Gauss Bonet Theorem

Journal of the Mathematical Society of Japan Vol. 9, No. 4, October, 1957

The Gauss-Bonnet Theorem for V-manifolds,

By Ichiro SATAKE

(Received Aug. 30, 1957)

Introduction.

The purpose of this paper is to generalize the Gauss-Bonnet theorem (established by Allendoerf er and Weil [1]) to the case of V manifolds. The notion of a V -manifold has been introduced by the author in a previous short paper [7]. The outline of this paper is as follows. In 1 we shall give fundamental concepts concerning V manif olds and V -bundles. Our

principal idea is the following : while an ordinary manifold can be considered as an inverse injective limit of Euclidean spaces, a V manifold is considered as that of Euclidean spaces allowing a finite

N group of automorphisms. More precisely speaking, let {Ua}aEA be a system of Euclidean spaces (of the same dimension), A being a directed system such that for any a, QUA there exists a r U A with r a, r I9 and assume that for any a, 9 U A, a s ,9 we have

N N

an injection' 'spa from Ua into UQ such that we have Ara=2raA a for a ~ / r. Then the injective limit M= U ~oa(Ua) of the inverse

a

N N

system {Ua, ~Qa}, ~oa denoting the canonical injection Ua-~]k[, is a N manifold. (ca(Ua) being homeomorphic to Ua, M is locally homeomor-

phic to a Euclidean space.) Now a slight modification of the above definition will lead to a V manif old. Namely replacing the word an injection' by ' a finite number of injections' we obtain a V

N N

manifold M=U cpa(Ua), which is locally homeomorphic to Soa(Ua) N N N

Ga\Ua, Ga being a finite group of automorphisms of Ua, composed of 2aa Similar considerations can be applied also to the definition of VVbundles. Namely a V -bundle B can be considered as an injective limit (allowing a finite group of automorphisms) of an (inverse)

N system of direct products of a Euclidean space Ua and a fixed manifold F (called fibre) with respect to injections of a special form. As examples of V -bundles the notions of tangent vectors and differ-ential forms will be introduced at the end of this section.

The Gauss-Bonnet Theorem for V -manifolds. 465

In 2 a short resume of Riemannian geometry on a V manif old is given. After these preparations it is quite easy to see that Chern's proof [3] of the Gauss-Bonnet theorem can be transferred almost literally to our case. But in order to make clear the point in which our proof differs from Chern's and also the meanings of the notations used, we shall restate the main arguments of his proof in 3. In the Gauss-Bonnet formula for a V -manifold M, thus obtained, appears the Euler characteristic x(M) of M as a VVmanifold in place of the ordinary Euler characteristic. This number will be of some interest, because, on the one hand, it is also related with Hopf's formula [4] on vector fields over M and, on the other hand, it appears in the theory of automorphic functions of one variable

(e. g. in the Riemann-Rock formula concerning the dimension of the space of automorphic forms). Finally in 4 we shall show that the Gauss-Bonnet formula can be also applied to Siegel's modular varietyn-Mn\s~n, ~dn being the space of all symmetric complex matrices Z=X+iY of degree n with the imaginary parts Y> O and Mn Siegel's modular group of degree n operating on This case was treated already by Siegel in his Symplectic Geometry' [9], which gave a motive to this investiga-tion. Siegel considered the Gauss-Bonnet formula only for the case of where is compact and (~ is a discontinuous group of symplectic transformations without fixed point. Our results will complete his work in the sense that we extend his formula to the case of compact where C3 allows fixed points, as well as to the non-compact caseWe conclude this paper by an appli-cation of the Gauss-Bonnet formula for to the determination of the least common multiple Nn of the orders of isotropy subgroups of M /{ E2n}. The Gauss-Bonnet formula for 3n gives a lower estimation of Nn, while an upper estimation of it will be obtained by a method of Minkowski [6].

1. V-manifolds and V-bundles.

1. Definition of V-manifold. Let M be a Hausdorff space. A

(C-) local uniformizing system (abbreviated in the following as 1. U. s.) N {U, G, ~} for an open set U in M is by definition a collection of the following objects:

466 I. SATAKE

N

U : a connected open set in Rm (m-dimensional Euclidean space),

N

G : a finite group of C-automorphisms of U, with the set of fixed points of dimension S m-2,

N

cp : a continuous map from U onto U such that c oa = co for all a E G, inducing a homeomorphism from the quotient space

N

G\U onto U. N N

Let {U, G, ~o}, {U', G', cp'} be 1. U. s. for U, U', respectively, and let N N

Uc U'. By a (C-) injection 2 : {U, G, cp} -~ { U', G', cp'} we mean a N N

C -isomorphism from U onto an open subset of U' such that rp = yd o2. Every a EG can be then considered as an injection of

N N N N

{U, G, cp} into itself. Also if A : {U, G, gyp} -- {U', G', cp'}, 2' : {U', G', '}-{U", N N N {U", G", ~"} are injections, A'o2 becomes an injection {U, G, ~o}-~{U",

N N

G", co"}. Hence if A is an injection {U, G, cp} -~ {U', G', c'} and o' E G', N N

then a'o 2 becomes also an injection {U, G, co}--~ {U', G', ~o'}. Conversely we have the following

N N

LEMMA 1. Let A, i be two injections {U, G, gyp} --~ {U', G', ~o'). Then there exists a uniquely determined a' E G' such that p = a' A. PROOF. Let p U. As we have '(4)) = co(d) = c'(2(p)), there

exists a a' E G' such that ,a(p) = a'(2(p)). Choosing 2(p) not to be a fixed point of G', the automorphism o' EG' is uniquely determined.

N As the set of non-fixed points of G' in 2(U) is, by the above as-sumption, connected and everywhere dense in 2(U), the relation ,u(~) = o'(A(p)) holds for all p U. Hence we have ,u = a' o 2 with a uniquely determined a' E G', q. e. d.

N

It follows, in particular, that if 2 is an injection {U, G, co}->

N {U', G', c'} and a E G, there corresponds uniquely a a' E G' such that Aoa=o'oA. The correspondence a-+a' becomes clearly an isomorphism from G into G'.

