A mathematician and his mathematical work : selected papers of S.S. Chern

" (n - 2>" - 1) '" (n - 2k - 1) ,

k

=

[~J

0 1 '" " '2

1

If n is even, say = 2p, then we have

d8 p _ 1 = -Yp_1

where

If n is odd, say = 2q

+ 1, then d8 q_ 1 = -Y q-l

•

But in this case we have also

so that

We define* 1).2p +}..>,,!
if n = 2q

+ 1 is odd,

or, for a formula covering both cases, (9a)

(_l)n

II =2 n ?THn-

l)

[Hn-I)]}. (;

1

(-1) >..!r(Hn-2>"+1»
* Our present form n differs, in the case of even n, from the corresponding one in our previous paper by a sign. There are several reasons which indicate that the present choice is the appropriate one .

124

677

CURVATURA INTEGRA

and (10)

n=

I

(-I)P

22

1

P1rP

I

p.

L E'1···'nn'1'2 ... n'''-1'.'

o

if n

, if n

= 2p

is even

is odd.

Our foregoing relations can then be summarized in the formula -dII = n.

(11)

2n l

We remark that II is a dllIerential form of degree n - 1 in M - • Over a simplicial chain of dimension n - 1 in M 2n- 1 whose simplexes are covered by coordinate neighborhoods of ~n-l the integral of II is defined.

§2. Remarks on the Formula of Allendoener-Weil .Ai3 we have shown before, the formula (11) leads immediately to a proof of the formula of Allendoerfer-Weil. We shall, however, add here a few remarks. LetObe a point of Rn, and let Oe~ .. . e~ be a frame with origin at O. A point P of R n sufficiently near to 0 is determined by the direction cosines Ai (referred to Oe~ .. . e~) of the tangent of the geodesic joining 0 to P and the geodesic distance s = OP. The coordinates Xi of P defined by (12)

are called the normal coordinates. In a neighborhood of 0 defined by s shall employ s, Ai to be the local coordinates, where

~

R we

(13)

.Ai3 the components of a vector \) through P we shall take the components referred to Oe~ ... e~ of the vector at 0 obtained by transporting b parallelly along the geodesic OP In the neighborhood s ~ R of 0 let a field of unit vectors \) be given, whose components are differentiable functions of the normal coordinates except possibly at O. The forms k , k ~ 1, being at least of degree two in dx" there exists a constant M such that

x"

k

~

1,

where S is the geodesic hypersphere of radius s about O. Let I be the index of the vector field at 0, which is possibly a singular point. By Kronecker's fonnula we have (14)

I

=

o~ n-l

1

WIn"

S

• Wn-l . n ,

125

678

SHIING-SHEN CHERN

where On_l denotes the area of the unit hypersphere of dimension n - 1 and is given by 21l.!n

On-l = r (~n) .

(15)

It follows that there exists a constant Ml such that

or that (16)

I

= (-It lim 8 -0

1

II .

S

Let the Riemannian manifold R n be closed. It is well-known and is also easy to prove directly that it is possible to define in R n a continuous vector field with a finite number of singular points. Draw about each singular point a small geodesic hypersphere. The vector field at points not belonging to the interior of these geodesic hyperspheres defines a chain in M 2n - 1 over which n can be integrated. From (11) and (16) we get, by applying the formula of Stokes, (17)

where I is the sum of indices of the vector field. Hence the sum of indices of the singular points of a vector field is independent of the choice of the field, provided that their number is finite. By the constmction of a particular vector field, as was done by Stiefel and Whitney [2], we get the formula (18) n

where x(R ) is the Euler-Poincare characteristic of Rn. In particular, it follows n that x(R ) = 0 if n is odd. The same idea can be applied to derive the formula of Allendoerfer-Weil for differentiable polyhedra. Let pn be a differentiable polyhedron whose boundary n iW is a differentiable submanifold imbedded in Rn. Let apn be orientable and therefore two-sided. To each point of apn we attach the inner unit normal vector to apn, the totality of which defines a submanifold of dimension n - 1 in 2n M - 1• The integral of II over this submanifold we shall denote simply by

1 is

II.

Then the formula of Allendoerfer-Weil for a differentiable polyhedron

a/,n

P"

(19)

where x'(pn) is the inner Euler-Poincare characteristic of pn.

126

679

CURVATURA INTEGRA

To prove the formula (19), we notice that the field of unit normal vectors on iJP" can be extended continuously into the whole polyhedron P", with the possible exception of a finite number of singular points. Application of the formula of Stokes gives then

r n = - lap" r II + (-1)" J ,

jpn

where J is the sum of indices at these singular points. That J = (-1)" x' (pn) follows from a well-known theorem in topology [3]. It would also be possible to deduce this theorem if we carry out the construction of Stiefel-Whitney for polyhedra and verify in an elementary way that J = (-1) nx' (P") for a particular vector field. §3. A New Integral Formula m

Let R be a closed orientable differentiable (of class ~ 3) submanifold of dimension m ~ n - 2 imbedded in Rn. The unit normal vectors to R m at a point of R m depend on n - m - 1 parameters and their totality defines a submanifold of dimension n - 1 in M 2nthis submanifold.

l

Denote by

•

f

nm

II the integral of II over

Our formula to be proved is then

(20)

where the right-hand member stands for the Euler-Poincare characteristic of R m , which is zero if m is odd. As a preparation to the proof we need the formulas for the differential geometry of R m imbedded in Rn. At a point P of R m we choose the frames Pel" . en such that el, ... , em are the taI'fgent vectors to Rm. We now restrict ourselves on the submanifold R m and agree on the following ranges of indices 1

~

ex, {3

~

m,

m

+1~

r, s

~

n,

By our chQice of the frames we have Wr

=

0,

and hence, by exterior differentiation,

L a

which allows us to put (21)

with (22)

W raWa

= 0

1 ~ A, B ~ n -

1.

127

680

SHIING-SHEN CHERN

Consequently, the fundamental formulas for the Riemannian Geometry on R m , as induced by the Riemannian metric of R n , are dWa

=

L

=

L

WfJWfJa,

fJ

(23)

m

dWafJ

Wa-y

-y~l

w,,~

+Q

afJ ,

where (24)

To evaluate the integral on the left-hand side of (20) we introduce a differentiable family of frames Pal'" a" in a neighborhood of R m , satisfying the condition that aa = fa and that exactly one of the frames has the origin P . The relation between the vectors am+! , ... , a" and em+l , . .. , e" is then given by the equations

er

(25)

L

=

u" n, ,

s

where u" are the elements of a proper orthogonal matrix. In particular, the quantities U nr = U r may be regarded as local coordinates of the vector e" with respect to this family of frames. We now get all the normal vectors to R m at P by letting U vary over all values such that (U )2 = 1. The forms W"a, W"r which occur in IT can be calculated according to the formulas

Lr r

r

W"a

= de,,· fa =

L

U r Ora,

W",

= de,,·e, =

L

du. ·u ra

(26)

n

8

+ L

U.UTtOat,

.s,t=m+l

where the product of vectors is the scalar product and where we define (27)

It is evident that k

For k

~

= 0,

2k> m.

m/ 2 we have by definition k

=

L

EAI·· · An_IOAIA • • • • OA2k-IA2k WA2k+I" ••• WAn_I" •

Each term of this sum is of degree m in the differentials of the local coordinates on R m and of degree n - m - 1 in the differentials du,. It follows that the non1, ... ,n - 1 occur among vanishing terms are the terms where the indices m A2k+! , • . • , A n - 1 • We can therefore write

+

128

681

CURVATURA INTEGRA

where A..- m _ 1 is the surface element of a unit hypersphere of dimension n m-1. The integration of ~I: over R m is then carried out by iteration. In fact, we shall keep a point of R m fixed and integrate over all the unit normal vectors through that point. This leads us to the consideration of integrals of the form }..,+1 }.~A I>-m-1 U",+1 .. , U..

J

over the unit hypersphere of dimension n - m - 1. It is clear that the integral is not zero, only when all the exponents >'m+1, ..• , >. .. are even. But for the integrals obtained from ~k we have >.. = m - 2k. It follows that, if m is odd, we shall have

L

f

R"'

~k =

0,

and hence

This proves the formula (20) for the case that m is odd. More interesting is naturally the case that m is even, which we are going to suppose from now on. It was proved that [4]

(28) (n -

m)(n -

m

+ 2)

2>'m+J .. ·2>... )0..-m_l ... (n - m + 2>'...+1

+ ... + 2>' ..

- 2)'

where the symbol in the numerator is defined by (29)

0) = 1,

2>.) = 1.3 ... (2)' - 1).

To evaluate the integral of ~k over R m we have to expand the product

We introduce the notation

where the last symbol stands for a product of 8's, whose first indices are a2k+1 , ... , am respectively and whose second indices are respectively 2>'",+1(m + 1)'s, 2>'m+2(m + 2)'s, and finally 2>'.. n's. Let it be remembered that b.(k; >'m+1 , ... , >'n) is a differential form of degree m in R"'. Expanding ~k and using (28), we

129 682

SHIING-SHEN CHERN

shall get

1

(31)

( I r -1



Rm

k

=

-

(n - 2k - l)!On-m- l 2'!-=-ffl---;k:---(n--m -'-)-(n---n-l-+~2-)-..:..~.'::':--':' (n---2- k-'-- - 2)

where the summation is extended over all AT ~ 0, whose sum is ~ - k.

nail

It is now to be remarked that for the curvature forms of the Riemannian metric on Rffl we have to substitute Oar for War in the expressions (24). n being m the form on R whose integral over R m is equal to the Euler-Poincare characterffl istic x(R ) by the Allendoerfer-Weil formula, we have 12-

1 "" ...am(12a\a, = (-1) ~m 2m7rlm(.! ),L...fa\ 2m.

"" L...OalTOa ,T)

or, by expansion , _

(32) 12

l!m

k

1

= 2Tn 7r Im L ( - I ) k' L A' k=O . hm+t+ -· · +Xn= lm-k m+l·

1 ·· ·

A ,ll (k;Am+l , ···,An). n ·

By a straightforward calculation which we shall omit here, we get from (9), (31), (32), and (18) the desired formula (20). m So far we have assumed that m ~ n - 2, that is, that R is not a hypersurface n of Rn. In case m = n - 1 the unit normal vectors of R - 1 = R m in R n are, under our present assumptions concerning orientability, divided into two disjoint families. It is possible to maintain the formula (20) by making suitable conventions.

In fact, we suppose that the integrals

1n-\

II over the families

of inward and outward unit normal vectors are t aken over the oppositely oriented manifold R n - I • Then we lvJ.ve

where the integrals at the left and right hand sides are over the families of inward and outward normals respectively. If n is even, we have

If n is odd, we have

130 683

CURVATURA INTEGRA

Both cases can be considered as included in the formula (20). In particular, if n is odd and if R"-l is the boundary ijp" of a polyhedron P", we have also, by (19),

r

II

J(Rn-l)-

= x'(pn).

Comparing the two equations, we get

which asserts that the inner Euler-Poincare characteristic of a polyhedron in an odd-dimensional manifold is -! times the Euler-Poincare characteristic of its boundary, a well-known result in the topology of odd-dimensional manifolds. It is interesting to remark in passing that, so far as the writer is aware, the formula (20) seems not known even for the Euclidean space. §4. Fields of Normal Vectors

We consider the case that R2" is an even-dimensional orientable Riemannian manifold of class ~ 3 and R" a closed orientable submanifold of the same class imbedded in R2". By considering normal vector fields over R", Whitney [5] has defined a topological invariant of R n in R 2n , which is the sum of indices at the singular points of a normal vector field (with a finite number of singular points) over R". Let us denote by if; this invariant of Whitney. To prepare for the study of this invariant we make use of the discussions at the beginning of §3. To each point P of R" we attach the frames Pel· . . e2n such that el, .. . , en are tangent vectors to R" at P. Then we have, in particular, (33)

dW;j

=

L

WikWkj

+ 8ij,

where n

L

8 ij = Q ' i -

(34)

WiaWja

Q=l

the indices i, j running from n + 1 to 2n. The differential forms 8 ij arc exterior quadratic differential forms depending on the imbedding of R n in R 2n. They give what is essentially known as the Gaussian torsion of R" in R 2n. We put, similar to (10), (35)

8 =

(-1) p 22 P 1P

rlO,

7r

'"

,L....- Ei ,· · · i n 8 " i2 . . .

p.

8in-lin ,

. n = 2p if

l~

even,

if n is odd.

·With these preparations we are able to state the following theorems : 1. If R n i s a closed oricntable submanifold imbedded in an orientable Riemannian manifold R 2n , the Whitney invariant if; is given by (36)

if;

=

f

Rn

8.

131

684

SHIING-SHEN CHERN

2. It is always possible to define a continuous normal vector field over a closed orientable odd-dimensional differentiable submanifold (of class ~ 3) imbedded in an orientable differentiable manifold of twice its dimension . The first theorem can be proved in the same way as the formula of AllendoerferWeil. We shall give a proof of the second theorem. For this purpose we take a simplicial decomposition of our submanifold R n and denote its simplexes by u; , i = 1, ... ,m. We assume the decomposition to be so fine that each u; lies in a coordinate neighborhood of Rn. Accordingto a known property on the decomposition of a pseudo-manifold [6], the simplexes u; can be arranged in an order, say u{' , ... , u,;: ,such that ut ,k < m , contains at least an (n - I)-dimensional side which is not incident to u{' , . . . , Uk-I. We then define a continuous normal vector field by induction on k. It is obviously possible to define a continuous normal vector field over u{'. Suppose that such a field is defined over u{' + ... + U[:-I. The simplex q;: has in common with q{' + ... + uk-I at most simplexes of dimension n - 1 and there exists, when 1c < m, at least one boundary simplex of dimension n - 1 of q;: which does not belong to q~ + ... + U;:_I. It follows that the subset of u;: at which the vector field is defined is contractible to a point in q;:. By a well-known extension theorem [7], the vector field can be extended .throughout q;: , k < m. In the final step k = m the extension of the vector field throughout u,;: will lead possibly to a singular point in q';:. Hence it is possible to define a continuous normal vector field over R n with exactly one singular point, the index at which is equal to the Whitney invariant if;. If n is odd, we have, by (36), if; = 0, and the singular point can be removed. This proves our theorem. INSTITUTE FOR ADVANCED STUDY, PRINCETON, NEW JERSEY, AND 'ISING H UA U:-
132 ANNALS OF MATHEMATICS

Vol. 47, No. I, January, 1946

CHARACTERISTIC CLASSES OF HERMITIAN MANIFOLDS By

SWING-SHEN CHERN

(Received July 10, 1945) INTRODUCTION

In recent years the works of Stiefel/ Whitney,2 Pontrjagin,3 Steenrod,4 Feldbau,s Ehresmann,6 etc. have added considerably to our knowledge of the topology of manifolds with a differentiable structure, by introducing the notion of so-called fibre bundles. The topological invariants thus introduced on a manifold, called the characteristic cohomology classes, are to a certain extent sus7 ceptible of characterization, at least in the case of Riemannian manifolds, by means of the local geometry. Of these characterizations the generalized Gauss-Bonnet formula of Allendoerfer-Weil8 is probably the most notable example. In the works quoted above, special emphasis has been laid on the sphere bundles, because they are the fibre bundles which arise naturally from manifolds with a differentiable structure. Of equal importance are the manifolds with a complex analytic structure which play an important role in the theory of analytic functions of several complex variables and in algebraic geometry. The present paper will be devoted to a study of the fibre bundles of the complex tangent vectors of complex manifolds and their characteristic classes in the sense of Pontrjagin. It will be shown that there are certain basic classes from which all the other characteristic classes can be obtained by operations of the cohomology ring. These basic classes are then identified with the classes obtained by generalizing Stiefel-Whitney's classes to complex vectors. In the sense of de Rham the cohomology classes can be expressed by exact exterior differential forms which are everywhere regular on the (real) manifold. It is then shown that, in case the manifold carries an Hermitian metric, these differential forms can be constructed from the metric in a simple way. This means that the characteristic classes are completely determined by the local structure of the Hermitian metric. This result also includes the formula of Allendoerfer-Weil and can be regarded as a generalization of that formula. Concerning the relations between the characteristic classes of a complex manifold and an Hermitian metric defined on it, the problem is completely solved by the above results. It is to be remarked that corresponding questions for Rie1

STIEFEl"

(24) .

The number in the bracket refers to the bibliography at the end of the

paper . 2

WHITNEY, (29), (30) .

