y (!!i)]() dV'(t) More .m: W k()( S w ds ei ''Y. By th e ch· am ru1e, dW'(s) ds - a dt. k
portantly, we have seen above that 1'(s) = a(t) in PTM. Thus (wki),'Y = (wki),a, (ei),'Y = (ei)'a and, by the chain rule, ~; = a~~. Equation (4.3b) now follows . From (4.3a), it follows that if a(t), a
~
t
~
b , is a geodesic, then so is its reverse,
I'(s) := a(a + b - s), a
(4.4)
~ s ~
b.
In conjunction with the standard existence and uniqueness result for the ODE in (4.2), property (4.3a) also implies that the time a map along the geodesic emanating from p with initial velocity v is the same as the time 1 map along that emanating from p with initial velocity avo This observation, familiar from Riemannian geometry, sets the stage for the introduction of the exponential map, which we will turn to after some technical preliminaries, whose proofs can be found in Spivak ([21], voU). In a coordinate patch, given any v = vi a~. E TqM, we let (4.5)
Ivl
:= [oii vivi]t and IIvll(q,y) := [9ii(q, y)vivi]f .
One can show that associated to any pre-compact (relative to the manifold topology of M) neighbourhood U are two positive constants k and K such that
(4.6)
644 166
DAVID BAO AND 5.5. CHERN
for all q E U, v E TqM, and y E PTqM . Using (4.6) and the standard theory for the ODE (4 .2), one finds that Associated to each P E M is a neighbourhood U and an
> 0 such that
f
for each v E TqM (q E U) satisfying Iivli(q,y) < there exists a unique geodesic Ov: (-2,2) (4 .7)
with 0,,(0)
= q and 0~(0) = v
->
f ,
M
.
Due to (4.7), the exponential map v 1-+ ov(l) =: expq(v) is defined in a neighbourhood of (P,O) in TM, for each P E M. As in Riemannian geometry, its derivative (4.8)
(expq) .. is the identity map at 0 E TqM .
Applying (4.8) to the map (q,v) finds that
1-+
(q,expq(v)) from TM into M x M, one
Associated to each P E }.II is an open neighbourhood W and an
f
>0
such that any two points in W (4.9)
are joined by a unique geodesic of length
IIJJ.(P)II ~ IIJt(P)1I > 0 => pEA) . Hence A = [0,1] and our inequality is true on 0 :::; t :::; l. This ends the proof of the first Rauch comparison theorem. Quite a few other results in Riemannian geometry also generalize to the Finsler setting. Some of these are of a decidedly global nature. They include Myers' theorem (which says that completeness and uniform positive lower bound on the Ricci curvature imply compactness and finite fundamental group) , Synge's theorem (any compact, connected, orientable, even-dimensional Riemannian manifold with positive sectional curvatures is simply connected), and the Cartan-Hadamard theorem (any complete, simply connected Riemannian manifold with non-positive sectional curvatures is diffeomorphic to Euclidean space). Modulo the calculus of geodesics and covering spaces, the proofs of these theorems depend primarily on the formula for the second variation of arc length. Auslander [3] gave a generalization of the above theorems to Finsler spaces. He used the Cart an connection in his treatment. As a result, his curvature forms all have three pieces, two of which are analogous to the R and P tensors in our curvature formula (2.33), together with a third term (abbreviated as S) that is absent in our case because of torsion-freeness. When the second variation of arc length is expressed in terms of the Cartan
656 178
DAVID BAO AND S.S. CHERN
connection and its curvature, the S part does not enter. As for the P term, it was recently pointed out to us by Zhongmin Shen that it actually drops out; a direct calculation by us confirmed his claim. Since our formula of the second variation contains no P term either, we expect the same theorems to hold in our context as well. We now turn to the issue of whether geodesics are minimal among nearby curves with the same end-points. If a geodesic (J, with velocity T, contains conjugate points, then by (4.27), leW, W) < 0 for some W E n~, hence L"(O) < 0 by (4.24) . So (J is not minimal. The question is then: Will (J be minimal (among nearby curves with the same end-points) if it contains no conjugate points? With the absence of conjugate points, there is a standard argument in Riemannian geometry (see, for example, Gallot, Hulin, and Lafontaine [16), or Spivak [21] vol.4 and voLl) which reduces this question to whether short segments of (J are minimal, and we have already answered that in the affirmative at the beginning of section 4. Let us conclude by enumerating a few more global issues: (i) The classification of Finsler spaces with constant flag curvatures. Here, a modification of the scheme used by Wolf [23] is perhaps useful. (ii) The formulation and the proof of the Gauss-Bonnet-Chern theorem, using the connection defined in section 2. In 2 dimensions, this has already been done for Landsberg surfaces, see Chern [11]. In higher dimensions, a paper of Lichnerowicz's [18] proved a generalized version of this theorem using the Cartan connection, but only for Finsler spaces of Berwald type (namely, those for which the S part of the curvature vanishes). Preliminary investigations indicate that a satisfactory unified picture does emerge from the works of these authors (see Baa [4]). (iii) The proof of a sphere-pinching theorem. Zhongmin Shen called our attention to. the works of Dazord ([12], [13]) and Kern [17), which sketched the proofs of the homotopy sphere theorem and the differentiable sphere theorem, respectively. Both of these authors used the Cart an connection; but we believe that their methods will adapt readily to our setting. Alternatively, an approach advocated in Tsukamoto [22], which circumvented the Toponogov theorem, is perhaps also viable here. (iv) A detailed understanding of Finsler spaces for which the R part of the curvature vanishes. Finally, there has been some speculation that the L 4 -Finsler metric induced by the Finsler function F(x,y):= [~ijkl(X)yiyjykyl]t might provide
657 ON A NOTABLE CONNECTION IN FINSLER GEOMETRY
179
some geometrical insight for the Ginsburg-Landau model of superconductivity. The details remain to be worked out. A relevant reference here is Asanov [1].
Acknowledgements. We would like to thank: Giles Auchmuty for discussions about a possible extension of (1.15b) which turns out to be false; Robert Bryant for suggesting that we pursue the convexity of F in our proof that geodesics are locally minimal; Peter Li an-a Phil Yasskin for discussions about the relationship between the signs of I(U, U) and L"(O) (see the remark following (4.24)); Zhongmin Shen for discussions about the P term in Auslander's formula (for the second variation) and for supplying the references on the sphere-pinching theorems; Phil Yasskin for clarifying a certain property of the Cartan exterior differential; and one of the referees for some very constructive criticism regarding (2.5), (2.22b), (2 .31), and condition (5) in the construction of the transplant ~ w.
REFERENCES 1. Asanov, G.S., Fin3ler Geometry, Relativity and Gauge Theorie3 , D . Reidel 1985. 2. Auslander, L., The U3e of form.3 in variational calculation3, Pacific J . Math. 5 (1955), 853-859 . Remark on the U3e of fOTm3 in variational calculation3, Pacific J. Math. 6 (1956), 209-210. 3. Auslander, L., On curvature in Fin3ler geometry, Trans . Amer. Math. Soc. 79 (1955), 378-388. 4. Bao, D., A note on the Gauu·Bonnet· Chern theorem for Fin3ler 3pace3, in preparation. 5. BoIza, 0 ., Lecture3 on the Calculu3 of Variation3, G .E. Stechert and Co., New York 1946. 6. Canan, E., Le3 probleme3 d'equivalence, Select a de M. Elie Cartan, 113-136, Paris 1939. 7. Canan, E. , Le3 e3pace3 de Fin3ler, Actua.lites Scientmques et Industrie1les no. 79, Paris, Hermann 1934. 8. Cheeger, J. and Ebin, D., Compari3on Theorem3 in Riemannian Geometry, North Holland 1975. 9. Chern, 5.5., Local equivalence and Euclidean connection3 in Fin31er 3pace", Sci. Rep. Nat. Tsing Hua Univ. Ser.A 5 (1948), 95-121; or Selected Papers, vol. n, 194-212, Springer 1989. 10. Chern, 5.5., On Fin3ler geometry, Comptes Rendu Acad. Sci. Paris 314{1} (1992), 757-761.