N N N

LEMMA 2. Let 2 be an injection {U, G, co} r-+ {U', G', So'}. I f o'(2(U)) N N N

(1 2(U) = with a' G', then a'(2( U)) = 2(U) and o' belongs to the image o f the isomorphism G --> G' defined above. PROOF. Assume that a'(A(U)) fl 2(U)=1=c5. Then there exist j, qE U such that a' o 2( ) = 2(q). Then, since gy(p) = co(q), we have 4) = q with some z E G. Let 'c' be the element of G' corresponding to r (i. e. one such that 2 o v = z' o2). Then we have a'(2(p)) = z'(d()). Choos-ing A(p) not to be a fixed point of G', we have a'=-C' and hence

N N N N

a'(A(U))=r'(2(U))=2(z(U))=2(U), q. e. d. N N

It follows, in particular, that if 2 : {U, G, ~} -- {U', G', cp'} is an

The Gauss-Bonnet Theorem for VVmanifolds. 467

N N N N

injection and if Sc(U)=co'(U'), we have a'(t(U))=2(U) for all o'EG'. N N

(For otherwise U' = U o'(A(U)) would be disconnected.) Hence the o' EG' N N

C-isomorphism 2 : U--~ U' and the associated isomorphism G-~G' N N

become onto, and A-i becomes also an injection {U', G', co'}-{U, G, ~o}. N N

In this case, we call two 1. u. s. {U, G, cP}, {U', G', ~'} equivalent. After these preparations we shall give the definition of V manifold. DEFINITION 1. A (C-) V -manifold is a composite concept form-

ed of a (Hausdorff) topological space M and a family (called a defining family for a V manif old) of (C-) 1. u. s. for open subsets in M satisfying the following conditions.

(I) Every point p of M is contained in at least one -uniformized N open set (i. e. an open set U for which there exists 1, u, s. {U, G, cp}

N in such that q'(U) = U). If p is contained in two -uniformized open sets U,, U2, then there exists an -uniformized open set U3 such that p U3 C Ul (1 U2.

N N N N

(II) If {U, G, co}, {U', G', c9'} are 1. u. s. in such that ~o(U)c~o'(U'), N N

then there exists always a (C~-) injection 2 : {U, G, ~o} - + {U', G', co'}.

(2 is uniquely determined up to o' E G', by Lemma 1.) By what we have mentioned above, it follows from (II) that two 1. u, s. in for one and the same open set in M are always

N N N

equivalent. Also if {U, G, }, {U', G', yo'}, {U", G", "} are 1. u. s. in

` N N N N N

such that rp( U) C ~2'(U') C ~o"(U"), then an injection {U, G, ~o} ---~ { U",-G", N N N

~o"} is given by a composite of injections {U, G, 9} -*{U', G', cp'} -+{U", ~:"}

Two defining families , ' are said to be directly equivalent, if

,' are both contained in a defining family (satisfying (I), (II)), and equivalent, if there exists a chain of defining families i (1s i S r) such that r=' and that ~,~+1 (1SiSr-1) are directly equivalent. Equivalent families are regarded as defining one and the same Vmanif old structure on

1) Thus a V-manifold is, strictly speaking, a composite concept of a topological space M and an equivalent class of defining families. But in the following we consider a V-manifold M with a fixed defining family (i. e. a `coordinate V-mani-f old' (M, )). 2) It can be proved that for any defining family -, there exists a defining

family such that and that any open set in it! which is simply connected

and is contained in an -uniformized open set is f-uniformized.

468 I. SATAKE

N

Let M be a V manifold and p E M. Take a 1L u. s. {U, G, So} E such that p E So(U) and choose E U such that cp(s) =p. Then it can easily be seen by Lemma 2 that the structure of the isotropy sub-

group Gp of G at does not depend on the choice of U and p (and hence of ), and is uniquely determined by p. Hence we call some-times Gp simply the ' isotropy group of p.'

An ordinary C-manifold is nothing other than a C~-V manifold for which the isotropy group of each point reduces to the unit group. In general, let 11/1 be a V -manifold and consider the set S composed of all ' singular points' of M, i, e. the points of M with

N ti

non-trivial isotropy groups. Let p E S and {U, G, ~t}, p, Gp be as above. Then taking a suitable coordinate system around , Gp becomes a finite group of linear transformations. (Let {u1, , um} be a coordinate system around and consider the system v2 =

i - 1_ a., fr 1)u'oo, where aj3(a) = au 6 , NG~ =[G~ :1].) Hence q-1(S) N~_ auk p p

p p N

is expressed locally by a finite union of linear submanifolds of U. This means that S is a VV subvariety of dimension r m-2 of M.3' The isotropy group of an 'ordinary point' of S, i. e. a point

where S has locally a structure of V -manifold, depends only on the irreducible component Si of S containing that point and is called sometimes the isotropy group of S2. Clearly M-S is an ordinary

C-manifold (connected if M is connected).

N

It can be also proved that if M is a C~-manifold and is a

N properly discontinuous group of C'-automorphisms of M, then the

N quotient space C \M possesses a canionical V -manifold structure. REMARK. If we replace the word " a connected open set in R'~ " in the above definition of 1. u. s. by " a connected Cx-manifold ", we obtain an equivalent definition of V-manifold. This modification of the definition of 1, u. s. will be used in 3.

2. C- V-manifold map.

DEFINITION 2. Let (M1, 3, (MG,,) be two V -manifolds. We

3) A C'-V-subvariety X of M is a (closed) subset of M such that for any 1, u. s. N N

{ U, C, y} ~ for U, co-1(X f U) is a (G-invariant) Cam-subvariety of U in the usual sense. A V-subvariety X is decomposed into a locally finite union of irreducible subvarieties Xi , called the 'irreducible components' of X.


mean by a (C-) V mani fold mad h from (Ml, into (M2, 2) a system N of mappings {hU,} ({U1, G1, ~oj E1) as follows:

(i) There is a correspondence {U1, G1, T 1} -- { U9, G2, ~2} from 1 N into 2 such that for any {U1, G1, ~1} E 1 we have a C-map hU1 from

N N

U1 into U.

(ii) Let {U1, G1, c}, {U1', G1', c1'} E U1, {U2, G2, T2}, {U2', G2', T2'} E 2 N be the corresponding 1. u. s. (in the sense of (i)) and let So1(U1)C

c 1'(U1'). Then for any injection 2l : {U1, G1, T 1} ---* {U1', G1', T 1'} there N N

exists an injection 22: {U2, G2, 92} --~ {U2', G2', Co2'} such that

N ~ UN A2 o Ul-l121 .

It follows from (i), (ii) that there exists uniquely a continuous N map h from M1 into M2 such that for any {U1, G1, CP1} and for

N the corresponding {U2, G2, Sot} E2, we have

~o2 hU ==h ~o1 .

h is called a C-map M1-*M2 defined by a C-V-manifold map h = {h iJ :

(Mi, 1)-- (M~, 2).4~ (We use thus the same notation for a C'-V manif old map and the corresponding C~-map. But no confusion will arise.)