(18), (19). (21), (22) . , FELDBAU, (12). • EHRJ:)SMANN , (10) . 7 C HERN, [51, (6), (7). 3 PONTnJAGlN ,

• STEI!]NROD ,

8 ALLENDOERFER.WEIL, (2) .

85

133

86

SHIING-SHEN CHERN

mannian manifolds remain open. Roughly speaking, the difficulty in the real case lies in the existence of finite homotopy groups of certain real manifolds, namely the manifolds formed by the ordered sets of linearly independent vectors of a finite-dimensional vector space. The paper is divided into five chapters·. In Chapter I we consider the fibre bundles which include the bundles of tangent complex vectors of a complex manifold and which are called complex sphere bundles. To a given base space a complex sphere bundle can be defined by a continuous mapping of the base space into a complex Grassmann manifold and it is shown that this is the most general way of generating a complex sphere bundle. We take the Grassmann manifold to be that in a complex vector space of sufficiently high dimension and define a characteristic cohomology class in the base space to be the inverse image under this mapping of a cohomology class of the Grassmann manifold. We are therefore led to the study of the cocycles or cycles on a complex Grassmann manifold, a problem treated exhaustively by Ehresmann.9 A close examination of Ehresmann's results is therefore made in Chapter II, in the light of the problems which concern us here. In fact, we are only interested in the cocycles of the Grassmann manifold which are of dimension not greater than the dimension of the base space. If the Grassmann manifold is that of the linear spaces of n (complex) dimensions in a linear vector space of n N dimensions, there are on it n basic cocycles such that all other cocycles of dimenslon ~ 2n can be obtained from them by operations of the cohomology ring. The cycles corresponding to these cocycles are determined and geometrically interpreted. In Chapter III we identify the im2.ges of these cocycles in the base space with the cocycles obtained by generalizing the Stiefel-Whitney invariants to complex vectors. A new definition of these cocycles is given, which is important for applications to differential geometry in the large. Chapter IV is devoted to the study of a complex manifold with an Hermitian metric. It is proved that the n basic cocycles in question can be characterized in a simple way in terms of differential forms constructed from the Hermitian metric. These results are then applied in Chapter V to the complex projective space with the elliptic Hermitian metric. Classical formulas of Cartan 10 and Wirtingerll are derived from our formulas as particular cases.

+

CHAPTER I COMPLEX SPHERE BUNDLES AND THEIR IMBEDDING

1. The complex sphere Various definitions have been given of a fibre bundle. For definiteness we shall adopt the one of Steenrodl 2 and follow his terminology. We are, however, going to restrict the kind of fibre bundles under consideration. 9

10

EURESMANN, CARTAN,

[8] .

[3] . [31] . [22] .

II WIRTINGER, 12 STEENROD,

134 CLASSES OF HERMITIAN MANIFOLDS

87

Let E(n; C) be a complex vector space of n dimensions/ 3 whose vectors will be denoted by small German letters. In E(n; C) suppose a positive definite Hermitian form be given, which, in terms of a suitable base, has the expression n

(1)

AA

=

L z'i, iz=l

where z' are the components of the vector Ain terms of the base and the bar denotes the operation of taking the complex conjugate. A vector A such that AA = 1 is called a unit vector. The group of linear transformations

i = 1, ... , n,

(2)

which leaves the form (1) unaltered is the unitary group and will be denoted by U(n; C). We shall call the complex sphere Sen; C) the manifold of all the unit vectors of E(n; C). It is homeomorphic to the real sphere of topological dimension 2n - 1. The letter C in these notations will be dropped, when there is no danger of confusion. In this paper we shall be concerned with fibre bundles such that the fibres are homeomorphic to the complex sphere Sen) and that the group in each fibre is the unitary group U(n). Such a fibre bundle is called a complex sphere bundle. The most important complex sphere bundle is obtained from the consideration of the complex tangent vectors of a complex manifold M(n) of complex dimension n and topological dimension 2n. By a complex manifold M(n) we shall mean a connected Hausdorff space which satisfies the following conditions: 1) It is covered by a finite or denumerable set of neighborhoods each of which is homeomorphic to the interior of the polycylinder

I z' I <

1,

i

= 1, ... , n,

in the space of n complex variables, so that Zi can be taken as local coordinates of M(n) . 2) In a region in which two local coordinate systems z' and z*' overlap the coordinates of the same point are connected by the relations (3)

t

where are analytic functions. It follows from this definition that the notions of M(n) which are expressed in terms of local coordinates but which remain invariant under the transformations (3) have an intrinsic meaning in M(n). This is in particular true of a tangent vector at a point, which we define in the usual way as an object which has n com13 Throughout this paper we shall mean by dimension the complex dimension . The dimension of a manifold in the sense of topology will be called the topological dimension, which is twice the complex dimension . The dimensions of simplexes, chains, cycles, homology groups, etc ., are understood in the sense of topology, so long as there is no danger of confusion.

13S

88

SHIING-SHEN CHERN

ponents Zi in each local coordinate system and whose components Zi, Z*' in two local coordinate systems Z*i are transformed according to the equations

z"

(4) It is clear that the space of the tangent vectors at a point is homeomorphic to E(n). We consider the non-zero tangent vectors and call two such vectors

equivalent if their components system satisfy the conditions

z\

Wi with respect to the same local coordinate Wi = pZ\

where p is a positive real quantity. This relation remains unchanged under transformation of local coordinates. Also it is an equivalence relation in the sense of algebra, being reflexive, symmetric, and transitive. Hence the non-zero tangent vectors can be divided by means of this equivalence relation into mutually disjoint classes. We call such a class of non-zero tangent vectors a direction. With a natural topology the space of directions at a point is homeomorphic to the complex sphere Sen). Furthermore, by using the so-called unitarian trick in group theory, it is easy to verify that the manifold of all directions at the points of M(n) is a complex sphere bundle with M(n) as the base space It will be called the tangent bundle of M(n). Although all the results in this chapter will be formulated for general complex sphere bundles, it is the particular case of the tangent bundle of a complex manifold that justifies the study of complex sphere bundles. 2. The Grassmann manifold and the imbedding theorems Consider the space E(n + N ;C) and the linear subspaces of E(n + N ;C) of dimension n. The manifold of all such linear subspaces is called a Grassmann manifold and will be denoted by H(n, N;C) or simply H(n, N). It is of dimension nN. The unit vectors of the complex sphere Sen + N) in E(n + N), which belong to a linear subspace E(n) of dimension n, constitute a complex sphere Sen), to be denoted by Sen + N) n E(n). Now let B be a finite polyhedron in the sense of combinatorial topology and let f be a continuous mapping of B into H(n, N) . From the mapping f we can define a complex sphere bundle 3' with B as base space as follows: 3' consists of the points (b, \J) of the topological product B X Sen + N) such that \J E feb) n Sen N), and the projection 7r of 3' onto B is defined by 7r(b, \J) =b. It is easy to verify that 3' is a complex sphere bundle over B, which we shall call the inducp.d bundle over B . The importance of the notion of complex sphere bundles induced by the mapping of the base space into a Grassmann manifold is justified by the following theorems: THEOREM 1. To every bundle 3' of complex spheres Sen) over a finite polyhedron B of topological dimension d there exists a continuous mapping f of B into H(n, N) with N ~ d/2, such that 3' is equivalent to thc b-undle induced by f.

+

136

CLASSES OF HERMITIAN MANIFOLDS

89

THEOREM 2. Let ~I and ~2 be two bundles of complex spheres Sen) over a finite polyhedron B of topological dimension d induced by the mappings it , f2 respectively of B into H(n, N), N ~ d/2. The bundles ~I and ~2 are equivalent wherJ, and only when the mappings fl and f2 are homotopic. 14 Similar theorems for real sphere bundles are known. It follows from these theorems that to a class of equivalent bundles of complex spheres Sen) over a finite polyhedron B of topological dimension d corresponds a class of homotopic mappings of B into the Grassmann manifold H(n, N), where N is an integer satisfying 2N ~ d. This class of mappings induces a homomorphism h of the cohomology groups of dimension ~ d of H(n, N) into the cohomology groups of the same dimension of B. A cohomology class of B which is the image under h of a cohomology class of H(n, N) is called a characteristic cohomology class or simply a characteristic class and each of its cocycles is called a characteristic cocycle. 3. Proofs of Theorems 1 and 2 The proofs of the Theorems 1 and 2 do not differ essentially from the real case· We shall therefore restrict ourselves to a brief description of the general procedure. We need the following two lemmas: LEMMA 1. (Covering homotopy theorem)15 Let ~ be a fibre space over a base space B, which is a compact metric space. Let S be a compact topological space and let J be the unit interval. Suppose a mapping h(S X 1) C ~ be given having the property: There exists a mapping H(S X 0) C ~ such that h(p X 0) where 7r is the projection of such that

~

= 7rH(p X 0),

into B.

PES,

Then there exists a mapping H(S X J) C

h(p X t) = 7rH(p X t),

PES,

~

O~t~1.

From Lemma 1 follows the lemma: LEMMA 2 (Feldbau) .16 Let ~ be a fibre bundle "ver a compact metric base space B. If B is contractible to a point, then ~ is equivalent to the topological prodw;t of B and one of its fibres F. To prove Theorem 1 we are going to define the mapping f whose existence was asserted by the theorem. We take a simplicial decomposition of B which is so fine that each simplex lies in a neighborhood, and denote by q~ , i = 1, ... , Cik , k = 0, 1, ... , d, its simplexes. We denote as usual by 7r the projection of ~ onto B. Our purpose is to define a mapping feB) C H(n, H) and a mapping f*(m C B X Sen N) such that

+

14 WHITNEY,

[29];

STEENROD,

15 HUREWICZ-STEENROD,

ferent versions . 16 FELDBAU,

[12].

[22].

[15] . The theorem is given in various papers, in slightly dif-

137

90 (5)

SHIING-SHEN CHERN

!*(p)

E

7I"(p) X

{f(7I"(p»

n

Sen

+

N) I

and that for a fixed 7I"(p) the mapping!*(p) is a homeomorphism preserving the scalar product. The definition of these mappings is given by induction on the dimension of the simplexes of B. The images f«(f~) E H(n, H), i = 1, ... , (fo, are defined in an arbitrary way and it is clear how!*(p) can be defined for all p E 15 such that 7I"(p) = (f~ , i = 1, ... , ao. We suppose the mappings be defined over the (k - I)-dimensional skeleton of B and consider any simplex (fk of dimension k. We take a neighborhood U which contains (fk and decompose the set 71"-I(U) into a topological product of U and a complex sphere So(n). Then we can define n mappings 'Pi«(fk) c 15, i = 1, ... , n, such that: I) 7I"'Pi(P) = p, p E (f\ 2) 'Pi(P), 'Pj(p), i =1= j, are orthogonal vectors on the complex sphere 7I"-I(p). We proceed to define by induction !*('Pi(P» = P X qi, which will satisfy the condition that qi , qj for i =1= j are orthogonal vectors of Sen + N). By hypothesis, !* ('PI (p) ) is defined for all p E ()(fk .17 Since O(fk is topologically a sphere of topological dimension k - 1 ~ d - 1 ~ 2N - 1 < 2(n + N) - 1, which is the topological dimension of the complex sphere Sen + N), and since 7I"(p), p E 'PI«(fk), is the cell (f\ it follows that !*('Pl(P», P E O(fk, is contractible in B X Sen + N). This means that there is a continuous mapping g(O(fk X t) c B X Sen + N), 0 ~ t ~ 1, such that the following conditions are'satisfied: I) g(O(fk X 0) is a point; 2) g(O(fk X 1) is identical with !*('PI(P». On the other hand, we can introduce in (fk the "polar coordinates" p, P, where 0 ~ p ~ 1 and P E o(l . For a point of (fk having the coordinates p, P we define J*('PI(p, p» = g(p X p).

Suppose now that J*('PI(P» = P X ql, .. . , J*('Pi-I(P» = P X qi-l, P E (fk, are defined, such that qk , qj , k, j = 1, ... , i - I , k =l= j, are orthogonal vectors of Sen + N). To defineJ*('Pi(p» we consider on Sen + N) the complex spheres Sen + N - i + 1) whose vectors are orthogonal to ql , .. . , qi-I. These complex spheres Sen + N - i + 1), depending on P, constitute a. complex sphere bundle over the simplex (fk. By Lemma 2, it is a topological product of (fk and a complex sphere So(n + N - i + 1) . By induction hypothesis, the boundary 0(/ is mapped into So(n + N - i + 1), by means of the vectors qi E Sen + N - i + 1) (p). Since the topological dimension k - 1 of O(fk is smaller than the topological dimension 2(n + N - i) + 1 of So(n + N - i + 1), the map is contractible and the mapping of O(fk can be extended continuously throughout (fk. It follows that a mapping h«(fk) C So(n + N - i + 1) and hence a mapping hl«(fK) C Sen + N) can be defined such that h,(p)

E

Sen

+

N -

i

+

l)(p),

k PE(f.

We then define J*('Pi(P» = P X h,(p) = P X qi , P E (fk. qi is orthogonal to ql , ... , qi-I .

Clearly the vector

17 We shall make use of the notation auk to denote both the combinatorial and the set. theoretical boundary of the simplex uk, as the meaning will be clear by context.

138 CLASSES OF HERMITIAN MANIFOLDS

91

To complete the induction on the dimension k let P* E lS such that 1r(P*) = P E ,/. Then p* has n components Ul, ... , Un with respect to 'l'l(P) , ... , 'l'n(P)· We define f(p) to be the linear space of n dimensions of E(n + N) which contains the complex sphere detennined by ql , ...• qn and r(p*) = p X q,

where q belongs to f(p) n Sen + N) and has the components Ul , ... , Un with respect to ql , ... , qn . Thus our induction is complete and it is easily seen that the mappings f and fulfill our desired conditions. It is also clear that the complex sphere bundle induced by the mapping feB) C H(n, N) is equivalent to lS. This proves our Theorem 1. Concerning Theorem 2 it is not difficult to prove that lSI and lS2 are equivalent if fl and f2 are homotopic. The converse is proved by defining a mapping feB X I) C H(n, H), withf(B X 0) andf(B X 1) coinciding with the given mappings fl and f2 respectively. Because of the equivalence of lSI and lS2 a complex sphere bundle can be defined over B X I in an obvious way. The rest of the argument consists of defining the mapping feB X I) by an extension process analogous to the proof of Theorem 1. We shall omit the details here.

r

CHAPTER

II

STUDY OF THE COCYCLES ON A COMPLEX GRASSMANN MANIFOLD

1. Summary of some known results Let H (n, N) be the Grassmann manifold of n-dimensional linear subspaces in E(n, N). Our main purpose in this chapter is to give a homology base for the cocycles of dimension ~2n of H(n, N). It is to be remarked that, H(n, N) being a manifold of topological dimension 2nN, there corresponds to each cycle of dimension 8 a cocycle of dimension 2nN - 8, and vice versa. There are two different ways to describe the cocycles of H(n, N), ,yhich are both useful to our purpose. To explain the first method let 0 ~ rp(i) ~ N, 1 ~ i ~ n, be a non-decreasing integral-valued function. Let L i , 1 ~ i ~ n, be a linear vector space of dimension i rp"(i) in E(n N), such that

+

+

LI

C

L2

C··· C

Ln.

Let Z(rp(i» be the set of all n-dimensional linear spaces X(n) such that dim (X(n)

n L i)

~ t,

i

= 1, ... , n,

where the notation in the parenthesis denotes the linear space common to X(n) and L,. Z(rp(i» is called a Schubert variety in algebraic geometry. It is a pseudo-manifold of dimension 8 = L~-1 rp(i) and carries an integral cycle of dimension 28 of H(n, N). Concerning the significance of the Schubert varieties for the topology of Grassmann manifolds the following theorem was proved by Ehresmann :18 18 EHRESMANN,

[8], p. 418.

139

92

SHIING-SHEN CHERN

THEOREM 3. The Grassmann manifold H(n, N) has no torsion coefficients and has all its Betti numbers of odd dimension equa.l to zero. Its Betti number of dimension 2s is equal to the number of distinct non-decreasing integral-valued functions cp(i), 1 ~ i ~ n, such that L~=lcp(i) = s. The integral cycles carried by the corresponding Schubert varieties Z(cp(l)) constitute a homology base for the Betti group of dimension 2s. The second method to describe the cocycles of H(n, N) is by means of differential forms. Let X(n) E H(n, H) and let el , ••• , en be n vectors in X(n) such that

1

~

i,j

To these n vectors we add N further vectors conditions

~

n.

en+! , • • • ,

en+N

satisfying the

1 ~ A,B ~ n+N.