658 180
DAVID BAO AND S.S. CHERN
11. Chern, 5.5., Hi$torica.l remark$ on Gau$$-Bonnet, In Analysis et Cetera, volume dedicated to Jiirgen Moser, 209-217, Academic Press 1990. 12. Dazord, P., Variete$ Fin$lerienne$ de dimen$ion paire 6-pincee$, Comptes Rendu Acad. Sci. Paris SU. A 266 (1968), 496-498. 13. Dazord, P., Variete$ Fin$lerienne6 en forme de 6phere$, Comptes Rendu Acad. Sci. Paris Ser. A 267 (1968), 353-355. 14. Fouion, P., Noufleauz intlariant$ geometrique$ de$ $1I$teme$ d1lnamique du $econd ordre. Applica.tion$ Ii l'etude de leur comportement ergodique., Theses d'etat 1986 . 15. Fouion, P., Geometrie de$ eqUatiOn6 differentielle6 du $econd ordre, Ann. Inst. Henri Poincare 45(1) (1986), 1-28. 16. Gallot, 5., Hulin, D. and Lafontaine, J., Riemannian Geometry, 2nd ed., Springer 1990. 17. Kern, J., Da6 pinchingproblem in futriemann6chen Fin6ler6chen mannigfaltgkeiten, Manuscripta Math. 4 (1971), 341-350. 18. Lichnerowicz, A., Quelque$ theoreme6 de geometrie differentielle globale, Comm. Math. Helv. 22 (1949), 271-301. 19. O ' Neill, B., Semi-Riemannian Geometry, Academic Press 1983. 20. Rund, H ., The Differential Geometry of Fin6ler Space6, Springer 1959. 21. Spivak, M ., A Comprehen$ifle Introduction to Differential Geometry, Publish or Perish 1975 . 22 . Tsukamoto, Y ., On Riemannian manifold., with po.,itifle curvature, Memoirs Fac. Sc., Kyushu Univ., Ser. A 15(2) (1961) , 90-96. 23. Wolf, J. , Space., of Con6tant Curvature, Publish or Perish 1974.
Department of Mathematics University of Houston Houston, TX 77204-3476 Mathemat ical Sciences Research Institute 1000 Centennial Drive Berkeley, CA 94720 Received September 10, 1991
659 Reprinted from Global Analysis in Modern Mathematics, Publish or Perish, 1994.
CHARACTERISTIC CLASSES AS A GEOMETRIC OBJECT Shiing-shen Chern
1. Riemannian geometry and Erlanger program Following Felix Klein [8] I take 1870 as the starting year. A year earlier, in 1869, E. Christoffel and R. Lipschitz solved the fundamental problem in Riemannian geometry, the form problem: To decide when two ds 2 's differ by a change of coordinates. In the solution, Christoffel introduced the covariant differentiation now called the Levi-Civita connection. In 1872 Klein announced his Erlanger program which defines geometry as the study of a space with a group of transformations. This has given geometry a unifying principle, which has dominated geometry for at least a century. Klein was aware that the Erlanger program does not include Riemannian geometry, as a generic Riemannian metric admits only the identity isometry, but the two views progressed in parallel. One of the implications of the Erlanger program was the development of non-euclidean geometry by the Cayley-Klein metric; non-euclidean geometry became a chapter of projective geometry. To illustrate the scope of Klein's great idea, let me give an example. As the group plays the central role, the same analytic argument could give geometrical theorems which are radically different. For example, the theorem that the three heights of a spherical triangle meet in a point can be "translated", using dual numbers, to the following theorem of Morley-Petersen (1898): Consider a simple skew hexagon in space whose adjacent sides are perpendicular. There is a line that meets perpendicularly the three common perpendiculars of the three pairs of opposite sides. Riemannian geometry received a great impetus through the theory of general relativity in 1915. Several books, now classics, appeared in the period 1924-26. On the other hand, following in the tradition of classic differential geom-
221
660 SHIING-SHEN CHERN
etry, there is a rich field of studying the geometry of submanifolds under a group other than the group of rigid motions. Of these the "'all-embracing" group is the projective group and, beginning in 1906 by E. J. Wilczynski and continued by the Italian school led by G. Fubini and E. Cech, there were considerable activities on projective differential geometry. The differential geometries of the other groups, such as the affine group and the various sphere groups, were also studied, principally by W. Blaschke and his school. An account can be found in Blaschke's classic treatise [3]. There are beautiful results, but the subjects suffer the disadvantage of being isolated. Activities in Riemannian geometry extend to efforts in finding a space which can be adapted as the universe of a unified field theory. The problem has too many constraints and there was no happy conclusion. In the thirties differential geometry fell to a low point, waiting for the dawn of a new era of global differential geometry.
2. Connections after Elie Cartan A generalization which embraces both Riemannian geometry and the Erlanger program is the "espaces generalises" of Elie Cartan. In modern terminology this is a fiber bundle with a connection. The latter gives an infinitesimal relationship between the fibers. This obviously includes the geometry in the sense of Klein, while Riemannian geometry can be looked on as the geometry of the tangent bundle with the Levi-Civita connection. Let me give the basic analytical facts of a connection. Let
7T:E-+M
(1)
be a principal bundle with structural group G a Lie group. The tangent space at the unit element e E G has a Lie algebra structure. It will be denoted by L(G) and the multiplication by a bracket. The map
(2)
g -+ aga- 1 ,
a, g
E
G,
a fixed,
leaves e invariant and induce the adjoint map (3)
ad(a): L(G) -+ L(G).
Using the connection, a covariant differentiation D can be defined, which converts an L(G)-valued exterior differential form of degree k into one of degree k + 1.
222
661 CHARACfERISTIC ClASSES AS A GEOMETRIC OBJECT
Over a neighborhood U of M, rr-I(U) can be described by the local coordinates (x. gu)' x E U. gu E G. Then a connection is given by a linear differential form (4)
where dg u ' g;;1 is the right-invariant Maurer-Cartan form in G and ()u(x. dx) is an L(G)-valued linear differential form in U. w is an L(G)-valued I-form in E, given locally in rr-I(U) by the expression (4). By covariant differentiation one finds (5)
(6)
Dw
= Q = ad(gu)8u. DQ=O.
where 8 u is an L(G)-valued exterior 2-form in U. Notice that w, Q are globally defined in E, with the local expressions (4), (5) respectively. They are called respectively the connection and curvature forms. Equation (6), which follows from (5) by exterior differentiation, is called the Bianchi identity. The form of equation (5) leads us to the consideration of the multilinear symmetric real-valued functions P(X 1 • • ••• X s), Xj E L(G), I :::: i :::: s. It is called invariant if
(7)
P(ad(a)XI •...• ad(a)Xs) = P(X 1 ••••• X s).
for all a E G. With multiplication defined in an obvious way all invariant polynomials P form a ring I (G). The form (8)
P(Q) = P(Q ..... Q) '-v-"
is an exterior differential form of degree 2s , which is globally defined in M. It will be called a characteristic form . As we will describe below, it plays an important role in both local and global problems. We wish to observe that if G = GL(q) , L(G) is the space of all (q x q) matrices. When a geometric structure is given, a fundamental problem is to see whether a connection can be defined intrinsically. This is the case with the Riemannian metric with its Levi-Civita connection. Generally the problem is non-trivial; in fact it is not always true that an intrinsic connection exists. In the case of the geometry of paths of Veblen and Thomas a normal projective connection exists. Recently Bao and I showed that for the Finsler metric an intrinsic connection exists whose base space is the space ofline elements of the manifold [1].