It is possible to define an equivalence relation between V mani-fold maps quite similarly as in 1. Then the C~-map MI--~M2 defined by a V manif old map (Mi, 1) -~ (M~, 2) depends only on the equi-valence class of the latter. Considering R (the set of all real numbers) as a V-manifold defined by a single 1, u. s. {R, {1}, 1}, we define a C~-function on a V -manifold M as a C-map M-- R.

3. V-bundle. Let M, B be two V manifolds with a C'-map 7r: B--M. Let further F be a C'-manifold and G be a Lie group operating on F as a (C-) group of transformations. (we don't assume G to be effective.) DEFINITION 3. A pair of defining families (,), being a defining family of M and that of B5', is called a pair of defining

4) The notion of C-map thus defined is inconvenient in the point that a com-posite of two C-maps defined in a different choice of defining families is not always a C'-map. 5) Here the 1, u, s, in g, (or at least those in ) are understood in the modi-

fied sense as was stated in Remark at the end of 1.

470 I. SATAKE

families for a (coordinate) VVbundle (B, M, iv, F, G), if it satisfies the following conditions:

N N

(i) There exists a one-to-one correspondence {U, G, ~o} H {U*, G*, N N

cp*} between and such that U*=UxF and denoting by it the N N

projection U*-~ U, we have

7r0S9*=99 o7rU. N N N N

(ii) Let {U, G, q}, {U*, G*, c *} ; {U', G', co'}, {U*', G*', 9*} be two N N

pairs of corresponding 1, u. s. in (, *) and let ~(U)cSp'(U'). Then N N *(U*) c*'(U*') and there exists a one-to-one correspondence E--* ~*

N N N N

between injections t : {U, G, co} -* { U', G', c'} and ~* : {U*, G*, q *} -~ { U*' A N N

G*', *'} such that for (p, q) E U* = Ux F we have

A*(p, q)=(2(), pg(P)q)

with gy(p) E G. The mapping g~ : U-G is a C'-map satisfying the relation

(1) g~A(p) =g~(A(p)) .g(P)

for any injections i n " .~ "~ 'r ,~ " y nject o s { U, G, c } { U, G, ~ } {U", G.r, ~9"}. ((1) is satisfied automatically, if G acts on F effectively.)

A composite concept of B, M, ir, F, G and a pair of defining fami-lies (, *) satisfying the above conditions is called a (coordinate) V -bundle. (For simplicity we call sometimes B a (coordinate) V bundle.)

It follows, in particular, from (ii) that there exists a one-to-one correspondence c H Q* between G and G* such that 6*(p, q) _ (o(p), g6(p)q). This correspondence is clearly an isomorphism. Denoting by Gp the isotropy subgroup of G at and by Gp* the corresponding subgroup of G*, we can prove easily that for p=So(p)

-1(Y)NGp*,pxFN {g,(p); cEGp}`F.

Thus a V -bundle B is not always a bundle in the usual sense over M. Equivalence relation between pairs of defining families (,*) is defined quite similarly as in 1. (A V -bundle is a composite con-cept formed of (B, M, ir, F, G) and an equivalent class of pairs of defining families.) Especially it can be easily verified that two pairs of defining families (,1*), (,2*) are directly equivalent (1. e. ({, }, { 1*, 2*}) becomes also a pair of defining families), if

N and only if there exists a C-map 8i: U--~G such that


c i*(A q)=T2*(L', 6a(p)q) N N

for any {Ux F, G1*, T1*} 1*, {Ux F, G2*, Sot*} E2 * corresponding to N the same {U, G, co} and that

(2) g2'(p) =~U(~(p))g~l'(p)~U(p)-1 N N

for any injection A : {U, G, cp} -*{U', G', cp'}, gl~, g$2) denoting the map g2 in (ii) corresponding to 2*, respectively. (The necessity of these conditions is evident. If, conversely, they are satisfied, define

N N

the injection {UxF, G1*, T1*} (E U1*).{UxF, G2*, p9*} (E2*) corre-N sponding to the identical injection of {U, G, cp} onto itself by the map (p, q)-C3, w(p)q).) Let (B, M, ir, F, G) be a V -bundle with a pair of defining families (j, *). A V manif of d map f = { fU } : (M, ) -- (B, *) is called a (C~-) cross section of this V -bundle, if the correspondence-->* in Defini-tion 2 (i) is given by the correspondence in Definition 3 (i) and if ~v f=1 (then clearly 7ro f=1). To give a cross section f : (M, )-~ (B, *) is therefore to give a cross section (in the usual sense) fv

N N N N

of each U* = Ux F such that for any injection 2: {U, G, } -- {U', G', gyp'} we have fU, o2 = 2* o fU. (In particular, fU is G-invariant in the sense that fv o6=6*ofU for all 6EG.)

A direct product V-bundle (i. e. a V -bundle for which g, =l in Definition 3 (ii)) has clearly many cross sections. Conversely let

(B, M, ir, G, G) be a principal V -bundle, i. e. a V -bundle such that F= G and g~ G acts on G as a left transformation, and assume that it has a cross section f={fU}. Then, denoting fU(~)=(p, o (p)), we have

gA(P) = N N

for any injection 2 : {U, G, co} --~ {U', G', c'}. This means that this

(coordinate) V -bundle is directly equivalent to a direct product V bundle over the same (coordinate) V -manifold.

4. V-bundle map. The notion of V -bundle map can by defined quite similarly as that of V -manifold map. Namely we have DEFINITION 4. Let (B1, Ml, 7r1, F, G), (B2, M2, nG7 F, G) be two V bundles with defining families (1, 1*), (, 2*), respectively. A system of mappings h* = {h~, } ({U1, G,, co j ) is called a (C''-) V bundle map if the following conditions are satisfied:

N N

(i) There exists a correspondence { U1, G1, Cpl} -- { U5, G , cp2} from

N

1 into 2, such that for any {U1, G1, ~ } we have a C-map /z1

472 I. SATAKE

N N N N

from U1 x F into U2 x F and a C -map hU, from U1 into U2 such that

q)=(hU,(p), rU,Q)q)

with rU,(p) E G. 1E7 , iS a C-map from Ul into G. N N N

(ii) Let {U1, G1, T 1}, {U1', G1', ('1'} be L u. s. in 1 such that T 1(U1) N N N

c c 1'(U1') and {U, GL, cot}, {U2', G2', ~2'} be the corresponding 1, u, s. in N N

~. Then for any injection 21: {U1, G1, T1}-~ {U1',G1', q1'} there exists N N

an injection d2 : {U, G, 92} -~ {U2', G2', ~n 2'} such that ~ nT

* 1,,_1* * ~2 ohUl-hUl, 0~1

(Hence also A2 ~hU, = hoA1). We assume further that = rU1, (~1(P))g~,(P)rU,(PY 1 .