(6)

When there is a differentiable family of the vectors

el , •.. ,

en+N ,

we put

(7) which are linear differential forms satisfying the conditions

(8) Among 0AB the forms 1

~

i

~

n,

n

+1~ r

~ n

+ N,

constitute a set of 2nN linearly independent forms at each point of H(n, N). Let e be a form in Oir , Bir with constant coefficients. e is called an invariant form jf it remains unchanged under the groups of transformations n

"a

0*,·, -- L-.

(9)

i= 1

0; 0ir.

n+N

07r =

(10)

L

bra 0., ,

$- n+l

where (ai;), (bro) are arbitrary unitary matrices. It is called exact, if de = o. It is well-known that on a differentiable manifold of class two a co cycle with rational coefficients can be expressed by an exact differential form, and conversely. For convenience we shall therefore call an exact differential form a cocycle. Then we have the following theorem of E . Cartan: 19 THEOREM 4. Every invariant form of H(n, N) is exact. The B etti number of dimen sion 2s of H(n, N) is equal to the number of linearly independent (with constant coefficients) invariant differential forms of degree 2s. The set of these forms constitutes a cohomology base of dimension 2s. 19 CARTAN,

[3];

EHRESMANN ,

[8], p . 409 .

140

93

CLASSES OF HERMITIAN MANIFOLDS

2. The basic forms Let r be an integer between 1 and n. For reasons which will be clear later we shall be particularly interested in the n cycles Zr , r = 1, .. . , n, carried by the Schubert varieties defined by the functions 'Pr(i)

(11)

'Pr(i)

=N = N,

=

i

1,

1, ... , n - r

i = n - r

+

1,

+ 2, ... , n.

+

The cycle Zr is of dimension 2(Nn - n r - 1). We shall find the invariant differential form which gives the cocycle of dimension 2(n - r 1) corresponding to Zr. For this purpose we put

+

n+N

e' i = .=n+1 L O•• O'i

(12)

I

1 ~ i,j ~ n,

and (13)

where o(i1 ... in-r+l j j1 ... jn-r+1) is zero except when j1, ... ,jn-r+l form a permutation of i 1 , . •• , i n - r +1 , in which case it is 1 or -1 according as the permutation is even or odd, and where the summation is extended over all indices i1 , ... , i n - r +1 from 1 to n. It is easy to verify that r is an invariant form on H(n, N). Our problem is then solved by the following theorem : THEOREM 5. The invariant differential form r defines a cocycle of dimension 2(n - r 1) on H(n, N), which corresponds to the cycle Zr in the sense that, for any cycle r of dimension 2(n - r 1), the relation

+

+

+

KJ(rl, Zr) =

(14)

i

r

holds,20 whenever both sides are defined. To prove this theorem we notice that both sides of the equation (14) are linear in t, so that it is sufficient to prove (14) for the cycles of a homology base of dimension 2(n - r 1). By Theorem 3, these are the cycles carried by the Schubert varieties Z('P(i» such that

+

n

L 'P(i) i=l

= n - r

+ 1.

Since the function 'P(i) is non-decreasing, we must have (15)

'P(l) = ... = 'P(r - 1) =

o.

20 The notation KI, due to Lefschetz, means the Kronecker index (or the intersection number) .

141 94

SHIING-SHEN CHERN

Let ~k, k 1, ... , m, be the cycles of a homology base of dimension 2(n r 1) defined in this way, and let ~l be the cycle defined by (16) !p(I) = ... = !p(r - 1) = 0, !p(r) = ... = !p(n) = L

+

Then for the cycles

~k,

k

~

1, we must also have

= O.

!p(r)

It is therefore sufficient to prove that

KI(~k , Zr) =

(17)

1

=

k

4>r.

1, ...

,m,

tk

for the cycles ~k which are well chosen so that both sides of (17) are defined. Let us first assume that k ~ 1. By definition, there exists a fixed linear vector space L(r) of dimension r such that any X(n) of the Schubert variety carrying ~k satisfies the condition dim (X(n) U L(r)) ~ r, which means that X(n) ::::> L(r).

On the other hand, any X(n) of the Schubert variety carrying Zr has the property that it intersects a fixed linear space L(N n - r) of dimension N n - r in a linear space of dimension ~ n - r L We take in E(n + N ) a frame tl, . .. , en+N and let L(r) and L(N n - r) be spanned by the vectors tl, ... , er and er+l, ... , en+N respectively. There is clearly no X(n) which conn - r) a linear space of ditains el, ... , tr and has in common with L(N mension n - r + 1, which means that

+ +

+

+

+

k

~

L

To show that the integral on the right-hand side of (I7) is also zero, we choose the frame tl, ... , en+N such that tl, . .. , er belong to L(r) and el, ... , en belong to X(n). It is obviously possible to choose el ; .. . , er to be fixed . Under this choice we have 8;j

= 0,

1

~

i,j

~

r,

and hence 4>r

= O.

Thus the relations (I7) are proved for k ~ L To define ~l we take two linear vector spaces L(r - 1), L(n + 1) of dimensions r - 1, n 1 respectively, such that L(r - 1) C L(n 1) . An element X(n) of ~l is then defined by the conditions

+

+

L(r -

1) C X(n) C L(n

Let el , ... , en+N be a frame in E(n in question to be so chosen that

+ N).

+

1).

We suppose the linear vector spaces

142 95

CLASSES OF HERMITIAN MANIFOLDS

el , ... , e.-I C L(r - 1), el, ... , e1'+l C L(n e

T ,

•••

,

+ 1),

en_I, en-I, ... , e1'+N C L(n

+N

- r).

By this choice L(n + 1) and L(n + N - r) have in common a linear vector space of dimension n - r + 1, namely the one spanned by e en-I, en+l . It follows that ZT and have in common only one X(n), which is spanned by el, ... , en-I, en+l' This intersection is to be counted simply, and we have21 T ,

rl

•• •

,

KI(ZT. rl) = 1. It now only remains to evaluate the integral in the right-hand side of (17) for k =, 1. The linear vector spaces L(r - 1) and L(n + 1) being fixed, we choose a fixed frame 01 , . . • , On+N in E(n + N) such that 01, •••• OT-l belong to L(r - 1) and 01, ..• , 0n+l to L(n + 1). If X(n) E rl , we choose the frame el, ... , en+N, whose first n vectors belong to X(n) such that A = 1, ... , r - 1, n + 2, ,n+N, (18)

B, C = r, ... , n

It follows that

where UBe are the elements of a unitary matrix. (hi

(19)

=

dek·ei

= 0,

k = n

Oik = -Oki = 0,

+ 1,

+ 2, ... , n + N,

i = 1, ... , n

+ N,

and hence that 1

~

i, j

~

n.

For simplicity we shall write e for en+l, UB for Un+l,B , and Oi for On+l,i. We remark that X(n) is completely determined by the vector e, whose components with respect to OT, . .. • On+l are UT , • •. , U1' +l. Our purpose therefore is to transform the form T into an exterior differential form in U T , • •• , Un+l , from which the integration can be carried out. With this purpose in mind, we have 0,

= de·e, =

(f dUBOB)(~1 B=r

U,eae)

C==r

1'+1

(20)

= O.

L

B=T

dUBU,B ,

i

= r, "', n,

i = I, ... , r - 1,

= 0,

and i,j

all other 21

Cf.

eij

being zero.

EHRESMANN,

= r, ... , n,

It follows that <1>. is equal to

[8], p. 421.

143

96

SHIING-SHEN CHERN

(21)

(21Ty=ly-n-l "0(' ( ) , • £.oJ u tl n-r+l.

= (_I)Hn-r)(n-r+l)

.

. .

. )0 0In-r+l 'I iJ

0 '"

0 0- ..• 0-

... t n- r+l ,Jl .••

(n -

+ 1)

r

(2?Ty _It- +l T

n r

r

•••

O'n-r+IO;.-r+1

n·

From now on we shall agree on the following ranges of indices: 'Y ~

a, (3

~

n,

From (20) and we get, by solving for du; , (22)

=

du,

L: u",O" + U,On+l .

We notice that

o.

(23)

Consider now the form n-r+l

(24) itr

=

L:

dur ··· dUr+k-l dUr+k-l ... dUn+! dU T

•••

dUr+k-l dUr+k+l ... dUn+! ,

k~O

which we shall prove to differ from
+

'" (1)Hn-r)(n-T+J) (n - r I)! ,T, '*'r = (21TY-1)n r+1 "'r·

(25)

To integrate
=

dU r ... dUn dU r ... dUn,

which is to be integrated over the domain Un+l

This integration is then easily achieved.

Then itr

Ur

=

Vr

UT

=

VT

>

O.

In fact, we put

+ y=I Wr , -

V=I W

T

•

= (_1)!(n-T)(n-T+1)( -2 y -1t- T+1dv r dW r ... dUn dW n ,

and the integral is over the domain D:

144

97

CLASSES OF HERMITIAN MANIFOLDS

U~+l

+ v; + w; + ... + v! + w! =

1,

Un+l>

O.

But the integral

is the volume of the domain bounded by the unit hypersphere of dimension 2n - 2r 1, which is equal to 7r n -,+l/(n - r I)!, Hence we have

+

+

( _1)hn-rHn-r+l)

(27r' V- ~)n-r+1 - 1 (n - r

+

1)1 '

and finally,

1 tl

q"

=

1.

Our Theorem 5 is therefore completely proved.

3. The basis theorem The importance of the invariant differential forms q,r, 1 ~ r ~ n, lies in a theorem we proceed to prove, which asserts that every invariant differential form of degree ~ 2n in H (n, N) is a polynomial in q,r with constant coefficients. The exact statement of our theorem is as follows: THEOREM 6. Every invariant differential form of degree ~ 2n in H(n, N) is a polynomial in q,r, 1 ~ r ~ n, with constant coefficients. If the form defines an integral cocycle on H (n, N), the coefficients are integers. The theorem follows easily from the so-called first main theorem on vector invariants for the unitary group, which we state as follows: LEMMA 3. Let 1>1, •.• , Il m be a set of vectors in E(n) under transformations of the unitary group U(n). Every integral rational invariant in the components of Ilk, 1 ~ k ~ m, is an integral rational function of the scalar products Ilitik' 1 ~ i, k ~ m. It is known thae 2 under the unimodular unitary group such an invariant is an integral rational function of the scalar products and of determinants of the form [Ill· .. Ilnl or [b1 .. . tinl. But under a general unitary transformation of determinant eV - 1a the determinants [Ill· .. Ilnl and [01 ... bnl will be multiplied by eV=lo and e-V - 1a respectively. It follows that an invariant will involve the determinants only in products of the form [Ill· .. Ilnl · [ij~ ... ii~], which can however be expressed as a determinant of scalar products:

Thus the lemma is proved. 22

WEYL, [28], p. 45 .

145

98

SHIING-SHEN CHERN

To prove Theorem 6 let 'l1 be an invariant differential form of degree 28 ;;;;; 2n in H(n, N), which is therefore an exterior form in (JiT , 8iT , 1 ;;;;; i ;;;;; n, n + 1 ;;;;; r ;;;;; n + N, with constant coefficients. The form 'l1 being in particular invariant under the transformation (J;r = ev'=ia (JiT , it follows that 'l1,when reduced to its lowest terms, will contain in each term exactly 8 factors each of (JiT and BiT' Let us fix our attention for the moment to the group (10) . We take from 'l1 all the terms of the form

with a fixed set of the indices iI, ... , i. , jl .... ,j., and call their sum 'l11 . Since the indices iI, ... , i. , jl , ... ,j. are now fixed, we shall drop them for simplicity. Now it is well-known that there is an isomorphism between the ring of exterior forms and the ring of multilinear forms with alternating coefficients. To 'l11 corresponds, in the complex vector space of N dimensions, an alternating multilinear form of degree 28. Since 'l11 is invariant under the unitary group (10), the same is true of its corresponding alternating multilinear form. By our Lemma the latter is an integral rational function of the scalar products. It follows by the isomorphism that 'l11 can be expressed as a polynomial in sums of the form LT (JiTB jT = -8ij . Consequently, 'l1 is a polynomial in 8ij, 1 ;;;;; i, j ;;;;; n; with constant coefficients. Let us now put

r = 1, ... ,n.

(26)

By the same argument as above, we can prove that 'l1, being also invariant under the group (9), is a polynomial in P r , 1 ;;;;; r ;;;;; n, with constant coefficients. On ~the other hand, it is easy to show, by induction on r, that P r is a polynomial in <1>1, ... ,r, with constant coefficients. Hence the first part of our theorem is proved. To prove the second part of the theorem consider the products of the form (27) such that (27a)

Al

+ 2A2 + ... + nAn = 8.

These forms constitute a basis for all invariant differential forms of degree 28 ;;;;; 2n on H(n, N). Since 8 ;;;;; n, their number is equal to the number of partitions of 8 as a sum of integral summands. By Theorem 3 this is equal to the Betti number of dimension 28 of H(n, N). It follows that the products in (27) are linearly independent, and that every invariant differential form of degree 28 of H(n, N) representing an integral cocycle is equal to a linear combination of the products (27) with integral coefficients.

146 CLASSES OF HERMITIAN MANIFOLDS CHAPTER

99

III

THE BASIC CHARACTERISTIC CLASSES ON A COMPLEX MANIFOLD

1. A second definition of the basic characteristic classes Let 111 be a complex manifold of dimension n. We consider the complex sphere bundle defined from.. the tangent vectors of 111 and imbed it, according t.o Theorem 1, in a Grassmann manifold H(n, N), N ~ n, by means of a mapping of 111 into H(n, N). It follows from Theorem 2 that the inverse image of a cohomology class of dimension ~2n of H(n, N) induced by this mapping is an invariant of 111 (or rather of the analytic structure of 111), which we have called a characteristic cohomology class of 111. From Theorem 6 we see that of all the characteristic cohomology classes of 111 those which are inverse images of the cohomology classes of H(n, N) containing the cocycles if>r • 1 ~ r ~ n, playa particularly important role, because all the others can be obtained from them by operatiom; of the cohomology ring. We therefore call these n classes the basic characteristic classes, the inverse image of the class containing if>r being the rth basic class. Our first aim is to identify these basic classes with the classes obtained by generalizing to complex manifolds the well-known procedure of Stiefel-Whitney.23 In order to understand the situation we recall briefly the results of StiefelWhitney for real sphere bundles, emphasizing the differences between the real and complex cases. From a bundle of real spheres of dimension n - lover a polyhedron as base space Stiefel and Whitney considered the fibre bundle O\'er the same base space whose fibre at each point is the manifold Yen, r) of r(l ~ r ~ n) linearly independent points of the real sphere at this point. It was proved that all homology groups of dimension
STIEFEL,

[24J;

WHITNEY,

[29J .

147 100

SHIING-SHEN CHERN

r».

a sphere of dimension n - r into a fibre and hence an element h of Hn-T(V(n, The delicate point is to get from this element h an integer or a residue class mod. 2. This is possible if a generating element of Hn-T(V(n, r)) is defined. When Hn-T(V(n, r)) is cyclic of order two, it has only one generating element, so that no further assumption is necessary. When Hn-T(V(n, r)) is a free cyclic group, we assume that a continuous field of generating elements of Hn-T(V(n, r)) can be defined over the whole M, which is possible if M is orientable. The element h is then equal to the generator of Hn-T(V(n, r)) so defined, multiplied by an integer or a residue class mod. 2. Taking this integer or the residue class mod. 2 as the value of a cochain for the simplex qn-T+l, we get an integral cochain or a cochain mod. 2. It was proved that the cochain is a cocycle and that its cohomology class is independent of the choice of the mapping from which it is defined. This cohomology class is the class of Stiefel-Whitney. It is to be remarked that the definition can be given under more general conditions, but we shall be satisfied with the above resume. The situation is simpler in the case of complex sphere bundles. From a bundle of complex spheres Sen) we consider the fibre bundle jJ(T) over the same base space whose fibre at each point is the manifold U(n, r) of r(1 ~ r ~ n) linearly independent vectors in E(n). It can be proved thae 4 all homology groups of dimension <2n - 2r + 1 of U(n, r) vanish and that the homology group H2n-2T+l(U(n, r)) of dimension 2n - 2r + 1 is a free cyclic group. To define a generator of H 2n - 2T +l(U(n, r)) we take in E(n) an ordered set of r - 1 mutually perpendicular unit vectors el , . .. , er-l. The unit vector er in E(n) perpendicular to el , .... eT-l describes a complex sphere in the E(n - 'r 1) perpendicular to el, ... , er-l . The complex sphere Sen - r 1) in E(n r -t- 1) is topologically a real sphere of topological dimension 2n - 2r -t- 1. Its two orientations define two cycles belonging respectively to the two generating classes of H2n-2T+l(U(n, r)). The cycle carried by the oriented real sphere Sen r + 1) is completely determined by the orientation of E(n - r + 1) considered as a real Euclidean space of topological dimension 2(n - r + 1). This orientaen in E(n - r + 1). It tion is independent of the order of the vectors e follows that the fibre bundle jJ(T) is orientable in the sense of Steenrod, which means that there is an isomorphism in the large between the (2n - 2r + 1)dimensional homology groups of the fibres of jJ(r), Or that a continuous field o'f generating elements of H 2n - 2T +l(U(n, r)) can be defined over the whole manifold. The fibre bundle jJ(r) has two opposite orientations and we shall from now on make a definite choice of one of them. Using this "oriented" fibre bundle, we shall be allowed to replace an element of H2n~2r+l(U(n, r)) at a point by the integer which, when multiplied by the generating element at this point, is equal to the element in question. With these explanations understood, we have:

+

T ,

• • •

+

,

24 EHRESMANN, [9J. This fact is easily proved by making use of the covering homotopy theorem, the complex case being even simpler than the real case.