223
662 SHIING-SHEN CHERN
3. Global differential geometry Global problems in differential geometry have a long history. For example, particular emphasis was paid by Blaschke in his treatise [3]. But a systematic development needs a foundation of differential topology, and among those who contributed were H. Whitney, G. de Rham and H. Hopf. A fundamental tool is the exterior differential calculus. It was developed by Elie Cartan in 1922. Inspired by remarks by Poincare, de Rham showed its topological significance by proving the following theorem in 1931: Let M be a differentiable manifold. Let A k (M) be the space of its differential forms of degree k and Ck(M) C Ak(M) the subspace of closed k{orms. Then
(9) where the right-hand side stands for the k-dimensional cohomology group of M with real coefficients. Since a differential form is also a local entity, this gives a
relation between local and global properties. In his seminal paper [7] Heinz Hopf first drew attention to the relation between curvature and topology. He laid emphasis on the high-dimensional Gauss-Bonnet formula, which expresses the Euler-Poincare characteristic of a compact even-dimensional Riemannian manifold as an integral of a polynomial of the components of the Riemann-Christoffel tensor, a generalization of the classical 2-dimensional case. Hopf himself did the hypersurface case in his thesis in 1925. In 1940 C. B. Allendoerfer and W. Fenchel proved the formula for submanifolds imbedded in a high-dimensional euclidean space, the former making use of the Weyl tube formula. Andre Weil was attracted by the formula and in a tour de force he and Allendoerfer proved it for a Riemannian polyhedron in 1943. Finally in 1944 Chern gave a simple intrinsic proof, making essential use of the unit tangent bundle and with no use of an imbedding. This turns out to be a special case of the characteristic homomorphism, now commonly known as the Chern-Weil theory. This can be described as follows. Following section 2 we associate by (8) to an invariant polynomial P E / (G) of degree s an exterior differential form of degree 2s in M . This form is closed and by de Rham's theorem defines an element of H 2s (M; R). This leads to a homomorphism ( 10)
w: /(G) -+ H*(M; R)
where the right-hand side is the cohomology ring of M with real coefficients; w is a ring homomorphism. Applying the construction to the classifying
224
663 CHARACTERISTIC ClASSES AS A GEOMETRIC OBJECT
space BG, we have (lOa)
Wo:
I (G) -+ H*(BG; R)
which is an isomorphism, if G is connected, compact, and semi-simple. This result gives a bundle-theoretic interpretation of I (G), being the cohomology ring of the classifying space. Our main theorem is: If the bundle is induced l7y the mapping f: M -+ BG,
(11)
then
(12)
W
= f*
0
Wo o
This therefore identifies a characteristic form with a bundle invariant. The cohomology class of a characteristic form in the sense of the de Rham theory is called a characteristic class. Consider the special case G = GL(q; C). Then L(G) is the space of all (q x q )-matrices and the curvature form Q is a matrix-valued two-form , defined up to the change ad(s)Q = sQs- 1
The differential form
det
(I + 2~
Q) =
1 + Cl (Q)
+ ... + cq (Q) ,
where Ck (Q) is of degree 2k , 1 ::::: k ::::: q, is well-defined on M. These are the so-called Chern forms and their cohomology classes the Chern classes. We will denote the latter by q(£). Even the first Chern class Cl (£) has played a role in various problems in mathematics. The Gauss-Bonnet formula arises from the case that the group G = S0(2n) and P is the Pfaffian of an anti-symmetric matrix. If the bundle is the tangent bundle of a compact orientable manifold M of dimension 2n, the characteristic class in question, when properly normalized, gives the Euler-Poincare characteristic of M . The expression of characteristic classes by characteristic forms is of increasing importance in applications. For instance, it is well-known that
225
664 SHIING-SHEN CHERN
characteristic classes play an important role in the index theory of AtiyahSinger. By using the heat kernel, this can now be refined into a result on the characteristic forms (cf. [2]). In finding a combinatorial formula for the Pontryagin numbers, Gelfand and McPherson have extended the above theory to the important case of piecewise linear manifolds [6]. Another important case is when the manifolds in question are complexanalytic. Then the exterior differential calculus has two operators a and with the usual properties. Bott and I have developed the "transgression" in holomorphic vector bundles, which seem to be useful in several problems, including value distributions in several complex variables and algebraic number theory [4].
a,
BIBLIOGRAPHY 1. D. Bao and S. Chern, On a notable connection in Finsler geometry, Houston J. of Math. (to appear). 2. N. Berline, E. Getzler and M. Vergne, "Heat Kernels and Dirac Operators, Springer, 1992. 3. W. Blaschke. "Vorlesungen uber Differentialgeometrie, Bd. 1," 1921; Bd. 2, 1923; Bd. 3, 1929. Springer. 4. R Bott and S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math. 114 (1965),71-112; also Chern, "Selected Papers, Vol. n," Springer, pp. 399-440. 5. S. Chern, "Complex Manifolds Without Potential Theory, 2nd edition," Springer, 1979. 6. I. M. Gelfand and R. D. McPherson, A combinatorial formula for the Pontryagin classes, Bull. Amer. Math . Soc. 26 (1992),304-309. 7. H . Hopf, Differentialgeometrie und tvpologische Gestalt, ]ahresberichte der Deut. Math. Vereinigung (1932) , 209-229. 8. F. Klein, "Entwicklung der Mathematik im 19. ]ahrhundert," Springer 1926-27; Chelsea 1956. h
226
665 Reprinted from The Sophus Lie Memorial Conference, Scandinavian Univ. Press.
Sophus Lie and diffferential geometry Shiing-Shen Chern 1
Abstract This is t he 150th anniversary of Sophu::; Lie (l Scl2 -99). I think he was a gr cH t mathema.tician even without Lie group::;. In t.he flllluwillg I shall
discu,;s two of his contributions to differential geometry which haw dcYeloped and promiscd to hav(' a futmc.
I
Contact Transformations and Lie Sphere Geometry
1. Contact transformations Let lH h e a manifold of d imension n, T*!vI its cotang(,nt Imndlc. a nd PT' JI it:' projectivized cotangent bundle. It is well kllO\\"ll that PT"_1I has a cont act st ru cture, i.e. , a linear differential forlll n. defi ned up to a factor. sllch that 1 0: 1\ (do:t - i=- 0, ( 1) the latter meaning tha.t the left-h a nd side is never zero. A local diffeomorphism of PT' lII preserving the contact structure is called a cont ac t transformation. If 1'\ i ::; i ::; n, are local coordiuates in .11 . \n' ca n take a = d.r" - P1d.r l _ •.. - Pn _l dxn-1, so that :z;i, 1)" , 1 ::; (\ ::; 17 - l. are local coord inates in PT* 111. Then a contact transformation ca n be written
I
1 ::;
Po satisf~' ing
0:,
{3 ::; n - 1,
1 ::; i, j ::;
11
the co ndition
I I tl _ . . '-PnI -I d X In- I = P(d X n -PI di dJ: In -Pier X
_ ..