(This condition is satisfied automatically if G acts on F effectively.) Thus h* = {h*}, h = {hyJ are V manif old maps from (B1, 1*) into

(B, 2*), from (M1,1) into (M2,2), respectively, and the correspond-ing C~-maps h* : B1--B2, h : M1-+M2 satisfy the relation ire oh* =h on1.

If, in particular, M1=4, 1 = 2 and all hU1 are the identity, then it follows that all hU1* are onto, one-to-one and h*-1= {hf,*-1} becomes also a (C-) V -bundle map. In this case, we call these two (coordinate) VVbundles equivalent.6~ The following theorem is an analogue of the existence theorem for ordinary fibre bundles. The proof is also quite similar as in the ordinary case. THEOREM 1. Let M be a V mani fold with a defining family , F

a (C-) manifold and G a Lie group operating on F (as a C~-group of N N

transformations). I f we have a system o f C'-maps g~ : U--~ G ({U, G, gyp} N N

E , 2 being any injections : {U, G, c } ---> {U', G', cp'}) satisfying the condi-tion (1), then we can construct a (coordinate) V-bundle B over M as described in Definition 3. If two systems of mappings {g,l)}, {g,2)} satisfy the relation (2), then there exists uniquely a V -bundle map h* = {hi*} from the V -bundle corresponding to {gal} } onto the one corresponding to {g,2)} such that the correspondence -- in Definition 4 is the identity and that we have hU *(p, q) = (p, 8U (p)q) for (p, q) Ux F. (Thus {hUl * } defines an equivalence between these two (coordinate) V -bundles.) REMARK. We can define the notion of V -bundle, replacing in

N N N

Definition 3 (i) the words U* = Ux F" by U* is a fibre bundle

6) In case B1=B2i this notion of equivalence is somewhat different from that given in 3.


N over U with the fibre F and the group G " and in (ii) that for ti N N N N

(p, q) E U* = Ux F " by " that is a bundle map U*.+ U*~ inducing N N

the map 2: U--+U'." (Of course, these words require more explicit indications in case G acts ineffectively on F.) We need this modi-fication in 2.1.

5. Examples. Let (M, ) be a V manifold. Assuming that N N

every U({U, G, cp} E ) is contained in Rm, we fix a coordinate system

N {u1, , um} in each U once for all. Let F= Rm (vector space of dimension m over R) and G=GL(m, R) (group of all non-singular

N N

matrices of degree m.) For any injection .(: {U, G, c} -- {U', G', gyp'} put

gA(P) = au~2 (Jacobian matrix of d at p) , 8u3

N N

{ui}, {u'i} being the (fixed) coordinate systems in U, U', respectively. Then the system g,, satisfying the condition of Theorem 1, defines a V -bundle (T, M, n, F, G) with a pair of defining families (,*). This V-bundle is called the tangent vector bundle over M. Let us note that the 'fibre' 7-1(p) (pEM) is not always a vector space. Let p E ~o(U), {U, G, cp} and choose 5 E U such that p() =p. Then, it 1(p) {g); a EGp}\Rm, Gp denoting the isotropy subgroup of G at j,. NOW ~U-1(p) =p x Rm can be identified with Tp (the tan-gent space to U at p) by the correspondence

(X1

a

p x H X= x2 . x m i aui

Then, denoting by Tf p the linear subspace of Tp formed of all Gp-N invariant vectors (i. e. vectors invariant under gQ(p) (6 EGp)), we see that 2-1(p) contains a vector space Tp=q,*(TpGp), which is independent of the choice of U and p. An element of T~ is called a tangent vector to M at p.

A cross section of the V -bundle T is called a (contravariant) vector field over M In the above notations, AEU being a G-invariant

U* N N cross-section of = UxF (I. e. a G-invariant vector field over U in the usual sense), we have 3j() E T3; and so ~E(p) Tp. Thus 3(p) being a tangent vector at p for any p E M, the set of all vector fields over M forms a vector space. More generally we can construct an (r, s) tensor bundle over M by means of the system {g}:

474 I. SATAKE

au'1 au' gA(p) = __ x . . x x... a

u' au'i

S r

x denoting the Kronecker product of matrices. (In this case, F= Rm(r+s), and G=GL(m, .R) operating on F as an (r, s)-tensor represen-tation.) We can also consider skew-symmetric or symmetric tensor bundles over M.

In particular, consider the skew-symmetric (h, 0)-tensor bundle over M. As in the case of the tangent vector bundle, 7-1(p) (p E M) contains a vector space Dp, which is isomorphic to the space of G; -invariant skew-symmetric (h, 0) tensors at (~o(j) =p). Dp can be

m regarded as a dual space of Tp . It should be noted that Dp is n=o

not always an exterior algebra over Dp. A cross section w of D is called a differential form of degree h (or briefly h-form) on M Since w(p)ED p for any p E M, the set of all h-forms over M forms a vector space. By definition, to give an h-form w on M is to give

N N

a (G-invariant) h-form wU on each U ({U, G, ~9} E ) such that it holds N N

wU = wU, od for any injection : {U, G, cp} --~ { U', G', So'}. Using these 'local expressions', we can define the operations A and d just as in the case of ordinary manifold. Also if h : M-- M' is a V -manifold map we can define a form w' oh on M for any form w' on M'. In case M is orientable, i. e. in case we can choose the coordinate

'i A systems {ul,, um} such that we have det au >0 for any injec-

tion A, we can define the integral w of an n-form w as follows. If

M the carrier of w (i.e. the closure of {p; p E M, @(p)=1=0}) is contained N in an -unif ormized open set U= ~o(U), we put

(3) w= 1 _wv M NG U

NG denoting the order of G. In general, assume that there is a locally finite family of C'-functions { fi} such that the carrier of fi is contained in an -uniformized open set Vi and that f=1 on

i the carrier of w. Then we define the integral of cv by

SMSM

i


if the summation on the right side converges absolutely for any such 'partition of unity' {ft}. We can prove easily that this defini-tion does not depend on the choice of {f}. Finally let P be the principal V-bundle associated with T. (i, e.

a V-bundle with F=G=GL(m, R), gy(p)= a(_' ).) In the same au'

notations as above, (p, (xi)) E Ux GL(m, R) is in one-to-one corre-spondence with a 'frame' (i. e. a base of the tangent space) (X1,. ,

Xm) at p by the relation X~ = x~ a Hence P is called a frame i an

N N

bundle over M. Since G* has no fixed point in U* = U x GL(my R), P becomes a C-manifold in the ordinary sense.