148 101

CLASSES OF HERMITIAN MANIFOLDS

THEOREM 7. The rth basic characteristic class of a complex manifold M of dimension n can be defined as follows: Take a simplicial decomposition of M each of whose simplexes belongs to a neighborhood, and define over its skeleton of dimension 2n - 2r + 1 a continuous field of ordered sets of r linearly independent complex tangent vectors. To each simplex of dimension 2n - 2r 2 take a point in its interior and consider the manifold of the ordered sets of r linearly independent complex tangent vectors at that point. The field on the boundary of the simplex defines a mapping of the boundary into this manifold and hence an element of its (2n I)-dimensional homotopy or homology group, which is free cyclic. Attach 2r the corresponding integer to the simplex. The cochain so defined is a cocycle and belongs to the rth basic class. To prove this theorem we take on H(n, N) a definite Schubert variety V defined by the function in (11), which carries a cycle Z, dual to the cocycle 4>T' The Schubert variety being an algebraic variety on the algebraic variety H(n, N), it follows from the triangulation theorem of algebraic varieties25 that H(n, N) can be covered by a complex L such that V is a subcomplex. Let L* be the dual cellular subdivision of Land letfbe a mapping of Minto L*, which, according to Theorem 1, induces over M a complex sphere bundle equivalent to the tangent bundle of M. By the theorem on the simplicial approximation of mappings there exists a subdivision of M and a simplicial mapping fl of the subdivision into L* such that f and fl are homotopic. By Theorem 2 the complex sphere bundle over M induced by fl is equivalent to the tangent bundle of M. For simplicity of notation we can therefore assume f to be a simplicial mapping of Minto L* Let K be the skeleton of M of dimension 2(n - r + 1). j(K) is a sub complex of L* of dimension ~ 2(n - r + 1). The rth basic cocycle of M is by definition a linear function of the integral chains of dimension 2(n - r + 1) of K such that its value at a simplex IT of K is the intersection number of f(IT) and V. To give a description of the Schubert variety V we take in E(n N) a linear N - r) of dimension n N - r, and let L(r) be the linear subsubspace L(n space of dimension r which is totally perpendicular to L(n N - r) . Then V consists of all X(n) of H(n, N) satisfying the condition

+

+

+

+

dim(X(n)

n L(n + N

- r)) ~ n - r

+

+

+ l.

We take in L(r) r mutually perpendicular unit vectors ai, ... , aT' To each X(n) not belonging to V the projection of 01 , . . . , aT on X(n) will give r vectors which are linearly independent, and the construction fails exactly for the X(n) belonging to V. With these preparations consider a simplex IT of K. If the intersection number KI(f(IT) , V) = 0, the above construction will give a continuous field of r linearly independent complex vectors at each point of f(IT) and hence also at each point of IT, which shows that the integer attached to IT according to the statement of the theorem is also zero. It remains therefore to consider the case 25 VAN DER WAERDEN,

(26).

149

102

SHIING-SHEN CHERN

+

that KI(f(u), V) = E ~ o. In this case feu) is of dimension 2(n - r 1) . Since feu) and V belong to dual subdivisions, we have E = 1. Let Xo(n) be the linear space of H(n, N) common to Vandf(u). The orthogonal projection of al , ... , aT defines on each X(n) ~ Xo(n) of feu) r linearly independent complex vectors, which, by the resolution of all the X(n) belonging to feu) into a topological product, are mapped into the manifold p. of all the ordered sets of r linearly independent vectors in Xo(n). Our purpose is to prove that by means of the sets of vectors on the boundary of(u), of(u) is mapped into a cycle belongl)st homology group of p.. For simplicity ing to the generator of the (2n - 2r of language let us call index the integer m obtained by mapping the field on the boundary of(u) into a fibre U(n, r), the image cycle being in the homology class equal to m times the generator of H2n-2T+I(U(n, r)). Suppose first that a continuous field of ordered sets of n linearly independent vectors el, . . . , en is defined throughout feu) and let r linearly independent vectors h, ... , fT be defined over of(u) such that

+

+

n

fi =

L

k-I

fikek ,

1

~

i

~

r.

Regarding el, ... , en as fixed, these equations also define a mapping of df(u) into p.. We assert that their indices are equal. In fact, the existence of the field el, ... , en throughout feu) provides exactly a deformation of the field el , ... , en over df(u) into vectors e; , ... , e: which are constant. We assume that XG(n) has the property that the orthogonal projections of al , .. • , aT_I onto it are linearly independent, which is possible, after applying a small deformation if necessary. Since feu) is a simplex, we can define over feu) a continuous field of n linearly independent vectors, such that in every X(n) the first r - 1 of these vectors are the orthogonal projections in X(n) of ai, ... , aT _ 1 respectively. This continuous field is then deformed into a continuous field el, . . . , en over feu) such that in each X(n) the vectors CI , ... , en constitute a frame (that is, are mutually perpendicular unit vectors). It is well-known that the deformation can be so chosen that during the deformation the vector subspace determined by the first s vectors (1 ~ s ~ n) remains fixed. With these deformations performed, we proceed to study the orthogonal projection a~ of aT in X(n). The index of the field of orthogonal projections of al , .. . , aT on dfCu) is equal to the index of the field el , .. . , er_1 , a~ on df(u) and is also equal to the index of the same field, when CI, . . . , er - 1 are considered as constant vectors. "Ve also remark that the vector a~ is linearly independent of Cl , ... , er - 1 at every point ~ Xo(n) of feu). To show that the index in question is 1 we choose the continuous field of vectors en+l, ... , Cn+N over feu) such that eA, 1 ~ A ~ n N, is a frame in ECn N). Then we have n+N

+

aT

=

L

A-I

UA

eA ,

1,

+

150 CLASSES OF HERMITIAN MANIFOLDS

103

and

a:

n

=

L

Uriei.

i-I

According to our previous remark we can regard the vectors eA , 1 ~ A ~ n + N, as fixed and consider the mapping of f(q) - Xo(n) into Sen - r) defined by the vector whose components with respect to a fixed frame are Ur,r+l, • . • , Urn. Thus we see that the index is 1, and Theorem 7 is proved. It is also possible to introduce from a bundle of complex spheres Sen) the fibre bundles ~(r)* over the same base space whose fibre at each point is the manifold U*(n, r) of all ordered sets of r(1 ~ r ~ n) mutually perpendicular vectors of S(n) , The manifold U*(n, r) is an absolute retract of U(n, r), Theorem 7 still holds, if we replace everywhere the phrase "ordered sets of r linearly independent complex vectors" by "ordered sets of r mutually perpendicular vectors of S(n)", and, naturally also the manifold U(n, r) by U*(n, r). 2. A third definition of the basic characteristic classes We suppose in this section that the base space M, which is a complex manifold, is compact, From the tangent bundle ~ over M we construct the fibre bundle ~(r)* (1 ~ r ~ n) as explained at the end of the last section, Then the following theorem gives a third definition of the rth basic characteristic class: THEOREM 8. The rth basic characteristic class of Iv.[ is the cohomology class of M, each of whose cocycles 'Y has the following property: Under the projection of ~(r)* into M, 'Y is mapped into a cocycle 'Y*. There exists in ~(r)* a cochain {3*, such that o{3*

=

'Y* and that (3* reduces to a fundamental cocycle on each fibre of

~(r) *.

The last statement in the theorem needs some explanation. Let P be a polyhedron and Q CPa closed subpolyhdron of P. If 'Y is a cochain in P, its reduced cochain on Q is the cochain 'Y' such that 'Y" q = E 'Y' q, where E = 1 or 0 according as the simplex q belongs to Q or not. Moreover, the integral cohomology group of dimension 2n - 2r + 1 of a fibre being free cyclic, a fundamental cocycle on the fibre is a cocycle of dimension 2n - 2r + 1 which belongs to a generator of the cohomology group. It is also understood that the cycles and cocycles are defined in terms of simplicial decompositions of M and ~(r)*. To define the inverse mapping of the cocycles of M into the cocycles of ~(r)* induced by the projection 7r of ~ ( r ) * into M, we therefore take a simplicial approximation 71" of 71'. Let q* be a simplex of dimension 2n - 2r + 2 of ~(r)*. Then we define 'Y* ' q* = 'Y. 7r'(q*),

+

if 7r' (q*) is of dimElnsion 2n - 2r 2 and 'Y*. q* = 0 if 71" (q*) is of lower dimension , The cocycle 'Y* depends on 71", but its cohomology class is independent of it. The theorem asserts that any such cocycle 'Y* reduces to a fundamental co cycle on a fibre.

151 104

SHIING-SHEN CHERN

In order to prove Theorem 8 we need the following lemma: 4. With the notations of Theorem 8 let U be a neighborhood of M and let 71"-1(U) be its complete inverse image in \5(r)*. Let K be a finite complex and L its skeleton of dimension ~ 2n - 2r. Iff and g are two continuous mappings into 71"-1(U) of K and L respectively, there is a continuous mapping f* of K into 71"-1(U) which is homotopic to f and coincides with g on L. We denote by I the unit segment 0 ~ t ~ 1 and consider the topological product K X I. To prove Lemma 4 is to define a continuous mappingf(K X I) C 7I"-I(U), withf(K X 0) andf(L X 1) given. For this purpose we decompose K simplicially and arrange the simplexes of the decomposition in a sequence that every simplex is preceded by its faces. The mapping is then defined by successive extensions over the prisms constructed on the simplexes of the sequence. We resolve 7I"-I(U) into the topological product of U and a fixed fibre Fo and denote by A the projection of 71"-1(U) onto Fo. Let / be a vertex of K. Then f(u o X 0) andf(uo X I) are both defined in \5(r)*, and can be joined by a segment, on which the prism on U O is mapped. Using mathematical induction we suppose f be defined over all prisms on simplexes preceding u m of the sequence, and consider u m , m ~ 2n - 2r. By hypothesis, the mapping is defined over a(u m X I), which is topologically a sphere of topological dimension m. The mapping can be extended over u m X I , if and only if f(a(u m X I)) is homotopic to zero in ij(r)*, that is, by the covering homotopy theorem, if and only if v(a(u m X I)) is homotopic to zero in Fo. The latter is the case, because the m th homotopy group of Fo is zero. It follows that f is defined for the subcomplex L X I + K X 0 of the prism K X I. By a well-known elementary geometric construction26 f is then extended over K X I. Thus the lemma is proved. We proceed to prove Theorem 8. Let'Y be a cocycle of M belonging to the rth basic class defined by the construction of Theorem 7, with the bundle \5(r) replaced by \5(r)*. To explain Theorem 7 for this case, we take a simplicial decomposition of 1vI which is so fine that each simplex belongs to a neighborhood of M. Let K 2n - 2r +l = K be the (2n - 2r + I)-dimensional skeleton of the simplicial decomposition. There exists a continuous mapping 'It of K into ij(r)* such that 7I"'It(p) = P for every p f K . Let u be a simplex of dimension 2n 2r + 2. The mapping 'It defines a mapping of the boundary au of u into \5(r)* and the mapping X'It defines a mapping of au into Fo , and hence a cycle of dimension 2n - 2r + 1 of Fo. Let this cycle be homologous to a multiple 'Y(u) of the generating cycle of dimension 2n - 2r + 1 of Fo. According to Theorem 7 the cocycle 'Y(1/;) defillf~d by assigning the integer 'Y(u) to u belongs to the rth basic class. Moreover, it was also proved 27 that to a given cocycle 'Y of therth basic class there exists a mapping 1/; such that 'Y(1/;) = 'Y. We suppose 1/; to be so chosen. Let 71"' be a simplicial approximation of 1T" and let 'Y* be the inverse image of 'Y under 71"' as defined above. We shall show that 'Y* is the coboundary of an LEMMA

[1], p. 501: [21], p. 124.

26 ALEXANDROFF-HoPF, 27 STEENROD,

152 105

CLASSES OF HERMITIAN MANIFOLDS

+

integral cochain {3* of dimension 2n - 2r 1 of ij(rl*. To define (3* let r* be a simplex of dimension 2n - 2r 1 of ij(r)*, and let r = 11"'( r*). The discussion will be divided into two cases, according as r is of dimension equal to or less than 2n - 2r + 1. Suppose r be of dimension 2n - 2r 1. In this case 11"' establishes a simplicial and therefore topological mapping between rand .,.*. By Lemma 4 there exists a mapping 11"" ( r) C ij(r)*, which is homotopic to 11"'-\ r) and coincides with 'l' on the boundary or. We then take an oriented sphere of topological dimension 2n - 2r 1 arid denote by HI, H2 its two hemispheres. We map HI and H2 into r by the mappings hI and ~ of the degrees -1 and + 1 respectively, such that hi and ~ are identical on the "equator" HI n H 2 . A mapping I of the sphere HI + H2 into ij(rl* is then defined by the conditions

+

+

+

I(p) I(p)

= =

P E HI,

1I""h, (p), f~(p),

p E H2 .

This mapping I is by construction continuous. Taking its projection AI on Fo , we get an element of the (2n - 2r + I)-dimensional homotopy group of the fibre and hence an integer, on account of the orientability of the bundle ij(rl* . This integer we define to be {3* . r* . I t is to be remarked that {3* . r* in general depends on the deformation which carries 11",-1 to 11"", but only one such deformation will be utilized, and (3*. r* is thus well defined. Next let r be of dimension < 2n - 2r 1. We suppose without loss of generality that po = 1I"(Fo) E r. Let any simplex r' of dimension 2n - 2r 1 be mapped into r* by a non-degenerate .o rientation-preserving simplicial mapping. By Lemma 4 this mapping is homotopic to a mapping 1I"'''(r') C ij(rl* such that 1I""'(or') = qo, where qo is a point of Fo. We identify all the points on or', thus getting an oriented sphere of topological dimension 2n - 2r 1, which is mapped into Fo by the mapping A1I"'" (r') C Fo . This mapping defines an element of the (2n - 2r + I)-dimensional homotopy group and hence an integer, \\·hich is defined to be {3*. r*. It remains to show that the coboundary of the cochain (3* so defined is equal to -y*. For this purpose let q* be a simplex of ij(rl* of dimension 2n - 2r 2. It is sufficient to verify that -y*. u* = o{3* · u* = f3*. ou* . Suppose first that u = 1I"'(u*) is of dimension 2n - 2r 2. The mapping 7r" is defined for each simplex of AU and hence for AU itself, because it coincides with f on the (2n - 2r)-dimensional skeleton K 2n - 2r • We therefore have two mappings, Af and A7r" respectively, of AU into Fo such that they are identical on [(2n-2r . Our fibre Fo being (2n - 2r)-simple, this is a situation discussed by Eilenberg,28 who introduced several cochains, denoted in his notation by c(Af) , C(A7r"), deAf, A7r") respectively. In our notation they are given by

+

+

+

+

+

c(Af) . q

=

-y*. u* ,

C(A1I"") = 0, deAf, A1I"") ·ou 28 ElLE N BE RG,

[11], pp . 235-237.