· -Pn-Id.l' n- I ) . (:3)
lOepartJllent of :-- Ia( hemat ic5, University of California, Berkeley, California 9~""20 . \York dOlle with partial support of National Foundation grant 01\IS90- 01089.
129
666 p being a function of xj, P{3.
When we have a point transformation in (2), i.e., when X'i are functions of x j only, equation (3) determines p~ as functions of x j ,P{3. Thus a point transformation induces a contact transformation, but contact transformations are more general. Lie wrote a book on contact transformations. whose first volume, coauthored by G. Scheffers, appeared in 1896. The second volume was not finished; three chapters of it, edited by F. Engel, were published as a "Nachlass" in the l\latilematische A.nnalen in 1904. Contact transformations have a broader effect on geometrical structures. For instance, in the (x, y)-plane all the differential equations of the second order Y/I = F( .r. y. y I ) (4) are equivalent under contact transformations . as are the Finsler integrals
.I
F(.T:y.y')d.r.
(5)
In higher dimensions all first-ord er partial differental equations
( . aZ) = O.
(6)
F x' , z, axi
are equivalent under contact transformations. Thus it would be interesting to study the effect of contact transformations on a second-order equation i
a2Z ) = o.
az
1 :S i . j :S n,
F ( x , z, a----'" aa J.
x'
xt x
(7)
An important example of a contact transformation is the Legendre transformation. In three dimensions it can be written x'
= -q,
y'
= P,
z'
=z-
.rp - yq .
pi
= y,
q'
= -x .
(8)
Geometrically the transformation is based on the null system
-yx' + xy' +:' - :
=
o.
(9)
The Legendre transformation has many applications to nonlinear partial differential equations.
130
667
2. Lie sphere geometry Lie proved the remarkable theorem:
All contact transformations in the n-dimensional Euclidean space En which carry spheres into spheres form a finitedimensional group . The theorem is local, and spheres are to be understood in a generalized sense, i.e., oriented spheres including oriented hyperplanes and points, the latter being spheres of radius zero. To prove the theorem we introduce Lie sphere coordinates. For a sphere of center pEEn and radius r its Lie sphere coordinates are the homogeneous coordinates
(10) Xi+2
= p, .
Tn+3
= r,
1 :S i :S n,
where (11 )
They satisfy the relat ion
(x, x) =
-J.·i + ·d + ... + X; +2 -
X;+3
= O.
(12)
It can be shown that the tangency or contact of two spheres with the coordinates x and y is expressed by the polar relation
(.r. y) =
o.
(13)
This includes the generalized cases: for instance, the contact of a point and a sphere means that the point lies on the sphere, etc. From this the theorem can be proYed. The corresponding geometry is called the Lie sphere geometry. Its group is O(n + 1,2)/ ± I . We denote by p n+'2 the projective space with Xl," " X n +3 as homogeneous coordinates. Then equation (12) defines a hyperquadric Qn+l, whose points are generalized spheres of En Qn+l contains lines, but not lineare space of higher dimension. A line on Qn+l can be defined by the points Y, Z such that
(1". Y)
= (Z, Z) = (Y, Z) = O. 131
668 Its space is of dimension 2n - 1 and will be denoted by A2n-l. It has a contact structure defined by the one-form ex = (dY, Z)
(14)
Perhaps the most important objects in Lie sphere geometry are the Legendre submanifolds (15) which annihilate ex. They arise naturally from the unit normal bundle N n - 1 of an immersed manifold 111k ---; En; for x E Mk and v E Nn-l a unit normal vector at x, .-\(v) is the sphere tangent to Mk at x with center on the normal line in the direction of v. We call .-\ the Legendre map. It will likely playa role as important as the Gauss map in euclidean differential geometry. Given a Legendre submanifold (15) , we can define a second fundamental form and principal directions on N n - 1 , for details cf. (5) and T. Cecil and S. Chern, Dupin submanifolds in Lie spilere geometry, Springer Lecture Notes no.1369 , 1- 48 (1989), or Chern, Selected Papers, vol. IV, 269- 330, Springer, 1989. Generalizing the lines of curvature in classical surface theory, a priucipal curve is one which is everywhere tangent to a principal direction. Along a principal curve the lines .-\( v) form a developable. If all these developables are cones, the Legendre submanifold is called a Dupin submanifold. Consider the classical case n = 3, I.: = 2. The normal lines to the surface JI./2 have as envelope two focal sufaces. The Dupin condition means that they degenerate to curves. It is easy to show that they are conics. FollO\\"ing Dupin the surface itself is called a cyclide. The Dupin submanifolds are a natural generalization of the Dupin cyclides, and we have shown that they are best treated by Lie sphere geometry. In spite of their simple geometric characterization, in higher dimensions they demonstrate exotic phenomena and have not been COlllpletely classified. Their study is clearly an important problem in Lie sphere geometry. Dupin submanifolds are closely related to a global property of imbedded submanifolds in En , that of tautness. A compact submanifold k[k in En is called taut if the distance function dist( a, x), x E Mk, has the minimum number of critical points for all points a E En. Under 132
669 some mild conditions, Dupin submanifolds and taut submanifolds have been identified. The geometrical reason seems to be clear because both concepts are intimately involved with the spheres in En. Many important geometries can be built as a subgeometry of Lie sphere geometry. In fact, let Xo E pn+2 be a time-like point, i.e. (xo, xo) < 0, and call points the points x E Qn+l such that (xo: x) = O. This gives Mbius geometry, whose space has spheres and points. Similarly, let Xl E pn+2 be a space-like point, i.e., (Xl, Xl) > 0, and call hyperplanes the points X E Qn+l such that (Xl, X) = O. This gives Laguerre geometry, where we have spheres and hyperplanes. 'When both Xo and Xl are given, one time-like and one space-like, we get the non-euclidean hyperbolic geometry.
II
Surfaces of Translation and Web Geometry
3. Surfaces of translation Lie expanded an idea of I\longe and used surfaces of translation to study minimal surfaces. We recall that a surface of translation in E3 is defined by equations of the form (16) .r 1 = Ii (u) + 9i (V ) , 1:::; i :::; 3 whose parametric curves are called the directrix curves. For an oriented surface Min E3 described by the point x(u, v) we take the orthollormal frames xele2e3 , such that e3 is the unit normal vector. Then we call write
dx
+ W2 C2 ~(..JI + iW2)( e l
""leI
-
ie2)
+
complex conjugate.
The form (17)
defines the induced complex structure on 111. Using it , we have
Using the structure equations in E3, the exterior differentiation of this equation gives the interesting formula ( 18)
133
670 where H is the mean curvatue. This gives the theorem: M is a minimal surface (i. e., H only if the coordinate functions are harmonic. Suppose M is a minimal sufrace. Then the curve
y=
= 0) if and
J
ax
is holomorphic. Moreover, we have
(ax, ax)
=
(19)
O.