2. Riemannian geometry on a V-manifold.

1. Riemannian metric on a V -manifold. Let (M, ) be a V manifold with a Riemannian metric g. By definition, to give a Riemannian metric g on M is to give a Riemannian metric gU on each U ({U, G, c } ) such that for any injection A : {U, G, cp} ---~ {U', G', c'} we have

N N N N

gU(, J) =gv, (A(IE), A(J)) N N N N N

3, J being arbitrary vector fields on U and A(), A(sh)) the correspond-ing vector fields on A(U)c U'. (In particular, each gU is G-invariant.) By means of the Riemannian metric g we can define the unit

tangent vector bundle To and the orthonormal frame bundle Po over M

(by the 'reduction' of the structure group from GL(m, R) to 0(m) (group of all orthogonal matrices of degree m)). For instance, To

N N N

is defined as follows. For each {U, G, cp} E , let U* = T0(U) be the N unit tangent vector bundle (in the usual sense) over U (with F=

N N

S'1, G=0(m)); T0(U) can be considered as a subset of T(U) (the N N

tangent vector bundle over U). For each injection A : {U, G, q } --~ N N N

{U', G', rp'}, let A* be the corresponding bundle map U*~U*'. Then N defining c as a restriction to T0(U) of the corresponding map

N N

T(U) -* T and putting To = U cp*(U*), we obtain a V manif old To with N a defining family ={{U*, G*, q'*}}, (To, *) has clearly a structure of V -bundle with F=Sm-1, G=0(m) over (M, ) in the modified sense as stated in Remark at the end of 1,4.

476 I. SATAKE

2. Basic forms, connection forms and curvature forms. Let

{U, G, c } E, E U and (X1,., Xm) be an orthonormal frame at p (i. e. a base of Tp such that gU (Xi, Xj) =b). j). Let {u1,., um} be a

N coordinate system in U and put Xj = ~; x i , (X) = (x. Then the i du N basic forms over U are defined by

e 2= X~duj (1Sism). j

N They are 1-forms on P0(U) and the above formula is considered as giving their expression at p* = (p ; X1, , Xm). (If we consider them as covariant vectors at , (01,..., 8m) is nothing other than a dual base of (X1, , Xm).)

N N

If BUi, BU,i are basic forms over U, U', respectively and if ,( is N N

an injection { U, G, gyp} --~ { U', G', p'}, then we have DUi =BU, o2*. For let = (; X1,..., Xm) P0( U), A*(1~*) = (2(i) ; A(X1)..., 2(Xm)), {u'1} be a co-

ordinate system in U' and X' = x'i a . , X'i = x'i -1. Then i au

/ 8U,2 at ~(~) _ X'idu'j = X'i au j duk

j jdu

= ~; X,duj = By1 at ]i.

This proves our assertion. (It follows, in particular, that 0y (1 s N i S m) are G*-invariant.) Hence the systems {0y} ({U, G, cp} E ) define 1-forms 6i on P0, which we call the basic forms over M.

Quite similarly connection forms wij (1 s i, j s m) over M are N defined by those wover U. They are 1-forms on Po with the

following characterizing properties:

d01 = wij 1 1 Bj, wi j - w ji

and have the local expressions as follows:

(4) wi j U = - ~..1 X1 dxjk - L..l Xki P du' X0'. k k,l,h

Also curvature forms S2ij are defined by the 'structure equation':

dwij=~wtikllwkj+Qij (1si,jsm)

k and have local expressions as follows:


(5) -1 _ xipx q Rpqkl duk A du'. 3 ~ 3 2 p,4,k,1

As is well-known, we have the following relations

' ij A0 j = U ,

I

0)ik n'S2k j - 2ik n 0k j . k k

Denoting by RA the 'right translation' of Po by A = (a11) C 0(m) and putting A-l=(Aij), we have easily

01 R4 Al 0', RA= AikAjl W kI j k, t

0 RA = AikAjl 2k1.

3. Gaussian curvature. Now let m be even =2p. We put

() Q l...im i/ 1 l 1 ... ~JLi i 6 2p p 2 m-1 m 2 .7r (ii, ,2m)

im) running over all permutations of (1,..., m) and 611m ..1m denoting the sign of (1''m) ii(In case m is odd, we put simply Q =O.) Clearly Q is an m-form on Po invariant under the 'right trans-lations' of S0(m) (group of all orthogonal matrices of the deter-minant 1). Hence 2 can be considered as an m-form on M, if M is orientable (and one of its orientations is fixed), and as an m-form on the orientable covering of M (i, e. an m-form on M of the 2nd kind in the sense of de Rham), if M is not orientable. Now, assuming for simplicity M to be orientable, we have

(7) S2= 2 Kdw . Om

where

m+1

0m _ 27r2 _ 22p+ 1M 7r (volume of m-dimensional sphere Sm), r m + 1 (2p) !

2

_ 1 ~i...im i kl...kK p i 6 m R112k!k2 Rzm-limkm-lkm

(ki, ',km)

(Gaussian curvature of M),

478 I. SATAKE

dw = ,/g du1 dum (volume element of M),

(u1,', um), (X1,.., Xm) being taken to be concordant with the (fixed)

N orientation of U and g . , =gi3, g=det(gi3). For au~ au3

1Q A ... n Ziia im-lim

1 p ~il...im x~pl..xipmR k k ...R l..dukm m pp2 a pm-lpmkm-ikm duk. 2

)det(xi) iLJ ~pl...pm ekl...km Rpp k Rpm ipmkm ikm du1...dum = ( _ 1 ~ ~ 2kl, -- (-1) p p

g i ...Pk l...k p E m m Rp,pzk,k,...Rpm-ipmkm-ikm dw, 2 - -1

since det(x,i) =1/g . 6Z,...im Qj,ia A A Qim-lim is thus considered as an m-form on M. (7) follows now immediately.