=

f3*·ou*,

153 106

SHIING-SHEN CHERN

where the second relation follows from the fact that A1r" is defined for the simplex u bounded by au. From a theorem of Eilenberg we have odCA1/;, h") = c("A1/;) - c(h"),

or 'Y*' u*

=

(3* ·au*,

which is to be proved. Next suppose u = 1r'(u*) be of dimension <2n - 2r 2. If each simplex of au* is mapped by 1r' into a simplex of dimension <2n - 2r 1, {3*·au* is 1 and clearly zero. The other possibility is that u is of dimension 2n - 2r that exactly two of the simplexes of au* , say T~ and T:, are mapped by 1r' into 11. If T~ and T: are coherently oriented with the boundary i}u*, it follows by definition that {3*' (T~ T:) = O. On the other hand, it is not difficult to see that {3*' (i}u* - Tl* - T2) * = O. Hence we have {3*·i}u* = O. We have thus proved that 'Y* is the coboundary of a cochain {3*. To see that (3* reduces on a fibre Fo , we suppose the simplicial decomposition of \1(r)* so made that Fo is a subcomplex. Every simplex of Fo is mapped by 1r' into a point, so that we have 'Y*' u* = 0 for every simplex u* of dimension 2n - 2r 2 of Fo , which shows that {3* reduces to a cocycle on Fo . To show that {3* is the fundamental co cycle on Fo , it is sufficient to show that one is the value of its product with a cycle belonging to the generating homology class of dimension 2n - 2r -r 1 of Fo . For this purpose we take an oriented sphere of 1 and map it simplicially into Fo such that the topological dimension 2n - 2r map belongs to the generator of the (2n - 2r I)-dimensional homotopy group of Fo. The image of this map defines a cycle of Fo belonging to the generating homology class of the (2n - 2r I)-dimensional homology group, and its product with {3* is 1. It follows that (3* reduces to a fundamental cocycle on Fo. Our Theorem 8 is now completely proved.

+

+

+

+

+

+

+

+

3. In terms of differential forms Consider again the Grassmann manifold H(n, N) . We take as point of a new space a linear space E(n) and r vectors en- r+!, .. . , en belonging to E(n) such that eiej = O,j, n - r 1 ~ i, j ~ n . This space, to be denoted by R(r, n , N), is clearly a fibre bundle over H(n, N), the projection of a point of R(r, n, N) being the corresponding E(n) and each fibre being homeomorphic to U* (n, r). This fibre bundle R(r, n, N) is transformed transitively by the unitary group in the space E(n N). Let TV r be the rth basic class of H(n, N) and'Y E TVr be one of its cocycles. 'Y is mapped by the inverse mapping induced by any simplicial approximation of 1r into a cocycle having the properties asserted by Theorem 8. In particular, we can take for'Y the cocycle defined by the differential form r . The inverse image of r in R(r, n, N) under 1r is a differential form which for convenience we denote by r. The form r then defines a cocycle 'Y* in R(r, n, N).

+

+

154 107

CLASSES OF HERMITIAN MANIFOLDS

From Theorem 8 it follows thae 9 there exists a cochain (3* in R(r, n, N) such that l>{3* = -y* and such that {3* reduces to a fundamental cocycle on a fibre. In order to define (3* in R(r, n, N) by means of a suitably chosen differential 30 form, we shall make use of the following lemma proved by de Rham: LEMMA 5. Let 111 be a compact differentiable manifold of class ~2 and let K be a simplicial decompositwn of lIf, whose simplexes are a'; , i = 1, ... , aI" P = 0, 1, ... , n, and whose incidence relations are

au,p.

__

a~l

(1')

L

1'-1

1)ij

Uj

j=1

Then there exists a set of differential forms (1')

cP i ,

i

p = 0, 1, ...

= 1, .. . , aI"

,n,

such that the following conditions are satisfied: 1)

1

cp~p) =

l>ij,

Uj(p)

2)

A __ (p)

_"

""Pi

-

(1'+1)

L.J

1)ji

(1'+1)

cPj

•

We apply this lemma to the manifold R(r, n, N), and write for simplicity 1. The cochain {3* is defined by definition by a system of equations p

= 2n - 2r

+

1, . . . ,

i

hi,

(XI'.

Let ap

W

=

L

Xicp~P).

i=l

Then we have

1 p

U

w

=

x•

i

which shows that the differential form w defines the cochain (3*. we have

By construction

dw = r •

+

+

Now the unitary group U(n N) in E(n N) transforms transiti,oely the manifold R(r, n, N). Let s be a transformation of U(n N). If (J is a differen-

+

29 We have tacitly assumed at this point of our discussion that the Theorem 8, proved by a combinatorial construction for the simplicia.l approximations of the projection 1T, holds for 1T itself, when the cocycles are expressed by means of differential forms. It is, however, possible to avoid this assumption by observing that the cochain (3* exists in R(r, n, N) such that 0(3* = 'Y~ . That (3* reduces to a fundamental co cycle on a fibre then follows from the very definition of the characteristic cocyc\e on H(n, N). 30 DE RAHM, [20], p_ 178.

ISS

108

SHIING-SHEN CHERN

tial form in R(r, n, N), we shall denote by sO its transform by the transformation s. We also use the notation 0 '" 0 to denote that 0 is derived. LEMMA 6. Let f3* be a cochain of dimension 2n - 2r 1, whose co boundary is 'Y*. Let w be a differential form which defines f3*. Then sw - w '" O. First of all, the differential form sw - w is exact, since we have

+

d(sw - w) = ~r

s

+ 1 of R(r, n, N).

Let be a cycle of dimension 2n - 2r that

The cycle Sw Then we have

-

<1>, = O.

-

w being homologous to zero, let

1 r

sw - w =

1

.r-r

w

=

Z* be the chain it bounds.

1 =f w

z·

iiZ'

It is sufficient to prove

<1>,.

Let Z be the projection in the base space H(n, N) of the chain Z* in R(r, n, N). The boundary of Z is the projection of Since the Betti group of dimension 2n - 2r + 1 of H(n, N) is zero, the projection of S bound's in H(n, N) a chain which we shall call Zl ' Then the projection of bounds the chain SZl , and we have

ss - s.

ss

It follows that

Thus Lemma 6 is proved. ThEOREM 9. Under the projection of R(r, n , N) into H(n, N) the differential form <1>, is mapped by the inverse mapping into R(r, n, N). There exists a differenN) tialform 'If' which is invariant under transformations of the unitary group U(n operating in R(r, n, N) and whose exterioT derivative d'lf' is equal to <1>•. By Lemma 5 we can construct the differential form w in R(r, n , N) such that

+

dw = <1>,. For such a differential form w it follows from Lemma 6 that 5w - w '"

o.

Let dv be the invariant volume element of U(n dvover U(n N) is equal to 1. We put

+

IT =

1

U(n+N)

+ N) such that the integral of

sw dv.

156

109

CLASSES OF HERMITIAN MANIFOLDS

Then II is invariant under Ul..n

xII

=

1

U(n+N)

+ N) and we have

s·dw·dv = 4>r

1

U( ..+N)

dv ... 4>r,

which proves our theorem. To a point of R(r, n, N) we now attach the frames el , ••• , e,,+N in E(n N) such that el, ... , en determine the E(n) and that en-r+l , ••. , e" are the vectors in question. For clearness let us agree in the remainder of this section on the following ranges of indices:

+

+ 1 .;;:;; A, B, C ~ n, n + 1 ;;:;; i, j, k ~ n + N.

1 ;;:;; a, (3, 'Y ;;:;; n - r,

n - r

In a neighborhood of R(r, n, N) we can choose a differentiable family of such frames, one attached at each point of the neighborhood. By means of the family of frames the forms (JAa , BAa, (JAi , OAi, (JAB can be constructed according to the equations (7) . They constitute a set of linearly independent linear differential forms at each point of R(r, n, N). Our form'll", whose existence was asserted by Theorem 9 and which is invariant under U(n + N), is necessarily a polynomial (in the sense of Grassmann algebra) in the forms of this set, with constant coefficients. On the other hand, the form II, being itself in R(r, n, N), must be invariant under the transformation (J:"1 ~. = £...J ~a '1 · ·(Jn.J •. •

;

where a,j are the elements of a unitary matrix. e AB =

L

We put

(JAi (JiB

i

It follows from the first main theorem on vector invariants of the unitar) group that II is a polynomial in (J Aa , -"a, (JAB, e AB ,with constant coefficients. Moreover, on a fibre, that is, omitting all tenns in e AB , II becomes a fundamental cocycle. All these results can be summarized in the following theorem: THEOREM 10. There exists in R(r, n, N) a polynomial IT in (JAa, IJ Aa , (JAB, eAR , with constant coefficients, such that dIT = 4>r. When all terms involving e A8 in IT are omitted, the form defines a fundamental cocycle on a fibre. CHAPTER

IV

HERMITIAN MANIFOLDS

1. Fundamental formulas of Hermitian Geometry

Let M be a compact complex manifold. M is called an Hermitian manifold, if an intrinsic Hermitian differential form is given throughout the manifold. In each local coordinate system :;' the Hermitian differential form is defined by

157 110 (28)

SHIING-SHEN CHERN

ds

2

n

=

L

(Iii (Z,

i

z) (dz dzi) ,

i';=1

where, as well as in later formulas, we insert a parenthesis to designate that the multiplication of the differential forms in question is ordinary multiplication . We shall agree, unless otherwise stated, that the indices i, j, k take the values 1 to n. Our main result in this chapter is to establish that the n basic classes which. arise from the analytic structure of a complex manifold, are completely determined by the Hermitian metric, if the manifold in question is an Hermitian manifold. In particular, as \ye shall see later, the theorem for the class WI reduces to the formula of Allendoerfer-Weil, if \\'e interpret the Hermitian metric as a Riemannian metric for the real manifold of 2n topological dimensions. We begin by establishing the fundamental formulas for local Hermitian Geometry. For this purpose we determine in a neighborhood of 111 n linear differential forms
2

ds =

(29)

L

(
i=l

The forms
(30)

Wi

=

L

Uii
z = 1 ... ,n,

1==1

where Uil are the elements of a unitary matrix

U = (nil)' and which we take to be independent variables. sponding to Wk, so that Wi(Ck)

Let

Ci

be the dual base corre-

= h.

From the Hermitian differential form a scalar product of the contravariant vectors can be defined, and we have We shall call a frame the figure formed by a point P and n such vectors C;. With a natural topology the set of frames constitutes a fibre bundle over M. The forms Wi are intrinsically defined in the fibre bundle. By actual calculation in terms of a local coordinate system, we find that their exterior derivatives are of the form

where

158

111

CLASSES OF HERMITIAN MANIFOLDS

so that Wii

But the fonus

W'J

+

Wii

=

0.

are defined up to the transfonuation

where the quantities Aiik are arbitrary. It is easy to show that there exists one, and only one, set of fonus Wi; , such that the following equations are satisfied: (31)

Wi;

+ Wj;

=

0,

From the uniquenest:l of this set of forms follows the fact that they are intrinsically defined in the fibre bundle. The forms Wi , Wi; constitute therefore a set of linearly independent linear differential fonus in the fibre bundle. From equations (31) it is possible to draw all the consequences of local Hermitian Geometry. In fact, we put (32)

Exterior differentiation of the first set of equations in (31) will give dn; -

L

Wkn k;

k

+L

nkWk;

k

= 0,

where we have put (33)

We remark that

dn i

+ Lk nk-Wki is of the

form

Li.k if;ijkWiwk.

and that we can put

On the other hand, we have

or

which shows that

X iki

is a linear combination of n

X iki

=

L

1=1

w" , Wk :

n

aikil WI

+L

1~1

biikl WI.

It follows that

159 112

SHIING-SHEN CHERN

Substituting this expression of Xikj into the last equation, we see immediately that aikjl must be symmetric in the last two indices j, l. It follows that n,k is of the form

L

=

n ik

bikjlWIWj .

The equations for the exterior derivatives dWi , dWsj and the equations obtained therefrom by exterior differentiation we shall call the fundamental equations of local Hermitian Geometry. These equations will now be summarized as follows: dw,

L

=

WjWj,

=

dWij

+L +L

dni

L

W ik Wkj

k

L L

njWj, -

j

(34)

dn,j

n,kWk; -

W,j

L"

ij ,

Wjn ji

=0,

Wiknkj

= 0,

j

k

ni ; =

+n +n

i ,

j

k

R'j. lm WI Wm ,

+

Wj,

= 0,

n,j

Rji .ml ,

R ij•lm

=

+n

= O.

I.m~l

j,

In a well-known way the forms Wi, W ij can be interpreted as defining an infinitesimal displacement, by means of the equations

dp =

L

w,e i

,

i

(35)

Of importance are the Hermitian metrics satisfying the condition

n.

(36)

= 0,

which will be called Hermitian metrics without torsion. without torsion can be characterized by the condition

An Hermitian metric

d(l: w,w.) = 0,

(37)

i

and was studied by E. Kahler. 31 Kahler proved that in this case there exists locally a function F(z" iii) such that the metric can be written in the form 2

(38)

"i)2

ds = L...J

F

.:> • .:>-k i.k uZ uZ

i_k

(dz dz).

Hermitian metrics ",rithout torsion play an important role in the theory of automorphic functions of several complex variables. 31

KAHLER, [16J.

160 CLASSES OF HERMITIAN MANIFOLDS

113

2. Formulas for the basic characteristic classes As defined in Chapter III there are on 111 n basic characteristic classes, the rth one (1 ~ r ~ n) being of dimension 2(n - r + 1). We shall show that, if 111 is an Hermitian manifold, the~e classes are defined by the local properties of the Hermitian metric. For this purpose we put 1

(39)

'¥r

= (27rv _ I _l)n-r+\n - r

+

I)!

where n;; are the forms defined in (34) and where the meaning of the summation has been explained before. Then we have the following theorem, which is the main result of this paper: THEOREM 11. The farm '¥r defined by (39) is the farm carresponding to' Ihe r'h basic characteristic class W r in the sense that the praduct af any homalagy class r af dimensian 2(n - r + 1) with W r is equal to' the integral af '¥ r aver r:

r· Wr =

(40)

i

'¥r .

We first establish the following lemma: 7. Let t. be the differential farm II in Thearem 10, with every (J and 8 replaced by the carrespanding wand n with the same indices. Then dt. = '¥r . We observe that the equations for dWi; , dn i ; are exactly of the same fonn as the equations for d(Ji; , d8 ij , the only difference being that 8 ij is given by the equation (12). It follows that LEMMA

,,+N

dt. - '¥r

==

0 mod. nij -

L:

B-,,+1

(JiB (JB;.

By mathematical induction on the degree of dt. - '¥T it is easy to show that then dt. - '¥, = O.

To prove Theorem 11 we make use of the definition of Wr given in Theorem 7, with ~(r) replaced by ~(r)*. A sufficiently fine simplicial decomposition K of 111 is taken and a continuous mapping if; of its (2n - 2r + I)-dimensional skeleton into ~(r)* is defined, such that the image of every point belongs to the fibre over it. Let
r,

equation (40a) will follow

161 114

SHIING-SHEN CHERN

But, by Lemma 7,

The image 1/;(11 A) has a singular point and we see that the last integral is precisely the definition of ')1(11 A) given in integral form . Hence Theorem 11 is proved. 3. The case r

= 1 and the formula of Allendoerfer-Weil

In the case r = 1 we can take for the cycle ~ in (40) one of the fundamental cycles of the manifold M. Then we have ~. W l = x, the Euler-Poincare characteristic of M. On the other hand, the manifold M can be considered as a real differentiable manifold and the Hermitian metric can be used to define a Riemannian metric in the real manifold. It is to be expected that the formula (40) will then reduce to the formula of Allendoerfer-Weil. We shall show that this is actually the case, if the Hermitian metric is without torsion. To study the Hermitian metric as a Riemannian metric, we decompose each of the forms w. , Wij , fl' j into its real and imaginary parts, writing

W. (41)

+ Y-11/;., ().; + Y -l1/;'j, 8.; + Y -1 '!I'i·

= ().

=

Wij

fl'j =

From the last two equations of (34) we have .

(41a)

().; + ();. = 8.; + 8 i • =

0,

1/;.; -

1/;;.

= 0,

0,

'!I.; -

'!I j •

= O.

The Hermitian metric can then be written as a Riemannian metric in the form:

di =

L {((),/ + (1/;.il . •

We get, moreover, by separating the real and imaginary parts of the equations for dw., dw.;, the following equations: (42)

d8. =

L;

()j()j' -

L;

1/;i!J;;; ,

d1/;. =

Li

();1/;ji

+ Li

1/;i()j. ,

and (43)

It follows that, for the Riemannian Geometry of 2n dimensions thus obtained, the curvature forms can be conveniently described by the matrix

162 115

CLASSES OF HERMITIAN MANIFOLDS

(44)

or simply by

(!