Such a curve is called isotrophic or minimal. Thus we have the theorem of Monge: A minimal surface is a surface of trallslation whose directrix curves are conjugate isotropi1ic curves. Lie exploited this theorem plus the fact that the equations of an isotropic curve can be put in a finite form involving an arbitrary function. He obtained in this way many concrete minimal surfaces. An example is the surface
In this way it is possible to get real minimal surfaces, but they are more complicated. It must have been a divine inspiration that he considered surfaces of double translation. These are surfaces which are surfaces of translation in two different ways. For such a surface there are four families of directrix curves. Take a direct rix curve and cut its tangent lines by the plane at infinity. The resulting curve of intersection does not depend on the choice of the curve in the family. Thus in the plane at infinity we have four curves. Lie's theorem says that these four curves belong to the same algebraic curve of degree four . Such a question drew his attention because mathematicians are interested in exotic objects as well as the general theory. If the surface is given in the non-parametric form z = f(x, y),
the condition for it to be a surface of translation is a partial differential equation of the form R( x, y, z, p, q)r
+ S(x, y, z, p, q)s + T(x, y, z, p, q)t = 134
O.
(21)
671
A surface of double translation satisfies an overdetermined system consisting of two such equations. The study of its integrability conditions is a formidable task, on which Lie was unexcelled. The theorem also has a beautiful proof by Darboux. Lie tried to extend the result to high dimensions, without complete success. The extension was finally completed by W . Wirtinger in 1938, using Chow coordinates. The general problem is best viewed by web geometry, which we will discuss in the following section.
4. Web geometry Web geometry was founded by W . Blaschke in 1926. It is the local theory of a number of foliations. I would like to discuss one of its fundamental problems , which contains Lie's theorem as a special case and which is even now not completely solved. In Rn we consider d foliations of co dimension one in general position. If xi , 1 :::; i :::; n, are the coordinates in Rn, they can be defined by the equations 1:::; ex:::; d, (22) where the functions are supposed to be smooth and satisfy some generality conditions. Such a geometric object is called a d-web of codimension one. An equation of the form (23) poses a condition on the web and is called an Abelian equation. The maximum number of linearly independent Abelian equations is called the rank of the web. In 1934 the author of this paper proved that the rank of a d-web of codimension one in R n has a universal upper bound 1
-rr(d, n) = 2(n _ 1) {(d - l)(d - n)
+ s(n -
s - I)} ,
(24)
where s is defined by s
== -d + 1 mod
n - 1,
O:::;s:::;n-2.
(25)
-rr(d, n) has been called the Castelnuovo number. It is the maximum genus of an algebraic curve of degree d in the projective space pn , 135
672
which does not belong to a hyperplane pn-l. In particular we have n(2n, n) = n + 1, and the corresponding Abelian equations can be written
(26) n+l::;a::;2n
These common expressions define III Rn + 1 a hypersurface of double translation and Lie's case corresponds to n = 2. Abelian equations are of fundamental importance in mathematics, because they generalize the addition formulas of classical functions, such as the circular and elliptic functions . From the point of view of web geometry a fundamental problem is that of "linearization" : when a web of maximum rank is given, can the local coordinates be so chosen that all the leaves of the foliations are linear spaces? The answer is not always affirmative. However, by a lengthy argument, Griffiths and I proved the theorem: In R n a normal d-web of codimension 1 and rank n(d, n), d ~ 2n, is linearizable. The issue is wether the "normality" condition can be dropped. Recently Robert Bryant showed that this is indeed the case if n :S 12 (unpublished). Very likely the normality condition is unnecessary for all n. For an expository account of web geometry, cf. reference 4.
5.
Conclusion
It is clear that transformation groups play an important role in all areas of mathematics. In the last decades there have been vigorous developments on abstract Lie groups and Lie algebras. They probably only touch the tip of Lie's work. A global theory of intransitive Lie transformation groups will heavily involve deep analysis and geometry, and awaits attention and progress. Lie has many interesting individual theorems, such as the Lie quadrics in projective surface theory, the Lie transforms of surfaces of constant Gaussian curvature, etc., etc., Fortunately, his "Gesammelte Abhandlungen" are available.
136
673
References [1] Sophus Lie and Georg Scheffers, Geometrie der Beriihrungstransformationen, Leipzig, 1896; second edition, Chelsea, Bronx, New York , 1977. [2] Sophus Lie , "Drei Kapitel aus dem unvollendeten zweiten Bande der Geometrie der Berhrungstransformationen" , Math. Annalen 59, 193- 313 (1904), aus dem Nachlass herausgegeben von F. Engel. [3] Sophus Lie, Gesmnmelte Abhandlungen, Leipzig, 1922-1960. [4] Chern, S., "Web geometry". Bulletin A . M. S. 6 . 1-8, 1982; also Chern, Selected Paper, vol. IV, 51- 58, Springer, 1989. [5] Cecil, T ., Lie Sphere Geometry, Springer 1992.
137
Math. Ann. 302, 581 - 600 (1995)
Amalen
@ Springer-Vcrlag 1995
Projective geometry and Riemann's mapping problem ShiingShen Chernls*, Shanyu Ji2,* MSRI, Department of Mathematics, University of California, Berkeley, CA 94270, USA Department of Mathematics, University of Houston, Houston, TX 77204, USA Received: 1 March 1994 /Revised version: 8 September 1994
1 Introduction A main purpose of this paper is to point out the rble that projective geometry plays in complex function theory. In one complex variable a cardinal fact is that the group of all holomorphic transformations leaving the unit disc invariant is a group of linear fractional transformations. Also the Schwarzian derivative is a projective invariant. In Cn with the coordinates zg, 1 a $ n, consider a domain D bounded by a real-analytic hypersurface
The work of Bochner-Fefferman P o ] [Fe] relates the biholomorphic geometry of D to the CR-geometry of its boundary dD. B. Segre suggested the consideration of the complex hypersurfaces
regarding the w a as parameters. With these hypersurfaces as a generalization of hyperplanes Hachtroudi [HI in 1937 introduced a projective connection or a generalized projective geometry. Such a projective structure will be within the holomorphic category and should be very useful, but we do not know how to define one for general domains. Instead we will restrict ourselves to a particular problem, viz., the problem of a generalization of Riemann's mapping theorem to higher dimensions. The Riemann mapping theorem asserted that any simply connected domain in C that is not the whole plane is biholomorphic to the unit disc. However for domains in C" where n > l , the situation becomes very subtle and *~esearch partially supported by National Science Foundation
582
S.-S. Chern,
S. Ji
complicated. As a well-known example due to Poincarb, the unit polydisc in C2 is not biholomorphic to the unit ball B2. Even for domains with smooth boundaries, the complex ellipsoid D := {(z, IV) E C211z12+ lw14 < 1) cannot be biholomorphic to the ball B2 [B, p. 2371. There are a few results around this problem: Wong-Rosay's theorem about a domain with C2 smooth boundary and with a transitive automorphic group [Wo] [R], Fridman's approximate Riemann mapping theorem [Fr], Stoll's theorem about strict parabolic manifolds [Sto], Bochner P o ] and Feffeman's works [Fe] about reduction from a domain equivalence problem into a boundary equivalence problem. Besides, the result of Cartan-Chem-Moser [Ca] [CM] can be regarded as a local version of the Riemann mapping theorem. For more information, see the nice survey papers by Bell and Stanton [B] [St]. In 1975, the first author showed [C] that a projective connection, called Hachtroudi connection in this paper, can be intrinsically defined over the Segre family associated to a real-analytic hypersurface in C n f ' ,which is a generalization of an early work of Hachtroudi [HI and of the classical projective geometry. The solution of local equivalence problem for Segre families was also found in [C] (cf. [F]), which is an analogue of the solution of the local equivalence problem for real hypersurfaces in Cn+' by Cartan-ChernMoser [Ca] [CM]. And as a consequence, one has a standard local property: given a Segre family, its Hachtroudi connection is flat if and only if it is locally isomorphic to the Segre family of the unit ball B"" (i.e., complex hyperquadric), which is an analogue of the similar local result in the CR-manifold theory. In the CR-manifold theory, a domain D in en+'whose Cartan connection is flat over aD may not be globally biholomorphic to B"+', by the example of Bums-Shnider P S I . However, for a Segre family AD,when its Hachtroudi connection is flat, we obtain the following global result.