3. Chern's proof of Gauss-Bonnet formula.

1. Auxiliary forms on P'. In the following we assume, for the sake of simplicity, that M is orientable. Then the orthonormal frame bundle Po splits into two parts P', Po", which are principal V -bundles with G=SO(m). If we fix an orientation of M once for all, we shall be concerned exclusively with one of them, say P0'. Together with the projection ( ; X1, , Xm) E P'(U) -~ (p ; Xm) E

N T0(U), P' can be also considered as a principal fibre bundle with G=SO(m-1) (subgroup of SO(m) leaving fixed Xm) over T0. Any form on P' invariant under the 'right translations' of SO(m-1) can be considered as a form on To. Following an idea of Chern [3], consider the following forms on F0': m-1 +Qi i .,Qi i @i m...(~i m (0k[m_]) 1 2k-1 2k 2k+1 m-1 ~ (ii,...,im-1) 2 (8)

?.k =2(k+ 1) E2,."2m-1 SQ.....,Q20120 i`1Gi2k+1m -Z2k+2m (ii....,im-1)

osks m- -1 .


Being all SO(m-1)-invariant forms on P0', they are considered as

((m -1)-, m-) forms on T0. We have

l~ k =k 6il."im-1 dQa..la ,Q.....~, 22klZ2k2k+1m ...'.

i zaa, -2 vm-1'm

+(m-2k-1) eim-1'Q2k-h2k2k+1m...im_lm

~k _1+ m-2k-1k ('_ =0). 2k+2 1

It follows that

'k=dok, Oslo m []-i)

2

d' A=o (m--2A-1)...(m-2k-1)

If m is even (-2p), we have d8p_1=!Fp_1 and

` - m EZL.,im-1 Q. ...~. p1 i,ia im-lm

_ 5" 1".2m ..,Qim-lim -' ( 1)22p!

If m is odd (=2q+1), we have dq_1=l!q_1 and also dDq=!1!q__1. Hence

d(q_1-Qq)=0=Q.

These results can be unified as follows:

( m [m21] (-1)~ (9) H -_ 1) (1)

~

m-1

2" 2 A-0 2

2. Index of singularity of a vector field. Let ~E be a unit vector field over M with singularities at i. e. a cross section of TO on M- {p1, , p3}, Let {U, G, rp} E , p E U be such that ~(U)

h p0, gy(p) =pi and let ZEU be the corresponding unit vector field over N U. Then ZEU has a singularity at p. Let Ip(ZEU) be the index of

singularity of ZU at in the usual sense and put

480 I. SATAKE

(10) I()=_1_ Ip~(?U) ~a NGp

NG~ denoting the order of the isotropy subgroup Gp of G at j3. It p is clear that this definition of Ip.(3) does not depend on the choice of U and p and is uniquely determined by and pi. We call I .(3E) the index of singularity of at pi. (It should be noted that Ip.(?) is

z not necessarily an integer.) Now let Si be a sufficiently small geodesic sphere around pi and consider it to be oriented by the induced orientation with respect to the outer normal vector field on it. Then we have the following "integral formula of Kronecker ":

(11) 1 (0im...0)m-1m) =4.( ) m-1 S Si For, {U, G, cP}, j being as above, we have by (3), (10) and by the integral formula (11) in the ordinary case

the left side of (11)_i 1_ 5 _(cv m ~, . m-1m U) v NGp m-1 Si h (3U) =4(?),

lY p 2 Gp

N S1 being the corresponding geodesic sphere around p.

3. Gauss-Bonnet formula for a compact V-manifold. Let M be a compact, orientable, Riemannian V -manifold of dimension m.

We shall consider the integral S~. For that purpose, let be an

arbitrary unit vector field on M with singularities at p1, , p, 7) and Ki be geodesic balls around pi with sufficiently small radius p. (We take p so small that Ki (1 K~=q5 for i4 j.) Then we have

=lim ~, M p-+0 M-UK1 i

and by (9) and 'Stokes formula'

7} The existence of - such a vector field can be proved easily.


S Q=- d(17~) M- U Ki M- U Ki _~ ~~

Si being the boundary of Ki oriented as above. Now since Dk is of 2k-th degree with respect to dui as is seen from (4), (5), (8), we have

`3(p2k)

and so

lim qk =0 1 m-1 - p-'0 si 2

On the other hand, we have by (11)

~5o0=(m_1)! m...@m_im) '~i 5Si =(m-1)! Om-1 1(3).

From these we get the following fundamental formula of Chern

S

M z=1 a

The left side of this formula being independent of the choice of E and the right side being independent of the Riemannian metric g, we see that this number is determined only by the V manif old structure of M. We shall denote this number by x(M)- Let us now assume that M has a triangulation M= J i sk I (I sk I denoting the carrier of the singular simplex sk) such that all the irreducible components of S (subvariety of all singular points) be-come subcomplexes of {sk}.8 Then considering a canonical vector field 3 attached to this triangulation (the vector field of Stiefel-Whitney) with singularities at each barycenters pk of sk, we have

x(M) = 4(?) _ (-1)dim sk NS

k the summation being taken over all simplexes of this triangulation

8) It is very plausible that every C-V-manifolds allow such a triangulation, though no proof is yet obtained.

482 I. SATAKE

and Nsk denoting the order of the isotropy group of sk (i. e. that of the isotropy group of an inner point p of Ski, which is, by our assumption, independent of the choice of 5). Thus, x(M) being an analogue of the Euler characteristic, we call it the Euler characteristic of M as a V -manifold. It should be noted that generally x(M) does not coincide with the ordinary Euler characteristic x(M) and that x(M) is not necessarily an integer.

We have thus obtained the following results: THEOREM 2. Let M be an orientable, compact, Riemannian V Vmani fold

of even dimension m. Then we have

(13) 2 Kdw = v(M) x 0m M

Let us remark that in the preceding considerations the assumption of the orientability of M is inessential. Namely considering dw as an n-form on M of the 2nd kind (in the sense of de Rham). Theorem 2 holds also for non-orientable M. We remark also that a (C~-) V manifold has always a (C~-) Riemannian metric just as an ordinary C'-manifold has one. We have therefore THEOREM 3. Let M be a compact V mani fold. Then for any (unit) vector field E with singularities at p1,. , ps, we have

4() = xv(M)

THEOREM 4. If M is a compact VVmanifold of odd dimension, we have

x(M) = 0. Theorem 2, 3 generalize the theorems of Allendoerf er-Weil [1] and of Hopf [4], respectively, to the case of V manif old. Theorem 4 is trivial (by the duality of Poincare) in the case of ordinary manifold.