-'It)

8.

It only remains to compare the integrand of the Allendoerfer-Weil formula calculated from this matrix of curvature forms with the expression 'ltl defined in (39) .

Let 12 denote the integrand of the Allendoerfer-Weil formula. We observe that (- 2'71)"12 obeys the same expansion rule as a so-called Pfaffian function of the 2n t h order,32 which is an integral rational function in a number of independent variables, whose square is equal to the value <;>f a ske"'-symmetric determinant of order 2n. It follmys that -'It

8

I.

On the other hand, ,ye have 1

from which we get, after some reduction, (-lr(27r)2n 'lt~

= 18 + ~ -1 'It

_18+ V -l'lt - 8+ V-I'lt

8+

~-I'lt1

-8+~'lt1 = (-lr

I:

-'It

8

I.

Hence we have

and finally 12 =

'lt

l ,

by comparing the coefficients of one of the terms in both sides. CHAPTER

V

APPLICATIONS TO ELLIPTIC HERMITIAN GEOMETRY

1. Preliminaries

We are going to make use of the above results to derive some consequences for elliptic Hermitian Geometry. 32 PASCAL,

[17], pp . 60-64 .

163 116

SHIING-SHEN CHERN

The local Hermitian Geometry whose fundamental equations are given by (34) is called elliptic Hermitian, if we have

(45)

i

~

j.

This definition owes its origin to the type of geometry studied by G. Fubini,33 E. Study,34 and E. Cartan,35 which we shall prove to satisfy the conditions (45). The elliptic Hermitian Geometry of Fubini-Study is defined as follows: We consider the complex projective space of n dimensions P n , with the homogeneous coordinates z", where, as well as throughout this section, we shall use the following ranges of our indices:

o~

ct,

{J, 'Y ~ n,

1

~

i, j, k

~

n.

In P" let a positive definite Hermitian form be given: (46) which will serve to define the scalar product of two vectors in the affine space An+! of n 1 dimensions, with the (non-homogeneous) coordinates z". We normalize the coordinates z" in P n, such that

+

(47)

(zz)

=

1.

An Hermitian metric in P n is then defined by the Hermitian form (48)

di

=

(dz dz) -

(z dz)· (2 dz).

The group of linear transformations in An+! which leaves the form (46) in1). We take in An+l n 1 vectors Ak variant is the unitary group U(n such that

+

+

(49)

For a differentiable family of such sets of vectors we have (50)

dAa

=

LII

(J,.pA Il ,

where (51)

and where the forms (52)

33 FuBlNI, 34

STUDY,

[13). [25).

35 CARTAN,

[4).

(Jail

satisfy the equations of structure:

164 117

CLASSES OF HERMITIAN MANIFOLDS

Let B", be fixed vectors in An+! , satisfying the equations = ~afJ'

B",B{J

The coordinates of a vector Ao with respect to B. are defined by the equation

Ao

=

L: z'" B., a

from which we find

(dAo dAo) -

(Ao dAo)(Ao dAo)

=

(dz dZ) -

(z dZ)(z dz).

It follows that, if we regard Ao as defining the points in p" , the Hermitian form in (48) can be written as ds2

= (dAo dAo) - (Ao dAo)(Ao dAo),

and, by (50), as " ds2 = L..J (0 0,0- 0 ,),

(53)

;

This proves in particular that the Hermitian form in (48) is positive definite. To calculate the curvature of the metric (53) we shall make use of the equations (52). Using the notations of local Hermitian Geometry in the last Chapter, we put

=

W,

(JOi.

Then, by (52), we get

dw, = deo, =

L: Wj(Jji + W;«(Jii

-

(Joo).

;~;

From the uniqueness of the set of forms equations of (34) it follows that

Wi;

=

(J;; -

Wi;

satisfying the first and the ninth

(Joo •

We find then

dw ij =

L: WikWkj + (JiO(JOj,

i

k

and therefore i

~j,

~

j,

165 118

SHIING-SHEN CHERN

which are exactly the equations (45). Thus we have proved that it is possible to define in the complex projective space an elliptic Hermitian Geometry. 2. Fonnulas of Cartan apd Wirtinger When the Hermitian manifold is locally elliptic, it is possible to calculate the forms 'lI r in (39) more explicitly. In fact, we are going to prove the following theorem: THEOREM 12. In a locally elliptic H ermitian manifold let A be the exterior differential form corresponding tv t.he H ermitian differential form, that is, A = 2 Li WiWi in case the given H ermitian form is ds = Li (w ic7' Then we have (54)

'lI r

= (27r V

1

-It

r+l

(n + 1) r

A

n-r+l

,

1

~

r

~

n.

It is clear that the construction of A from ds 2 is independent of the choice of the base linear differential forms Wi in terms of which ds 2 is expressed. The theorem is proved by induction on n - r. If n - r = 0, that is, r = n, we have

n

1

'lin

= 27rV-l ~

Suppose the formula (54) be true for r

Aii

+1

= 27rV-l

+ 1, ...

, n.

A.

We have

Consider the second sum inside the braces. If i n - r ~ jn-r , we can replace the sum Lk Qin-rkQkin-r by Win_,Wi,,_,A = Qi.. _,in_,A. If i n- r = jn-r, the sum can be replaced by (Win_,W'n_, + A)A = Qin-,in-,A. By our induction hypothesis we get easily the desired formula (54). As an application let us determine the n basic classes of the complex projective space of n dimensions. The result is given by the following theorem: THEOREM 13. The Tth basic characteristic class (1 ~ r ~ n) of a complex projective space of n dimensions is dual to the homology class containing the cycle carried by a linear subspace of dimension

T -

1

mulhplied by (n ;

1).

To prove this theorem we consider the affine space .1.+1 of dimension n + 1 with the coordi~ates za such that the projective space P n under consideration is the hyperplane at infinity with which .1n~ 1 is made into a projective spacc of n + 1 dimensions. As before, za are homogeneous coordinates in P n . Suppose that we hav'e defined at each point 0 of .4 n+l different from the origin T vectors \)i , 1 ~ i ~ r, whose components v~ , ... , 07 are linear forms in the coordinates

166 119

CLASSES OF HERMITIAN MANIFOLDS

za of a. Then all the points on the line joining the origin 0 to A are projected from 0 into the same point p of P" and the vectors \). , 1 ~ i ~ r, at the points of 03 are projected from 0 into the same vector of p", which we attach to p. I t. is easy to verify that the r vectors thus defined at each point of P" are linearly independent when and only when the r + 1 vectors A, \)1 , ••• , \)r in An+! are linearly independent. We take 1 ~ i ~ r,

where a'i are constants. The a'i can be so chosen that none of the determinants of order r + 1 of the matrix

will vanish. It then follows that the vectors A, dependent when and only when all products

o~

\)1, . . . , \)r

£Yo, • •• ,

lXr

~

will be linearly

n,

thp indices lXo, •• • , lXr being distinct from each other. This is possible when and only when n + 1 - r of the coordinates z« will vanish. In other words, for this particular field of r vectors the points of P n at which the r vectors are linearly dependent are the linear spaces of dimension r - 1 defined by setting It + 1 r of the homogeneous coordinates of P n to zero. The number of

such linear spaces is

(n ~ 1)

and each of them is to be counted simply.

Hence

our theorem is proved. THEOREM 14. (E. Cartan).36 Let M be a closed submanifold of topological dimension 2n - 2r 2 of the complex projective space of n dimensions P" in /Chich an elliptic Hermitian Geometry is defined. Let m be the number of points oj intersection of M with a generic linear sUbspace of dimension r - 1 of P n. Then

+

m -

(55)

- (271T -

1

1n - r +1)

1

A,,-r+l

m

I n particular, if M is an algebraic variety of dimension n - r -r 1 in P n , m is its order. This theorem is an immediate consequence of the Theorems 11, 12, and 13. Related to these discussions is also a formula due to W. Wirtinger. In Hermitian Geometry, as in Riemannian Geometry, it is common to define an element of volume of topological dimension 2p by means of the equation ~2n~

(5G) 36 CARTAN,

[3], p. 206.

= ±

"L..,

(i,· · · ipl

W·

'I

••• W ·

'1'

W·'. ... W·'I'

,

167

120

SHIING·SHEN CHERN

the summation being over all the combinations of i 1 , 1

to a sign A2p is equal to -; A P •

p.

••• ,

i p from 1 to n.

Up

We define

(57)

From (55) follows the theorem: THEOREM 15. (W. Wirtinger).37 In the complex projective space of n dimensions with the elliptic Hermitian metric let V 2p be a p-dimensional algebraic variety of order m and volume V. Then

V =

(58)

(21rY -,-m. p.

INSTITUTE FOR ADVANCED STUDY, PRINCETON, AND TSING HUA UNIVERSITY, CHINA. BIBLIOGRAPHY L ALEXANDROFF, P., UND HOPF, H., Topologie I, Berlin 1935. 2. ALLENDOERFER, C . B ., AND WElL, A ., The Gauss-Bonnet theorem for Riemannian polyhedra, Trans . Amer. Math. Soc ., Vol. 53 (1943), pp. 101-129. 3. CARTAN, E ., Sur les invariants integraux de certains espaces homogenes clos et les proprietes topologiqucs de ces espaces, Annales Soc. pol. Math., Tome 8 (1929), pp. 181-225. 4. CARTAN, E., Le~ons sur la geometrie projective complexe, Paris 1931. 5. CHERN, S., A simple inlTinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Annals of Math., Vol. 45 (1944), pp. 747-752. 6. CHERN, S., Integral fOT'lliulas for the characteristic ' classes of sphere bundles, Proc. Nat . Acad . Sci., Vol. 30 (1944), pp. 269-273. 7. CHERN, S., Some new viewpoints in differential geometry in the large, to appear in Bull. Amer. Math. Soc. 8. EHRESMANN, C., Sur la topologie de certains espaces homoglmes, Annals of Math., Vol. 35 (1934), pp. 396-443. 9. EHRESMANN, C., Sur la lopologie des groupes simples clos, C. R . Acad . Sci . Paris, Vol. 208 (1939), pp. 1263-1265. 10. EHRESMANN, C., Various notes on fibre spaces in C. R. Acad . Sci. Paris, Vol. 213 (1941), pp. 762-764; Vol. 214 (1942), pp. 144-147; Vol. 216 (1943), pp . 628-630. 11 . ElLENBERG, S., Cohomology and continuous mappings, Annals of Math., Vol. 41 (1940), pp . 231-251. 12. FELDBAU, J ., Sur la classification des espacesfibres, C.R. Acad. Sci. Paris, Vol. 208 (1939), pp. 1621-1623. 13. FuBINI, G ., Sulle metriche definite da una forma Hermitiana, Instituto Veneto, Vol. 63, 2 (1904), pp. 502-513. 14. HOPF, H., see ALEXANDROFF, P. 15. HUREWICZ, W., AND STEENROD, N ., Homotopy relations in fibre spaces, Proc. Nat. Acad . Sci ., Vol. 27 (1941), pp . 60-64. 16. KAHLER, E ., UbeT eine bemerkenstwerte Henllilische Metrik; Abh. Math. Sem. Hamburg, Vol. 9 (1933), pp . 173-186. 17. PASCAL, E ., Die Determinanten, Leipzig 1900. 18. PONTRJAGIN, L ., CharactcTistic cycles on manifolds, C. R. (Doklady) Acad . Sci. URSS (N. S.), Vol. 35 (1942), pp. 34-37 . 37

WIRTINGER, [311.

168 CLASSES OF HERMITIAN MANIFOLDS

121

19 . PONTRJAGIN, L., On SOIllC topologic invariants of Riemannian manifolds, C. R. (Doklady), Acad . Sci . URSS (N. S.), Vol. 43 (1944), pp . 91-94. 20. DE RHA~I , G., Sur l'analysis sit1ls des !'arietes d n dimensions, J. Math. pures et appl., Tome 10 (1931), pp. 115-200. 21. STEENROD , X ., Topological method.s for the constrllction of tensor functions, Annals of l\Iath., Vol. 43 (19-12), pp. 116-131. 22. STBENROD, ~., The classification of sphere bundles, Annals of Math., Vol. 45 (1944), pp. 2n-t-311. . 23 . Sn:ENRoD, X ., see I1UREWlCz, W. 24 . STIEFEl" E ., Richtungsfelder lind Fernparallelismus in n-dimensionalen Mannigfaltigkeitcn, Comm . Math . Relv., Vol. 8 (1936), pp. 305-343 . 25. STUDY, E., Kurzeste Wege im komplexen Gcbiet, Math. Annalen, Vol. 60 (1905), pp . 321-377. 26. VAN DER W.~ERDEN, B . L., Topologische Begriindung des Kalkuls der abziihlenden Geometric, Math . Annalen, Vol. 102 (1930), pp. 337-362. 27. WElL, A ., see ALLENDOERFER, C . B. 28. WEYL, II., The Classical Groups, Princeton 193n. 29 . WHITNEY, H ., Topological properties of differentiable mantfolds, Bull . Amer. Math. Soc., Vol. 43 (1937), pp . 785-805 . 30. WIIITNEY, II ., On the topology of differentiable manifolds, Lectures in Topology, pp. 101-141, Michigan 1941. 31. WIRTINGER, W., Eine Determinantenidentitat und ihre Anwendung auf analytische Geb£lde in Euklidischer und Hermitischer Massbestimmung, Monatshefte flir Math. u. Physik, Vol. 44 (1936), pp. 343--365.

169 Reprinted from Science Reports National Tsing Hua Univ. 4 (1947).

~

,

SUR UNE CLASSE REMARQUABLE DE V ARIETES DANS L'ESPACE PROJECTIF

AN DIMENSIONS

Par M. Shiing-shen Chern (g{1tI.Ji' )

Department of Mathematics (Received

SePtem~r 6, 1!J4f})

RESUME On donne dans ceUe Note une propriettJ caracUristique d'une cia sse de varitJtes. dans l' espace projectif a n dimensions, etudiees auparavant par Cartan. Ces varietes fournissent une generalisation des surfac.es ayant un reseau. It "xiste une transformation entre ces varietes generalisant la transformation bien connu" de Laplace. INTRODUCTION La geometrie differentielle projective des reseaux conjugues1 ) a ete j'objet de recherches de bien des geometres. AnaI~tiquement cette theorie est -etroitement lire a. Ia theorie de l'equation aux derivees partielles de Laplace: (1)

od a, b, c' sont des fonctions de u, v. C'est.Ie probl~me de I'integration de I'equation (1) qui a conduit Laplace, Moutara, Goursat de fonder sur cette equation une theorie analytique profonde. Les interpretations geometriques de ces resultats sont dues Ii Darboux, Koenigs, Guichard, Tzitzeica, Bompiani, etc . , en ne citant que Ies noms les plus importants. Je vais signaler dans cette Note une c1asse de varietes Ii p dimensions dans I'espace projectif a n dimensions . qui jouerit de proprietes grneralisant les proprietes des surfaces sur IesquelJes IJ existe un reseau conjugue. M. Cart an a rencontre ces varietes dans un probl~rne de nature differente.'l Je me propose donc d'appeler ces variae5 Ies varietes de Cart an . §l.

NOTIONS PRE LlMINA IRES.

Nous dMinissons

d'apr~s

EQUA nONS FONDAMENT ALES.

Cartan comme

rep~re

projectifl} de l'espace projectif

Ii n dimensions I'ensemble de' n + 1 points analytiques (chacun it n+ 1 composantes) 328

170

329

SUR UNE CLASSE REMARQUABLE

A, At,···. An, dMinis Ii un facteur commun pr~. Un tel rep~re d~pend de fl(n+ 2) paramt~res

arbitraires.

Les ~quations du d~placement infiniMsimal sont de la

forme

(2) dAn=wno A-/-W111Al+",+wnn An.

01\ les W sont des formes de Pfaff a ces param~tres. En introduisant les notions "produit ext~rieur" et . 'd~riv~e exterieure" comme d..'habitude, on sait que les (J} satisfont aux ~quations

wij

=

[wlo Wj]+[wi t U!Jj] + ",+[WiH wniJ,

wio

=

[Win wooJ+[Wit wlftl+ ,,,+[w;n

qui s'appellent les

~quations

i. j=l • ....

H,

(3)

("nol.

de structure de I'espace .