Theorem 1. Let D c Cn+' be a bounded domain whose boundary
is a smooth connected recrl-crnalytic hypersurfizce, where r is a real-analytic jicnction defined on Cn+'.Then the jollowing three statements ure equivalent: (i) There is a hololnorphic mapping g : Cn+' + Cn+'such that the Jucobiun det(Dg) = constant and that the restriction g : D -,Bn+' is biholomorphic. smooth, liftable, non(ii) The Segre fimily AD is defined on degenerate and locally isomorphic to the Segre family MBn+1,and there is one +U(O) such that g(dD n U(0))c dBn+'n(l(o) biholomorphic mapping g : qo) and such that det(Dg) = constant, where U(o) and U(o) are open subsets in Cn+' with aD n U(o)0 and dBn+'n(l(o, 0. (iii) is dejned on C2n+2, smooth, liftable, non-degenerate, and the associated Hachtroudi connection is flat, and there is one local biholomorphjc g us in (ii). mapping Moreover, from (ii) or (iii), the local mapping g can be extended into the global mapping g in (i).
-
+
+
676 Projective geometry and Riemann's mapping problem
583
If we drop off the conditions det(Dg) == constant for the local mapping g in (ii) and (iii), its global extension g may not be holomorphic on ern + l . For example, for 0 < lal < I, the function g(z) := ~ :z : BI -+ BI defines a local CR-isomorphic mapping near a boundary point, but g cannot be extended as an entire function. The remaining question is: if D is a simply connected domain of realanalytic boundary whose boundary defining function is only defined on a neighborhood of 15, and if the Hachtroudi connection is flat wherever it is defined, is there a biholomorphic mapping from D onto Bn +!? Maybe some stronger conditions on D are necessary. More generally, how to define a projective structure over a domain with smooth boundary? There is a restricted notion of equivalence of domains: either two domains are equivalent if there is an biholomorphism of all of ern+! whose restriction to one domain maps it onto the other, or two domains are related if there is an entire mapping whose restriction to one domain gives a biholomorphic mapping onto the other domain, and extend this relation to an equivalent relation. Theorem I seems to be related to the second of these. Let us discuss some ideas about the proof. Since we have a local isomorphism, a naive approach is to make holomorphic extension. However the extension situation here is delicate. In fact, Bums and Shnider [BS] constructed a hypersurface in er2 which is real-analytic, compact, strictly pseudoconvex, spherical (i.e., aD is locally CR-isomorphic to the unit sphere), and a nonextensible local holomorphic map from the unit sphere into this hypersurface. Therefore, it should be very difficult to extend a mapping from .A8"+1 to j t D . SO we approach our problem by making holomorphic extension from .AD to .A8 "+1. To show Theorem 1, a basic idea is to regard Theorem I as an analogue of the following result:
f
Lemma 1. (Pincuk) [Pi, I, theorem 2]. Let aD c ern+! be a compact, simply connected, spherical, real-analytic hypersurface. Then aD is CR-isomorphic to aB n +!. We are going to present a new proof of Lemma 1 (see Sect. 5), and Theorem 1 will follow in a similar way. In our new proof of Lemma 1, the following classical Poincare theorem played an important role: Lemma 2. (Poincare). Let U C ern+! be an open subset with Un aBn+ 1 *0. Let f : U -+ ern + 1 be a holomorphic mapping such that
f(U n aBn + l ) C aBn + 1
r
Then is constant or the form
r extends to be an automorphism of Bn+
1
which is in
S.-S. Chem, S. Ji
584
where
For n = 1, Lemma 2 was first proved by PoincarC [PI in 1907. For n > 1, this result was proved by Tanaka [T, I]. It was also rediscovered by Pelles [Pel and Alexander [A]. In order to prove Theorem 1 along the same line, we shall establish the following generalization of PoincarC theorem for A B n + t . Theorem 2. Let U c Let F : U -, F(U)C form of
be uiz open connected subset with U n ABn+l +O. a biholo~norphicmapping wlziclz is in the
a2"+'be
such t11ut
F(U n A B n + ~) c ,KBn+~ . Tllen F extends to be a birutioizal mapping from IdB.n+~ to itself such that
where all a;, 6:: E C are complex numbers. Throughout this article small Greek indices will run from 1 to n and the summation convention will be adopted. The authors wish to thank James Faran for giving helpful comments and suggestions to improve our first version of the manuscript. In particular, the original proof of Theorem 2 was long and the proof presented here is suggested by Faran.
2 Segre family a Segre family we mean a complex analytic hypersurface in an open subset
c an+'x c"", defined up to a complex analytic automorphism in each of the two factors. Denote by zJ, lk,1 5 j, k $ n + 1, respectively the coordinates Q
Projective geometry and Riemann's mapping problem
585
in these spaces. We suppose the hypersurface A C 12 be defined by an equation
where the left-hand side is a complex analytic function in its two arguments. Suppose A is smooth. On A let
Let 8, f30L (resp. 8,OU) be linearly independent holomorphic I-forms in the zspace (resp. c-space). They are defined up to a transformation
where u 9 0, det(u;) 9 0, det(v!)
+0 .
Then we have defined a G-structure, where G is the group of all complex matices
(" &) uu
+
with u 0, det(u;)
0
vu
+0, der($) +0. We have
for some holomorphic functions
&. The condition
is invariant under the change (2). The Segre family is called non-degenerate, if (3) is satisfied. As a basic example of Segre family, we consider any domain D c Cn+' whose boundary is given by a real-analytic hypersurface
where r is a real-analytic function defined on C n f l .Then this associates a Segre family
2 contains a neighborhood of where the 12 is some open subset in C Z n f which we say that the Segre family AD the conjugate diagonal. When 12 = is defined on C2n+2.For example, when D = Bn+' is the unit ball, ABn+, = + wq - 1 = 0) is defined on C Z n f 2 .
679 S.-S. Chern, S. Ji
586
We denote
(5) A Segre family A is called liftable if there is a discrete set Ll C ([,,+1 such that for any compact curve y C ([,,+1 - Ll started from p, for any point PEA with "Ttl (P) = p where "Ttl (z", W, (II. '1) = (z'r, w) is the projection, there exists a compact curve yeA started from P with "TtICY) = y. We need this definition for some technical reason. This definition is equivalent to a local one: there is a discrete set Ll C ([n+ I and a constant f: > 0 such that for any ball B( q, f:) in ([n+ I with center q and with radius 1'" any compact curve y E B( q, e) - Ll started from a point p, any point PEA with "Tt1(P) = p , there is a compact curve yeA started from P with "Ttl CY) = y. As examples, .;I{8"+1 is liftable with Ll = {a}; if g : ([n+1 --+ ([n+1 is a locally biholomorphic mapping and D = g-I(Bn+l), then AD is liftable with Ll = g-I(O). The non-degeneracy of .;I{D implies that the real hypersurface iJD is nondegenerate. If in addition the Hachtroudi connection is flat, the Cartan connection over the real hypersurface iJD is flat [F, theorem 5.11], and thus iJD is spherical. For a general domain D, .;I{D may be degenerate (See Example 2).