4. The case of V-manifold with boundaries. We define the notion of VVmanif old with boundaries as follows. Let (M, ) be a

N V manif old. We assume that for each {U, G, cp} E there is given N N N

a closed subset Uo of U such that Uo is the closure of its interior N N

and that the boundary bUo of Uo is a part of a finite union of

N C~-subvarieties of dimension m-1 of U. We assume furthermore N that U ~o(b Uo) becomes a part of a IT subvariety B of dimension


N m -1 of M. Then we call Mo = U So(U0) a V-manifold with boundaries. The boundary bMo of Mo is clearly a part of B.

Now let Mo be an orientable compact Riemannian V manif old with boundaries. Then, considering (unit) vector fields on Mo with finite number of singularities in the interior of Mo coinciding with the inner (unit) normal vector field J1 on bMo, we can prove

just as in 3 the following THEOREM 5. Let Mo be an orientable, compact Riemannian V mani-

fold with boundaries and let Y1 be the inner unit normal vector field on bMo. Then we have

(14) S~ =xy(M0) _ II S Mo bMo bMo being oriented by the induced orientation with respect to the outer normal vector field on bMo.

xc(M0) is a rational number depending only on the V -manifold structure of Mo and may be called the inner Euler characteristic of Mo as a V -manifold with boundaries. In case Mo has a triangulation M=U ski such that all irreducible components of S and of bMo become subcomplexes of {sk}, xy(Mo) is given as follows

X v(Mo) _ (__ 1)dim sk 1 ' k N sk

the summation being taken over all simplexes of this triangulation which are in the interior of Mo.

4. Application to the Siegel space

1. Some results of Siegel. Let Sin be the generalized upper half-plane of degree n, i. e. the space of all complex symmetric matrices Z=X+iY of degree n with imaginary parts Y>0, and let Mn be Siegel's modular group, i. e. the group of all symplectic

transformations a = A B of degree 2n with integral coefficients, CD

operating on Sin in the following form

a (Z) = (AZ+ B) (CZ- D) - .

Then, Mn being a properly discontinuous group of analytic automor-

phisms of ~~n, the quotient space 9n =M0\ 0 becomes a complex

484 I. SATAKE

analytic (hence orientable) V manif old with respect to the family N N ={{U,G, ~o}}, where U is any connected open neighbourhood of any

PE S~n such that Q(U)=U for 6 with a()= ~ and a(U) (1 U=~b other-wise and c the restriction of the canonical map to U. "Un is of real dimension m=n(n+1)=2p.

An invariant Riemannian metric of (the symplectic metric) is given by

ds2 = Tr( Y-'dZY-1dZ) .

Concerning this metric Siegel [8] obtained the following results: A) The volume element is given by

dw=2p-ndv=2p-n f dx15dY1,

where X = (x~,), Y_ 1 = (Y2~). Calculating the Gaussian curvature K, we have

~= 2 Kdw=(-1)p - aan- - dv, Qm 2np!i

an denoting a certain combinatorial number depending only on n. an is an integer. (See Siegel [9], III, Theorem 5)

pt B) The volume of Un=Mn\~~n is finite and is given as follows.

n

S dv=2 [1 {(k-1)! v ' ~(2k)} S)3 n k=1 =2'1 7Zp ] ck 1) B2k

k=1 (2k) !

B2k denoting the absolute value of the k-th Bernoulli number. (See Siegel [9], VIII, Theorem 11) Combining A), B), we obtain

(15 ~=(-1)p2 an n (k 1) ! B k ) III 2

We shall prove in the next paragraph that Q expresses the Euler 5~characteristic xn of 93, as an open V manif old (in the sense specified S below). For some small values of n we have the following numeri-

cal table.


2. Gauss-Bonnet formula for 93. Since is not compact, we can not apply directly the formula (13) to But we can procede as follows. Let 5~(i) be the set of all ZE sin such that Yl ~ ~n and put K) =Mn\Mn We shall first prove the following LEMMA 3. 37,-K) is a compact V -manifold with boundaries. PROOF. Let Fn be Siegel's fundamental region of Mn in . Then by Siegel's reduction theory [S] we have for any ZE Fn, Q E Mn, ~a(Z) I S I ~ZI, Z denoting the imaginary part of Z. Thus 13Q(Z) ~n implies ~Z) ~ rn. Hence ~~(77) C Mn(Fn f s~(J)). From this follows

easily that 3n-K) becomes a V -manifold with boundaries. It follows also from. Siegel's reduction theory that if Z=X + i YE Fn

then __L S xi3 S 1 and Y is reduced in the sense of Minkowski 2 _ 2

so that, in particular, we have y11 sy22 S... O depending only on n. It holds

also y11 > ~3 for ZE Fn. Hence the set {Z; ZE Fn, Yj S ran} is 2

compact. It follows that -K() is compact, q. e. d. Now, applying theorem 5 to 3n-K~(77), we have

(16) -KQ =xV( Un K)) - H,

~

5 n ,() bK(~) denoting the outer unit normal vector field on b(7). We shall

next prove that

lim HoJ~=0. ~-+ bK, G~)

For that purpose, we shall give some lemmas.

486 I. SATAKC

LEMMA 4. We have

H o ~ = c~dw~ * ,

dw* denoting the volume element (with respect to the symplectic metric) on b~~(~) = {Z; J YJ =rjn} and c~ a constant depending only on r~. PROOF. The group C of all symplectic transformations of the form

Q(Z)=AZtA+B, JAJ=1

leaves bS (,7) invariant and operates on it transitively. Since H is clearly a symplectic invariant, H o 1J&J is a ~-invariant (m-1)-form on b5~(~) and so is also dw~*. From these follows the lemma im-mediately. LEMMA 5. The constant c~ in the preceding lemma is of the form

c~ _cl?-(m-1)

PROOF. Let us compare H J~ and dw* with H o " `1 and dw1*, respectively. We denote by z~ the symplectic transformation Z-->iiZ of ~n, which transforms b~(1) onto b~(j). Then, since H is a symplectic invariant, we have (ho J )oz~=17o "`1. On the other hand, we have clearly (dw~*)oZ~-r-ldwl*. These prove the lemma. LEMMA 6.

m

dw* =0(r~ 2 -1) bl~ , (77)

PROOF. Since we have

p-n

dw=2 2 11dzj.dw',

dw' denoting the volume element in the space of positive definite symmetric matrices Y with respect to the invariant metric ds2 Tr(Y-1dY)2, it follows easily that

p-n

dw* = 2 2 fl dxi j dw~ *

dw * denoting the volume element in the hyperplane {Y; YJ=rn} with respect to the above metric. Now from the proof of Lemma 3 we can see that bKa(i) is obtained from Fn fl b~~(?/) = {Z; ZE Fn, J YJ = rjn} by identifying the equivalent points on the boundary.