Cela ~tant, consid~rons dans I'espace une variH~ V pap dimensions') et faisons correspondre a chaque point A de cette vari~t~ un re~re dont les n+ 1 sommets seront Ie point A, puis p points AI, A 2 , .... Ap situ~s dans I'hyperpian tangent. enfin n-: p autres points Ap+l' ... . An., Pour tout d~placement infinit~simal dans cette famille de rep~res on a Wp+I='" =Wn=O. (4)

;=1 ..... p; a=p+1. ....

H,

o~

les Cija sont des fonctions des param~tres sur la vari~M et !lont sym~triques par rapport aux indices. i . j . Soit q Ie nombre des formes quadratiques (S)

qui iont lin~airement ind~pendantes. On sa it qu'on peut choisir Ie rep~re attach6 au point A de mani~re que les formes ""PH, .... tPp+q soient Iin~airement

171 M. SHUNG-SHEN CHERN

330

iDd~pendantes, les formes !PP+Hl, ••. , !Pn ~tant identiquement nulles. lin6aire de cones du second ordre de sommet A dMini par l'~quation

Le

r~seau

est appeM Ie reseau asymptotique relatif au point A . 11 jouit de la propri~t~ que toot cone du reseau est engendre par les tangentes aux courbes de la vaJ'iCte dont Ie plan osculateur est situe dans line varietc plane fixe a p + q-l dimensions qui contient l'hyperplan tangent et est contenue dans I'hyperplan osculateur de Vp . Cela pose, M. Cartan s'est propose de determiner toutes les varietes dimensions dont Ie rcseau asymptotique est reductible a la forme 6 ) Al

w~+

(7)

... +lp wp=o.

Pour une telle variete on pourra evidemment supposer choisi Ie point A de rrlani~re a avoir !PP+'i=~'

rep~re

attache au

;=1, ... , p.

Les varietes cherchees peuvent etre done regardees comme les solutions du de Pfaff: W .. =o,

a=p+l, . .. , n,

W~, p+~=wi,

;= I, . .. , p,

ap

(8) syst~me

(9)

(OJ,

p+~=o,

i:f=j; i, j=l, ... , p,

wi, 2p+J.=0, i=l, ". , p ; l= ' , .. .. n-2p, LA derivation exterieure de ces equations nous donne alors I-p

[Wt (wp+~, p+i-2wi~+wOO)] - ~JWkWki)=O, ;=1, ... , p, -[WI Wjt]+[wj wP+j, p+~ ]=0, i:f=j ; i, j=l, ... , p, [wi Wp+ t , 2p+l] =0,

(10)

;=1, .'. , p; },=1, . ' ., n-2p.

U appliquant la theorie des syst~mes de Pfaff en involution, l\L Cartan a demontre va,ietes che,cMes dependent de pCp-I) fonctions a,bit,a;,es de de"" "'pments. Nous appelterons une teile variCte une variCtc de Cartan.

qtJe les

172

3J1

SUR UNE CLASSE REMARQUABLE

Supposons que lea indjces i, j, k prenneot ItlS va1eurs I, ...• I> et qu'lls sont tous distincts. Supposons encore que 1 prend les valeurs 1•... , ,.-2/>. Des equations (10) on dcduit

(11) Wp+i, 'f+),

=

wp+i, p+i -

ei.t Wi,

+

2wii

= - 1;: aji Wj +

Woo

Ci wi'

J

oil les coefficients des w sont des fonctions des paramNrcs auxquels depend Ie rep~re . En utiHsant les equations de structure et tenant compte des Equations (11), on a Wi

t

= [.w i

W'

(w . . -

[ wi (aji

t

+

-

(U ii

k

"ji Wjj

•

w jU

+

_ .

((JP+j. i

f

a jk akj wk )

1 (12)

~ "jk T:1 ki (Ok )]

+

J.

"i k ak ' [ (,). Wj '

a' J w · )]. J. J .

aji Wjj -

[Wj (b jt (u ii -

- E

LJ

(Unn -

U

•

On trouve done. en derivant la premiMe equation de ( 11 ). daji

db)· ;

=

= -

aji

C Wjj -

bJ·jC -

J

+

Wjo

+

i j i Wi

Wii +2 (u. ··-

~ .( bjk "ki

k:f:"

Wnn )

JJ

+

+

~ ajk aki (U k -

+

woo) -

(Ok

gji w j • (1).1<+ '

y

bii akj ) (Ok

aji ~ aki

J. '

+ ( gii

•

-

+

"'i J

(13)

h)·j w . J

aij -

~ b 'k aM )<0 .•

k:f:"j J

On tire de la premihe equation de (12 ) une consequence importante. que chacunt des equations Wi

=

0,

i

=

1, ".,

p

•

C'est

(14)

est rompletement integrable . Geometriquement eel a signifie qu'i! existe sur 1a varietc p families de varietes a p-l dimensions tangentes en chacun de leurs points a I'une des variCtes planes AA1· .. Ai~1 A;H···Ap. i=I, ... , p. Il est facile de voir qu'une telle variete a p-l dimensions est aussi une variCte de Cartan.

173

M. SHIlNG·SHEN CHERN

332

52.

UNE PROPRIfnE CARACT~RISTIQUE DES VARI:ETEs DE CARTAN

Nous dirons qu'un6 variete a p dimensions possede un reseau generalise ou simplement un reseau s'il existe sur la variete p families de courbes telJes que par tout point de la variete il passe une et une seule courbe de chaque famille et telles que quand l'hyperplan tangent est dfplace Ie long d'une courbe d'une de ces families deux hyperplans tangents infiniment voisins se rencontrent a une variet6 plane a p -1 dimensions taogente a tOl,ltes les courbes des p-l autres families passant par Ie point considere. II est clair que cette propriete generalise celIe d'un reseau conjugue sur une surface (variete a deux dimensions). Attachons a chaque point A de )a vari~t~ un rep~re AA1 .. ·An tel que la plane a p dimensions [AAl' .. Ap] soit l'hyperplan tangent au point A. avons donc les ~uations (4). Supposons que les p familles de courbes sont donn~es par les ~quations

vari~t~

Nous

(15)

Nous allons montrer q.ue les conditions necessaires et sujjisantes pour qu',m,

variete

a p dimensions ait un rtiseau gb.eralise donne par les equations • i=j::j. i.j=1. .....

p. a=p+ 1 •....• n.

(15)

SOft#

(16)

avec ta condition supplementaire que les jonctions giip+l' ..... giin. pour chaque. ji%e. ne sont pas toutes nulles. La condition est manifestement n~cessaire. En effect. quand l'hyperplan tangent est deplace Ie long de la courbe (15). la condition que Ie point %A+ x1A 1+ ... + xpAp

appartienne

a )'hyperplan tangent

infiniment voisin est que Ie point

soit une combinaison lineaire de A,Alo .... ,Ap. bJuations

Cette condition s'exprime par les

CIU

p ~ jn =1

g ., x . w Jlfa

J

k

=0,

a=p+ 1, .... , ...

174

333

SUR UNE CLASSE REMARQUABLE Pour que ces ~quations soient identique~ dans notre th~or~me soient remolies.

al' ~l1ation xi = 0

it faut que les conditions.

Nous supprimerons la d~monstration de la suffisance de ces conditions dont les raisonnements sont analogues. On peut remarquer qu'il suffit de supposer que la condition concernant I'in~ tersection des hyperplans tangents infiniment voisins en les d~pla~ant Ie long d'une courbe soit·vMifi~e pour 1'-1 des families de courbes sur la var;~t~. La rr.eme propri~t6 .pour les courbes de la p.i~me famille en r~sulte com me leur con~~quence. Pour une

vari~t~

ayant un

r~seau

on a donc

0=1'+ 1. .... , n.

(17)

Soit q Ie rang de la matrice

( 18)

de sorte que q des formes <Pa soient )in~airement inc~pendantes. II en les hyperplans osculateurs de la vari~t~ sont a I' + q dimensions et que

r~suIte

que

1 <:: q ~ P , q .~ n-p. APT~S

la

ces discllssions pr~liminaires on peut ~noncer dans Ie th~or~me sUlvant caractMistique pour les vari~t~ de Cartan dMinies au num~ro pJ&Ment:

propri~t~

La ccndition n.'cessaire et suffisante pour qu'une variite so;1 une variete tU Cartan est qu'it existe sur elle un reseau generalise et que s,s hyperplans osculateur s soie11 t a 21' dimensions. II est ~vident que la condition est 5uffisante remarquons qu'on a

n~cessaire.

i= I, ...•

p.

Pour voir qu'elle est aussi

175 M. SHLING-SHEN cHERN

334

Dans ces

~quatiol!s

les poillts

g iiP+l A P+l +"'+g I,n . A n' ;=1· " .. p' sont

lin~airement

independants.

On peut mod:fier Ie rerMe attache au point A

en les choisissant com me les points Ap+l ,,,~.' A 2p du rep~re nouveau . Avec cette famille de rep~res nouveaux "n voit facilement que la vari~t~ consideree. est nne vari~t~ de Cartan.

§3.

LA TRANSFORMATION GENERALISEE DE LAPLACE

Dans ce num~ro nous allons ~tudier une transformation des vari~t~s de Cartan, qui jouit de proprj~tes analogues a la transformation bien connue de Laplace des

r~seaux

Commen~ons

conjugues .

a d~monter Ie theor?-me suivant:

Sur chague tangenfe AAi £l ,xiste p-I Mints (19)

ayant la proprietd que quand la. va.1ih6 ..tv dimensions [AA;1 ... Ai.,) placre Ie long de AA j

•

j"#

£1> "', i"" deux varihtl planes

ment voisines ont en commun la variet'; plane Pour demontrer ce

il

.....

i",'

th~or~mc

n v-I

+

J est que Ie

inJin;.

dimensions [Ail i ... Ai"j].

l

.. .

Ai" ] est de la forme

~ %t At t

La condItion pour qu'i1 appartienne aussi [AA~ ... Ai",

de-

supposnns que I'indice t prend les valeurs

Un point de la variete plane [AAi

x A

a v dimensions

est

a la

variele plane infilllment voisine de

point

x dA

+ 'E

Xt

dAI

t

soit une combinaison lineaire de A , Ai 1 ..... Ai 1-' . Lorsque [AAi 1 ... Aiv. ] d~place Ie long de AA j , cette condition s'exprime par I'equation

se

176 335

SUR UNE CLASSE REMARQUABLE Celd montre que I'intersection de [AA . . ... A. ) et de la '1

"v

variet~

plane infiniment

voisine se compose des points

OU " I}

, ".,

"sont arbitrlJ.ires. iv

Done Ie

th~or~me

est

d~montre.

Lorsque Ie point A decrit la variete V p, chacun des points Ajl (i, j

p; • "p j) decrit une

=

I, ' . . ,

vari~te

dont les points sont en correspondance biunivoque avec ceux de V p. Pour ces vari~M on a la propriet~ fondamenfals suivan~e: La varit'te dt'crite par Aji est en gdnhal {me variete de Cartan. Pour detnontrer ce theor~me on utilise les equations du deplacement in. finitesimal (2 ) et les equations (13;. On trouve alors

OU I'on a pose (21)

En supposant que

on voit que la variHe decrite par Aji est a p dimensions, dont I'hyperp)an tangent au point Aji est la varide plane [AAJi . .. Ai_Ii A i + li ··· APi B j , ] . Pour voir si elle cst une varicte de Cartan calculons ?A, dAhl (k"pi, j), dBjt mod. A, Akl (k=foi ), Bji . c'est·a~dire mod.

A, Ak (k"pi), Ap+i

+

bii A,.

Un

calcu)

facile nous donne

Cela montre que la variede de.crite par Ajl est bien une variHe de Cartan, si les points

177 336

M. SHllNG-SHEN CHERN

sont lineairemel1t independants. En generalisant la notion classique de la transformation de Laplace on peut d ire que les varietes decrites par Aji ' i =f:: j; i = 1,···, P sont les varietes transformees de Laplace de la variHe donnee. Vne variete de Cart an poss~de done p(p-l) variete transformees de Laplace qui sont en general aussi les variCtes de Cartan . Dans Ie cas !>pecial p = 2 on obticnt les deux transformes de Laplace d 'un reseau conjugue. On peut chercher a trouver les transformes de Laplace de la variete decrite par Ajt Eo ce qui concerne ce probl~me signalons les resultats suivant5 dont la demonstration est d'ailleurs facile: :u.s tangentes auX courbes du reseau generalise Aji etan! les droites A j ; A, Ajt Aki (k =f:: i, j), Aji BJi , les trans/ormes de Laplace sur Aj; A sont Ajl (l=Fi,j), A, ceux sur Aji Aki(k=f::i, j) sont

/M A,ii - /ji Aki, A ki , (aki - akl ) Aji

+ ( ajl

- aji ) Aki , t=f:: ': , j, k,

tandis que ceux sur Aji Bji sont dotmt:s par· des expressJ'ons pItts compliquces . BIBLIOORAPHI E 1. Les travaux sur Ia theorie des n~seaux sont nombreux. Citons seulement Ie livre: G . Tzitzeica, G.om'trie differentielle projectiv6 des rtseaux, Paris J\)24 .

2. E. Caftan, Sur les varittes de courbure constante d' un espace euc1idietl 014 "0".uclidien, Bull. Soc . math . de France, 47, 12~-1(iO (I Ul:I), 48, 13~-~OS (1!120). Cette Note lera citee comme; Cartan. vade/os. 3. Cartan. variit;s. nos. 1. 2. 4. Cartan. vadet[s. nos . 33-38. 5. Cartan. vartetes. nos . • 3-45.

178 Reprinted from Comm. Math. Helv. 25 (1951).

A Theorem on Orientable Surfaces in Four-Dimensional Space By SHIING-SHEN CHERN and E. SPANIER, Chicago 1. Introduction. Let M be a closed oriented surface differentiably imbedded in a Euclidean space E of four dimensions. Let G denote the Grassmann manifold of oriented planes through a fixed point 0 of E. It is well known that G is homeomorphic to the topological product 8 1 X 8 z of two 2-spheres. By mapping each point P of M into the oriented plane through 0 parallel to the oriented tangent plane to M at P, we define a mapping t: M -+G. If M, 81> 8 z denote also the fundamental cycles of the respective manifolds and t* denotes the homomorphism induced by t, we have

In a recent paper l ) Blaschke studied the situation described above by methods of differential geometry and proved that the sum 1~1 U2 equals the Euler characteristic of M. He also asserted that U 1 = U z • The object of this note is to give a proof of this assertion, as well as a new proof of the theorem on U 1 U z•

+

+

2. Review 0/ some known results on sphere bundles. Let B be an oriented sphere bundle of d-spheres over a base space X with projection / . The relation between the homology properties of B and X are summarized in the following exact sequence Z) ;

where each H denotes a cohomology group relative to a coefficient group which is the same for all the terms of the sequence. The homomorphisms that occur in the sequence can be described briefly as follows: 1) Blaschke, W., Ann. Mat. Pura Appl. (4) 28, 205--209 (1949). 2) GyBin, W., Comm. Math. Helv. 14, 61-122 (1942). Proc. Nat. Acad. Sci., U. S. A. 36, 248-255 (1950).

Ohern, S. S. and Spanier, E. H.,

205

179

f*

is the dual homomorphism induced by the projection f ; 'II' is a mapping which amounts to "integrating over the fiber"; the third homomorphism is the cup product with the characteristic class Q (with integer coefficients) of the bundle. From this sequence we see that if, for every coefficient system, the fiber Sa I"j..I 0 in B then the unit element 1 of the integral cohomology ring of X is in the image of 'II' and Q = O. Let E be oriented. Over the oriented surface M (E there are two vector bundles, the tangent bundle of tangent vectors and the normal bundle of normal vectors. By taking unit vectors we get two bundles of circles over M. According to a theorem of Seifert and Whitney 3) the characteristic class of the normal bundle is zero. Since this theorem holds in a more general situation and can be proved in a simple way, we state and prove the theorem for the general case ").

Theorem. Let M be an orientable manifold imbedded in a Riemann manifold M'. If M 0 in M', then the chara.cteristic class of the normal bundle of Min M' is zero. f""oJ

Prool. Let B be a small tube around M. B is then the normal bundle of M . We will show that no fiber S of B bounds in B. Assume that S = 00 in B mod p for some p. Let D be the set of normal vectors of length ~ E having S as boundary. Then 0 - D is a cycle mod pin M' intersecting M in exactly one point. This is impossible because M 0 in M'. The above theorem also follows easily from results of Thom 5). f""oJ

3. PlUcker coordinates in G . Let e1 , e2 , e3 , e, be an orthonormal base for E such that e1 1\ e2 1\ e3 1\ e,6) is the orientation of E. If R is any oriented plane of E, let 11' 12 be an orthonormal base in R such that 11 1\ 12 is the orientation of R. Then 11

1\

12

= au e1 1\ e2

+a +

e2 1\ e3 au e1 1\ e4 23

+a

31

+a

2,

+

e3 1\ e1 a 3 , e3 1\ e, e2 1\ e, .