3 Local criterion for non-degeneracy Let DC
([n+1
be a domain such that AD is smooth. Let (z(O)a w(O) r(O) '1(0» E AD P(0) .= . " ~" ,
with (6)
By the implicit function theorem, we have a unique holomorphic function p near the point (z(O)a , (~O), '1(0» such that
We may replace the equation r iJ Pa := iJ;'
iJ
p~ := iJ~ '
=0
iJp pll := iJ(/I'
by w - p(z'. , (a, '1)
p~ : =
iJ2 P iJz"iJ'1'
= O. Let iJ2 P
p~ := iJzaiJ(~'
iJ2 P Pa.11 := iJza.iJz/l
as functions of z", (a., '1. Note Pa.~
= Plla.·
(7)
By (6) and the implicit function theorem again, we have a unique holomorphic function '1 = '1(za. , w, (a.) near (z(O)a., w(O), (~O». Then the above functions Pa., P~ , p P, p~, Pp and Pa.P can be restricted on AtD as functions of variables z«, w,{o:.
680 Projective geometry and Riemann's mapping problem
Consider dpa. = P~/ldzfl + p~d (fl + p~d 1'/ on dw = Pa.dz'" + pfid(/I + p"dl'/, we have
dll
587 ([2n+2.
Since on AD we have
dw - Pa.dza. - p11d(fI
= --~---=--.!.. pq
,
and
dpa.
P~Pfl) p~ = ( P~/I- - dz lI + -dw+ pq p"
( p~ - pZpfl). - d~fI ' pq
(8)
Here we used a fact pq(p(O»=t=O by (6).
Proposition 1. Let Dc
([1/+1
be a domain such that AD is smooth. Let
p(O) := (z(O)a., w(O), (~o>, 17(0» E AD satisfying (6). Theil the /ollOll'ing three statements are equivalent: (i) ./ltD is non-degenerate at p(O). (ii) 8, 8a., 8a. are linearly independent near p(O), where (9) ( 10) (11)
lind
(iii) The determinant d etC q~ )(p(O» =t= 0
(12 )
Proof By (6), q~ is well-defined. By (8) and (7) we have d8
= i8a. /\ (~~ 8 + q~d(p) = i8a. /\ 8a. . lp"
(\3 )
(ii) ¢:} (iii): 8, 8a., 8~ are linearly independent near p(O) is equivalent to 8/\8 1 /\ .. ·/\8 n /\ . .. /\8 1 /\, . ·/\8/1 = i det(q~ )dw/\dz l /\ • • ·/\dzn /\d( 1/\' . ·/\d(1/ =t= 0, at p(o) . (ii) =? (i): By (13). (i) =? (iii): Because the non-degeneracy is independent of the coframes.
o We make a remark. For any point PE'//{D with (ra.,rw)=t=(O) and (rfi,rq)=t= (0), if (6) is not satisfied, we can take an isomorphic linear transformation of ([2n+2 in the form
F(za., W,(/Iol'/) = (a~za.
+ bw,cxza. + dw,Aa.(a. + BII,Ca.(a. +Drf) = (z*a., w*,(~,1'/*) (14 )
S.-S. Chem,
588
S. Ji
where a,, b, c,, d , A', B, Cu, D are constant. We can choose suitable a,, b, c", d , ) (6) Au,B, Cu,D such that the point P* = F ( P ) E F ( A D ) = M F ( ~satisfies under the new coordinates system. We make another remark. When ( 6 ) is satisfied, we can use either the i,) or the coordinates system (zu,W , pa). Let P E AD coordinates system (zU,+v, satisfying ( 6 ) and ADbe non-degenerate. Consider the holomorphic mapping
whose Jacobian matrix is
From ( 6 ) , (za,w,[,) is a coordinates system near the point P in AD.Restricting the mapping H on ADnear the point P = (zu,w,[,, v), it induces a holomorphic map (z', t v , [,) w (zz,+v, p,) whose Jacobian matrix
is non-singular thanks to Proposition 1 . So near the point P we can use new coordinates (zU,w,p a ) instead of the coordinates (zu,w,[,). Under this new coordinates, the 1-forms in (9), (10) and ( 1 1 ) become
where pup = pb, are holomorphic functions. By the way, the coordinates system (zU,+v,[,)was used in F], while (zu,w,p,) was used in [C]. The formula ( 1 5 ) is the same as the one [C, ( 3 ) ] where pap was denoted by rap Example I . Let D = Bn+' C C n f ' be the unit ball. Its Segre family ADis also called the complex lzyperyuadric. Let P E M D be a point. When ( 6 ) is satisfied at P, i.e., w+0 and v+0, i.e., zu[,+l, we have
9, = dp, = -
Here
ia i(dw - ppdzb) i(1 - Z V [ ~ )
+
~ ( 6 2-; rS[,61; zY5,) dry, and ( 1 -z'CI2
+ zb[,) 4i= - lv(6i -( 1 ?[,6! -z'TO2
Projective geometry and Riemann's mapping problem
589
Exumple 2. Let D := { ( z , w ) E C211z(2p+ IwI2P < 1)- where p E Z . D is ' the unit ball with respect to the norm Il(z,w)I12 = IzI2P + llv1*P in the Hilbert space C 2 . By functional analysis, it was known that D is biholomorphic to B2 if and only if p = 1. Let us show that ADis degenerate at some point (z,.?!) when p > 1. Since r = zprp + wPqP - 1, we have
Take P = ( 0 , 1 , 0 , 1 )E AD. Then r,(P) w=
+0, and rq(P)=t=0. Write
(1 - zpp)'lp
near P .
r]
By direct computation, we know
When p 2 2, q t ( ~=) 0. By Proposition 1 , ADis degenerate at P
4 Hachtroudi connection Let D c C"" be a domain such that ADis smooth and non-degenerate. We have defined a 1-form 8 on dflD by (1). Following [CM] [C], we define a 1-form w:=ue, U * O , U E C , (16) and we consider the holomorphic bundle &' over ADdefined by
Then the I-form o is intrinsically defined on B and hence a global holomorphic I-form on 8. For any point P = (zZ,l.v, la,r ] ) E AD,by the non-degeneracy, there is a neighborhood U of P in ADon which there are holomorphic 1-forms €la and 0, satisfying dB G dBa = 0, mod(B,Ba), dB dBa = 0, mod(B,B,), dB=iBar\Ba+Br\4
-
for some holomorphic 1-form 4. Let us define a holomorphic principal bundle g over dlu, where {(P,uB)lP E U,u+O). Consider all sets of I-forms on &'Iu,
such that (a) o is as above.
&'Iu
:=
S.-S. Chern, S. Ji
590
(b) ma is the pull back under projection of some multiple of 0" (i.e., oa= u;Ofi uaO for some holomorphic functions u; and ua). (c) w , is the pull back under projection of some multiple of 0, (i.e., wa = P vaO v!0p for some holomorphic functions va and v,). (d) dw = i o a A w , + w A 4. Then the set Y of all such (a,ma,ma,4 ) is a principal bundle over dlu with the structure group GI given by the complex matrices
+
+
with uivf = 6!. Therefore the I-forms
+
where u 0, uivf = us!, are globally and intrinsically defined over Y. The bundle Y -+ &Iu is indeed defined over Y -+ I. We denote YD by this principal bundle. The forms wu, o, and 4 are globally defined on YD.
Lemma 3. [Cj. There m e n2 + 2n + 1 more uniquely determined holomorphic 4~ and $ on Y D such that the forms I-fbrms @, r#$,
are linearly independent and satisjj
684 Projective geometry and Riemann's mapping problem
59 1
where
und
:s!i = RE, = T:' = Qz = 0. These forms @, $;,4Pund $ are completely determined these conditions.
dy
The first (n + 2)2 - 1 equations above are called the structure equations, and the forms @,! @", @,, Y are called curvatures. We denote
These forms a, in fact define the Huchtroudi connection X D , which is a Cartan connection, on the principal bundle gD (cf. IF, proposition 4.151 [C]) given by: = 2ioa , Gf = 20.1, TI: = mu,
i
?I:
- 6 : ~ := $!,
X:
= -i&,
4:; - X:
x ; + ~= !@,
=4 ,
TI:+,
(23
= -$$
W e say that the Hachtroudi connection TID is flat i f the curvature forms @,! @", @, Y all vanish on gD. Given two smooth, non-degenerate Segre families MD, and A D , , and given points P I E AD,and P2 E AD,.Let gDIand g D , be the associated principal bundles, and let and c y ) , j = 1,2,. . .,(n + 2)' - 1, be the invariant forms over gD,and Y D , , respectively. A local biholomorphic mapping F : J ~ D -+, MD, with F ( P I )= P2 is called a local isomorphism if for some (and hence every) coframe { 8 ( 2 ) , 0:')) in the G-structure of A D , near P2, the coframe {O(I) = F * @ Z ) , =~ * e ( ~ 6:')) ~ = ,F*OY)) is in the G-smcture of AD, near P I .
4')
685 S.-S. Chern, S. Ii
592
Lemma 4. Let
"I( D, and "I( D: be two smooth non-degenerate Segre families. Suppose PI E ,/I(D, and P 2 E A D2 . Let F be a local isomorphism from AD, to "I(D2 with F(P I ) = P 2. Then F is locally the restriction of a biholomorphic mapping <en + I x <en + I -+ <e n+ 1 x <e n+ 1 given by
(z~, w, (~, '1)
f-+
(1(1 )(z~, w), . . . ,J(n+ I )(z~, W ),J( 1)( (~, '1), . .. ,J(n+ 1)( (~, '1») .
Proof If both the points PI and P 2 satisfy (6), Lemma 4 is a known result [F, Proposition 2.7]. If (6) is not satisfied for PI, or P 2, or for both the points, we can take a biholomorphic linear transformations HI and H2 in the form (14) to get a commutative diagram "I(D,
!
F
-+
"I(D2
!
H,
H2
F-
"I( H,(DIl
-+
"I( H2(02)
so that (6) is satisfied for the points Hl(Pd E HI (,/I(D, ) = AH,(o'), and H2 (P 2 ) E H2 ("I(0,) = "I(D2' Thus by [F, Proposition 2.7] again, F* as well as F = H 2- 1 0 F* 0 HI is in the desired form. 0 Notice that by Lemma 4, Theorem 2 can be restated as: any local isomorphism F : ,/1(8"+' -+ "I(8n+' can be extended to be a birational mapping of ,,1(8"+' .
A local isomorphism F : ,,1(0, -+ ,/1(0: induces a local biholomorphic map. en::.. &/ &/ • h en::* a (2) -- a (I). I , 2 , ... , ( n + 2)2 -. I We pmg :II' -~ D, -+ -~ D2 Wit :II' j ,j j also call such g; a local isomorphism. Conversely, any local isomorphism g; : i5.!J D, -+ i5.!JD2 induces a local isomorphism F : ,,1(0, -+ ,,1(02'
5 A new proof of Lemma 1 The following lemma is standard from Frobenius theorem and the uniqueness of the fundamental theorem of ordinary differential equations.
Lemma 5. Let aj and ai be l-forms on n-dimensional real (resp. real-analytic, or complex) manifolds M andM, respective/y, j = 1,2, ... ,n. Let p E M and p EM. Suppose al 1\ .. . 1\ an =F 0 and al 1\ .. . 1\ all =F 0 in neighborhoods of p in M and of p in M, respectively. Suppose that on some neighborhoods of p and ft we have dak
= Ckj/aj 1\ ai,
dak
= Ckj/aj 1\ a/
for all k,j, I with some constants Ckj/' Then there exists a unique local smooth diffeomorphic (resp. real-analytic diffeomorphic, or biholomorphic) mapping F with F(p) =p and F*aj = aj, i = 1,2, ... ,n.
Let D C <e n+ 1 be a domain such that aD is a smooth, non-degenerate, real-analytic hypersurface. There is a natural CR-structure on aD. By [CM], we can define the half-line bundle E over aD, the principal bundle Y,JO over E,
Projective geometry and Riemann's mapping problem
593
and (n + 2)2 - 1 invariant I-forms uj on Ym with the structure equations and curvatures. We also define the Cartan connection nm (Comparing 8,YD, C j , XD defined in Sect. 4). We want to give another version of Lemma 2 (Poincart theorem) below in terms of the principal bundle YdD.
Lemma 6. Let dBn+' c C"" be the uizit sphere und YaBn+l the ussociated principul bundle. Let p I , ,p2E K,Bn+~be any t~vopoirzts. Then there is u unique I $12,g * a j = a,, V j d~ffeomorphismF : YrlBn+1-' Y,lBn+~ sutiSfYiizg F ( ~) = and the mupping 9 is induce0 by a unique uutoinorphism f E Aut(Bn+') kvlzich is in the form f (z", *v) = (f (')(zU,w), .. . ,f (n+')(zU,w))
Proof: It was well known that na~.+I is flat. Then the structure equations are the same in a neighborhood Ul of y l in Y , > B n + ~ and in a neighborhood U2 of ~2 in Yi3Bn+l. Apply Lemma 5, there is a unique diffeomorphism 9 from UI to U2 with F(pI) = ,p2 and 9 * o j = aj, j = 1,2,. . .,(n + 1)2 - 2, where we may shrink Ul and U2 if necessary. Then 9 induces a unique local CRisomorphism f from (dBn+', p l ) to (dBn+',pz), where pi is the projection of pj into (En+', j = 1,2. Since dBn+' is real-analytic hypersurface, f can be extended to a local biholomorphic mapping. By Lemma 2, f must be extended as an automorphism of B"+'. It was well-known that every component or f("+') of f is a linear fractional map as above. Therefore such f induces a local isomorphism F : Yi,Bn+l + YaB,,+~which coincides the previous mapping on Ul. Lemma 6 follows. Lemma 7. Let dD c c"" be a sphericul, real-anulytic Izypersurface, und ;I be a compact curve started jiom u point p on dD. The11 uny locul biholomorphic mupping g defined on a neighborhood U of p in a"+' with g(dDnU) c dBn+' can be extended ~~Iong y as u locally biholonzorplzic mupping.
Proof: Write y : [0, 11
-
do, t
++
y(t) with y(0) = p .
Suppose g can be extended at all points y(t), 0
4 t
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