These "Plucker coordinates" a i ; of R are independent of the choice of 2 and satisfy the two relations

11'

t

8) Seifert, H., Math. Zeitschr. 41 (1936) 1-17. - Whitney, H., Lectures in Topology, Univ. of Mich. Press (1941) 101-141. () We owe this simple description of the proof to Professor H. Hopf, who also called our attention to the problem settled in this paper. 6) Thom, R. , C. R. Paris 230, 507-508 (1950). 6) The wedge denotes Grassmann multiplication as in Bourbaki, N., Algebre Multi. linears, Hermann, Paris (1948).

206

180

a l2 a 34 Ea~i

+ a 23 au + a

=

31

(1)

a 24 = 0

(2)

1 .

Conversely, any set of six real numbers satisfying (1) and (2) are the Plucker coordinates of some oriented plane in E; hence, G is homeomorphic to the subset of six space consisting of aij such that (1) and (2) hold. We introduce a linear change of coordinates by

Then G is homeomorphic to the subset of six space consisting of (Xi' YI) such that E ~ = E if; = 1. Let Sl' S2 be the unit spheres in the x-space and y-space respectively. We orient Sl and S2 by the orientations (Xl' X2, x 3) and (YI' Y2, Y3) of the x-space and y-space. Let h: G -+ Sl X S2 be the homeomorphism defined above using the Plucker coordinates. Let IX; G -+ G map each oriented plane R into its normal plane R', oriented so that R, R' determine the given orientation of E. We want to determine the mapping hIXh-1: SlXS 2 -+SlXS 2 , If R has Plucker coordinates ail and R' has Plucker coordinates bij' it is easy to see that the following equations are satisfied

.I: a ik b;k

=

0

k

.I: a;; b",

(i

*

j)

= 1 ,

the last summation being taken over all even permutations of 1, 2, 3, 4. It follows from these that bii = a kl , where i, j, k, l is an even permutation of 1, 2, 3, 4. Therefore, we see that hIXh-1(x, y) = (x, -y)

where - y denotes the antipodal point to y.

4. T.he Theorem. Let M be a closed oriented surface in E. Let t: M -+ G and n : M -+ G be the maps defined by taking tangent planes and normal planes respectively. It is clear that t = IX nand n = IX t . Over G there is a bundle of circles obtained by considering as the fiber over an oriented plane through 0 the unit circle in that plane. Let Q denote the characteristic class of i;his bundle and let Qt, Q n denote the characteristic classes of the tangent and normal bundles of M. Then

207

181

The bundle of circles over G defined above is the Stiefel manifold V of ordered pairs of orthogonal unit vectors through 0 in E and is easily seen to be homeomorphic to S2 X sa. The following section of Gysin's sequence shows that Q is a generator of the kernel of 1* in H2 ( V), since HI (V) and HI (G) are trivial. To find the kernel of 1* we determine the homomorphism of the second homology groups. A generating 2-cycle in V is S2 X e4 • The points Z of S2 can be represented as vectors of the form ZI e1 Z2 e z Za ea. Then

+

+

a

I ( 1: Zi ei , e4) = 1: zi(e i 1\ e4) i=1

and so

Therefore, we see that 1* (S2 X e4 ) = SI - S2' If S; , S; denote cohomology classes dual to the homology classes SI' S2, then the kernel of 1* consists of all elements of the form u (S; S;) where u is an integer. Orient SI and S2 so that Q = S; S;. Orient M so that Qt· M = XM = Euler characteristic of M. Then

+

Qt

= t* (S;

+ S;) =

t* S;

+

+ t* S;

and Since Q n = 0, we see that (t* S;) . M = (t* S;) . M = (i) XM •

We summarize the above results in the theorem: Let M be a closed orientable surlace in lour space E. Let G be the Grassmann manifold of oriented planes through 0 in E and let t : M -+ G be the map into oriented planes through 0 parallel to the tangent planes of M. Since G is homeomorphic to SI xS 2, we have t* (M) = u 1 SI U 2 S2' Then S1> S2 and M can be oriented so that U 1 = U 2 = (t) XM where XM is the Euler characteristic of M.

+

5. Remarks. The above theorem expresses relations between differential topological invariants of surfaces imbedded in Euclidean space 208

182

and suggests a more general problem. To describe the general situation let Mk (Ek+l be a manifold of dimension k differentiably imbedded in a Euclidean space of k l dimensions. Let G(k, l) be the Grassmann manifold of k-dimensional linear spaces through a point 0 and G(l, k) that of l-dimensionallinear spaces through o. There is a natural homeomorphism IX: G(k,l) ~G(l,k).

+

Using tangent planes and normal planes to M we define mappings t:

M

~

G (k, l) ,

n:

M

~G

(l, k)

such that n=IXt.

The general problem is to study the relation between the homomorphisms t*; Hp (G (k, l»

~ HP(M) ) p

n* ; Hp (G (l, k»

~

=

0, 1, ...

HP(M)

We hope to study this question on a later occasion.

(Received 31 th Mars 1951.)

14

Commeotarii Mathematici Helvetici

209

183 Reprinted from Amer. J. Math. 74 (1952).

ON THE KINEMATIC FORMULA IN THE EUCLIDEAN SPACE OF N DIMENSIONS.* By

SHIING-SHEN CHERN.

Introduction. The idea of considering the kinematic density in problems of geometrical probability was originated by Poincare. It was further exploited by L. A. Santalo and W. Blaschke in their work on integral geometry [1], culminating in the following theorem: Let lo, II be two closed surfaces in space, which are twice differentiable, and let Do, Dl be the domains bounded by them. Let Vi, Xl = K i/4rr be the volume and Euler characteristic of D. and let Ai, M, be the area and the integral of mean curvature of li' i = 0, 1. Suppose lo fixed and II moving. Then the integral of K(Do·D 1 } =4rrx(D o 'D 1 ) over the kinematic density of II is given by the formula

(1)

IK(Do D1):S1

=

81T2(VoKl

+ AoMl + MoAl + Ko V

1 ).

This formula includes most formulas in Euclidean integral geometry as special or limiting cases. The purpose of this paper is to apply E. Cartan's method of moving frames and to derive the generalization of this formula in an Euclidean space of n dimensions. By doing this, we hope that some insight can be gained on integral geometry in a general homogeneous space. Moreover, one of the ideas introduced, the consideration of measures in spaces which are now called fiber bundles, will most likely find further applications. The main procedures of our proof have been given in a previous note [2]. We consider a compact orientable hypersurface l , twice differentiably imbedded in an Euclidean space E of n (> 2) dimensions. At a point P of l there are n - 1 principal curvatures K a, ex = 1, . , n - 1, whose i-th elementary symmetric function we shall denote by Si, i = 0, , n - 1, where So = 1 by definition. Let dA be the element of area of l , and let i

(2) These Mi are integro-differential invariants of l.

=

0, 1, ' . " n - l ,

In particular, Mo is the

* Received :May 10, 1951.

22.

184

228

SHIING-SHEN CHERN.

area and Mn-I is a numerical multiple of the degree of mapping of l into the unit hypersphere defined by the field of normals. Take now two such hypersurfaces lo, ll' whose invariants we distinguish by superscripts. The volume of the domain Di bounded by l i we denote by Vi, i = 0,1. Let lo be fixed and II be moving, and let ~l be the kinematic density of ll. We suppose our hypersurfaces to be such that for all positions of II the intersection Do· Dl has a finite number of components. Then the Euler-Poincare characteristic X (Do· D l ) is well defined. If In--l denotes the area of the unit hypersphere in E and if (3) the kinematic formula in E is

IK(D o · Dl)~l

(4)

=

In{Mn--l(OlV l

+ M1V-l(llVo + ~ ~ (k ~ 1) M k(OlMn--2_k(ll},

where (5)

K(D o · D l ) For n

(6)

=

=

In--lx(Do· Dd·

3 this reduces to the formula (1).

The formula for n

=

4 IS

IK(Do Dd~l =

16.".4 (M 3 (OlV l

+ M (llVo + M o(OlM (ll + 1.Io(llM (Ol + i-Ml(O)Mlll»). 3

2

2

1. Measures in spaces associated with a Riemann manifold. We shall first review a few notions in Riemannian geometry, in a form which will be useful for our later purpose. Let M be an orientable Riemann manifold of class > 3 and dimension n. Associated with M are the spaces Bh (h = 1, · .. , n) formed by the elements Pel . . eh, each of which consists of a point P of M and an ordered set of h mutually perpendicular tangent unit vectors el,· .. , en at P. When h = n, such an element will be called a frame. In the current t erminology Bn is a principal fiber bundle over 111 with the rotation group as structural group and Bh are the associated bundles [3]. We shall introduce a measure in B h. Since Bh is clearly an orientable differentiable manifold, this can be done by defining an exterior differential form of degree Hh+ 1)(2n-h)(= dim of Bh). There is a natural mapping .ph: Bn -') B" defined by taking as the image of Pel · . . en the element Pel · . . eh. It induces a dual homomorphism of the differ ential forms of Bh into those of Bn. This process has in a sense a converse. In fact, let (7)

h

+ 1
185

229

ON THE KINEMATIC FORMULA.

be a rotation of the last n - h vectors. A differential form of En which is invariant under the action of (7) can be regarded as a form of En. The well-known parallelism of Levi-Civita can be interpreted as defining 1)/2 linearly independent Pfaffian forms in En, which we a set of n(n shall denote by Wi, Wij( = - Wji), 1 < i, j < n. To give it a brief description [4] we start from the following useful lemma on exterior forms: Let Wi be linearly independent Pfaffiian forms, and let 7rij = - 7rji be Pfuffian forms such that 1 ~ WJ 1\ 7rj; = O. (8)

+

j

Then

7rij

=

O.

In fact, it follows from (8) that

.

7rji =

~

Ujik'llk.

k=,.

Then ajik is skew-symmetric in its first two indices, because the 7rji are, and is symmetric in its last two indices, on account of (8). Therefore Ujik = 0 or 7rji = O. For geometric reasons we denote by dP the identity mapping in the tangent space at P, which maps every tangent vector into itself. Then dP can be written in the form (9)

where the multiplication is tensor product, and the Wi are Pfaffian forms in E,. and are linearly independent. The fundamental theorem on local Riemannian geometry asserts that there exists a uniquely determined set of Pfaffian forms Wi, Wij in En, linearly independent, which satisfy (9) and (10)

dw i

=

~

Wj

1\ Wji.

J

In fact, the uniqueness follows from the above lemma. For our purpose we shall study the effect of the rotation (7) on these forms. Denote the new forms by the same symbols with asterisks. Clearly we have (11)

1<

CI.

< h, h

+1<

r, s < n.

Taking the exterior derivatives of both sides of these equations and making use of (10), we get 1 We shall, following Bourbaki, use wedge product to denote exterior multiplication. It will sometimes be dropped, when the meaning is clear. Parenthcses will be used to denote ordinary products of dilIerential forms.

186 230

SHIING-SHEN CHERN.

L

+ Lr wr* 1\ (wra* -

<011/\ (wpa* -wlla)

/3

(12)

L a

wa

!\ (
UorIJJao)

L

UorIJJoa) =

+ L wo* /\ 4>0'* =

0

0,

0

0,

•

where 4>.. * are Pfaffian forms skew-symmetric in the indices s, r. The system of equations (12) is of the same form as (8), and the above lemma is then applicable. It follows that (13)

W/3a* =

WPa,

If we put

(14) we see from (13) that na is invariant under the action of (7). is therefore true of the form (15)

L n •h

The same

IT na IT Wall IT W!.

=

a

a
i

This form is clearly not identically zero, and we define it to be the density in B h • It gives rise to a measure in B h • 2. Differential geometry of a submanifold in Euclidean space. As a further preparation we need some notions on the geometry of a hypersurface in Eucliuean space. As no additional complication is involved, we develop them for a sub manifold V of p dimensions, which is twice differentiably imbedded in E . We agree in this section on the following ranges of indices: (16)

1<

IX,

(3, y < p,

p

+1 <

r, s, t < n,

1
Since E is a Riemann manifold, the discussions of the last section are valid. In this case Bn is naturally homeomorphic to the group of proper motions in E. To study V we consider the submanifold of Bn. characterized by the conditions that P £ V and that the ea are tangent vectors to V at P. If we denote by the same notation the forms on this submanifold induced by the identity mapping, we have (17)

Wr=O.

From (10) it follows that dW r

=

L

Wa / \ Wc:r

=

a

Since the (18)

Wa

are linearly independent, we have

O.

187

231

ON THE KINE)fATIC FORMULA.

where the Arcx.8 are symmetric in

1%,

f3 :

(19)

From these Pfaffian forms it is possible to construct some significant " ordinary" quadratic differential forms. The first is a set (20)

which generalizes the second fundamental form in ordinary surface theory. The second is (21)

'It = ~ ("'ra) 2 r,a

=

~ AratlAra'Y ("'/l"''Y ) ,

r,a,fj,'Y

generalizing the third fundamental form. The latter seems to deserve some attention. However, so far as the writer is aware, it has not been considered in the literature. For a hypersurface we have p = n - 1, and we shall write <1>, Aatl for <1>", Anatl respectively. The n - 1 roots of the characteristic equation

are called the principal curvatures. In the case of the Euclidean space E we can also write products, thus: (23)

"'i =

dP·

Wi, Wij

as scalar

e.,

3. A formula on densities. The situation we are going to consider consists of two hypersurfaces ~o, ~1 in E, with ~o fixed and ~1 moving, which intersect in a manifold V"-" of dimension n - 2, such that at a point of Vn-2 the normals to ~o, ~1 never coincide. We denote by , 0, 71", the angle between these normals and by :S1 the kinematic density of ~1. An (n- 2)-frame on V,,-2 has a density on each of Vn-", ~o, ~1' to be denoted by Lv, Lo, L1 respectively. Our formula to be proved can be written . . . .. (24) LV~l = sin n- 1 <j>L oL 1 .

+-

Throughout this section we shall agree on the following ranges of indices: (25)

1
1<

1%,

f3 < n - 2,

1::;.1,R
• an be the fixed frame aTld 0'0', Let 00 1 . a'" the moving frame. , For a given relative position between 00 ·0" and 0'0', . a'" let Pc, · . en-, be an (n-2)-frame on Vn-". We complement this into a frame Pc,· . en such that en is normal to ::So and also into a frame P'e'l .. e'fl

188 232

SHIING-SHEN CHERN.

such that e'" is normal to ~1 at P, and P' = P, e'a = ea. e'n.-1, e'n we have then the relations

e',,-1 = cos e....1 + sin en,

(26)

e'" =

-

sin e....1

Between e....1, e..,

+ cos e".

From this we derive the following useful relation

de'f>-l· e'" = dq,

(27)

+ den-1 . en.

Let us now express the relations between the frames so introduced by the equations

(28)

P' = 0'

+ ~ x'ia'i, •

We shall denote the differentiation by d' when O'a'l· .. a'" is regarded as fixed. In other words, d' is differentiation relative to the moving frame. Then we have, from (28), ~

dO' = dP - d'P -

(29)

•

X.dO'i .

It follows that, on neglecting terms in da';,

II(dP · ea)II(dO'· a',;) = II(dp· ea)II(dP· a'.- d'P· a'.) a

i

a

i

= II(dP · ea)II(dP· e'i- d'P· e'i) (30)

a

•

=== ± II(dP· ea)(d'P· e'a)(dP · e'n-1

-

d'P· e'n-1)(dP· e'" - d'P· e'n)

a

=== -t- sin II (dp· eA)(d'P· e' A). A

These are to be taken as congruences mod da'i. In particular, the last step follows from the fact that e'n is normal to ~1 at P and that the product of n factors involving dP is zero, because the locus of P is a hypersurface ~o. In order to get a further reduction of the left-hand side of (24) we start from the formula (31)

de'i - d'e'i =

~

u';kda\.

k

From the invariance of the kinematic density under a rotation it follows that

II (da'i . a'i) ;< i

Then we have

=

II( (de'; - d'e'.) . e'i). ;< i

189 ON THE KINEM:A.TIC FORMULA..

233

IT (dea ' ep) IT (da', . o'j) = IT (dea ' ep) II (de', - d'e'.) - e'j .<1 a