Hence we have by Siegel's reduction theory

c dw* = dw bK, G) Fnnb1 & )

1 P n 2

2 -2^ f dxt j dw' i~ j _ 1 ~Y~=fin

2 Y: reduced

p-n rtti -1 =(2 -2 \ dwi~*)i , q. e. d. FYI=1

Y; reduced

From Lemmas 4, 5, 6, we get

m

S x11,-O(?12), bK,,G ) which proves

lim Ho 1L=0. bK~(,)

On the other hand, since the subvariety S of all singular points of 9n is a union of a finite number of irreducible subvarieties (as is seen from the fact that Fn is bounded a by a finite number of alge-braic surfaces9)), the denominator of xV (fin-K~(~)) is bounded when i-+ co. These facts together with (16) prove the following THEOREM 6. x, (Yn-K)) becomes constant for sufficiently large ~. Denoting this number by xn' we have

(17) 2 Kdw=xn . Om Zn

3. An application.

We shall now apply the above result to the study of the least common multiple Nn of the orders of all isotropy subgroups of M/{ E}. First, it is clear that Nnxn is an integer. This gives a lower estimation of Nn. An upper estimation of Nn is given by a method of Minkowski [6] as follows.

We denote by Mn the 'homogeneous' Siegel's modular group, i. e. Mn=Sp(2n, Z), and put

9) This follows from Siegel [9], VI, Theorem 8.

488 I. SATAKE

M(l) = {a; a Mn~ a = E2n (mod l)},

l being any prime number. Then

Mn/Mn(l) ^ ~Sp (2n, l) (the symplectic group of degree 2n over GF(l)).

We shall first prove the following LEMMA 7. I f 14=2, Mn(l) contains no finite subgroup. I f 1=2,

any finite subgroup of Mn(2) is conjugate in Mn with a subgroup consist-ing o f elements o f the form

A 0 1 0 ~_ =

0 A 0 1

PROOF. The first half of this lemma is well-known (and is easy to prove).i0) Also it is easy to see that Mn(4) contains no finite subgroup. It follows that if G is a finite subgroup of M(2), we have a2=1 for all a E G (hence, in particular, G is abelian). There-fore to prove the lemma it will be enough to show that any aE Mn(2) with o2=1 is conjugate in Mn with an element of the form described above. Let a E Mn(2), a2=1 and 1v be an eigen vector of a (with the eigen value 1). We can take m so as to be a primi-tive integral vector. Then there exists a z E Mn with iv as the first 1 column vector.11> Then v-laz having 0 as the first column vec-

0 tor, we have by the conditions of symplectic matrices

+ 1 to b tb1

o A* b2 B* v_ 1az = ,

0 0 1 0

0 C* b D*

'A* B*

where a*_( IEM_1(2), na*2 =1. Hence, using the induction C* D*

on n, we can find zl ~Mn such that

10) Cf. Minkowski [5], [6], Siegel [9], VII, Theorem 10. 11) See Siegel [9], VIII, Lemma 15.


A B +_1 z1 Qz1= , A==(

0 to-1 0 + 1

Then from 62 =1 and 6 E Mn(2), we have (in a suitable choice of z1)

Er 2P -2(Q1 tP+PQ2) 2Q1 A= , B=

0 -E/ 2Q2 0

so that putting S T Er P 0 Q2

~2= J S= , T= , 0 tS-1 0 En-r \-Q2 0 .

we obtain Al 0 Er 0

z2zi 16Z1T2 1- ), A=( q. e. d. 0 Al 4 - En-r/

It follows that, denoting by G any finite subgroup of Mn, the restriction of the canonical homomorphism Mn-- Sp(2n, l) to G is an isomorphism, if 142, and has a kernel of the order dividing 2, , if 1=2. Hence, denoting by Nn the half of the greatest common divisor of the orders of Sp(2n,1) (l=1=2) and of the 2n times order of Sp(2n, 2), we have Nn Nn. Now since

the order of Sp(2n, l)=lna (12n-1) (1-n-2-1)...(12-1), we have

Nn= G. C. M {1na(12n__1)...(12-1) (l2), 2na+n(22n-1)... 22-1 } . 2

This number can be calculated easily by means of the following lemma of Minkowski. LEMMA 8.12) The greatest common divisor of (12n -1) (122 -2 -1).. (12 -1), l being all prime numbers, is equal to

Zx~ _ l_Ci

() f t : prime

l running over all prime numbers and [ ] denoting the symbol of Gauss. It is also known that

12) See Minkowski [6].

490 I. SATAKE

2nd =22nb2b2.bn ,

bk =jl ll+oralk = the denominator of B2 , 1-112k k

l running over all prime numbers such that l-112k and ordlk de-noting the exponent of the highest power of l dividing k.

Now we have

2n 3).

(We must replace 2n by 2n_1 for n=1, 2.) This formula seems to involve a new relationship between the numerators of the Bernoulli numbers. For some small values of n we have the following table.


From this and the other tables we obtain finally

N1= 6, N2=720 or 1440, N3=725760 or 725760x2.

REMARK. The same method can be applied, for instance, to the case of fin'=I'n\Pn1, Pn1 being the space of all positive definite sym-metric matrices of degree n with the determinant 1 and I'n the unimodular group (with the determinant 1) operating on P1. But, it is proved that the Gaussian curvature of Pn1=SL(n, R)/SO(n) is =0 for n>2.13' In fact, in virtue of the invariance of the alge-braic relations, it is sufficient to prove this in replacing SL(n, R) by any other real form of SL(n, C), say by SU(n), and for this latter case the Gauss-Bonnet theorem (for compact Riemannian manifold) can be applied. On the other hand the Euler charac-teristic of SU(n)/SO(n) can be calculated and proved to be =0 for n>2. Hence application of the Gauss-Bonnet theorem to 93' gives us nothing new. On the other hand, the least common multiple of the orders of the isotropy subgroups of P /{E} has been already calculated by Minkowski [6] with the result that it is equal to -n. 2

Added in Proof. The author was informed recently that Hir-zebruch has calculated an in (15) explicity, obtaining the result that

an _ lI n (2k-1)!

-

p. (k_1),

. It follows that

xn - ( 1)p n B2k 2 k=1 k

13) The author owes this remark to Professor A. Weil, to whom he wishes to express here his hearty thanks. The fact that x(SU(n)/SO(n))=0 (n> 2) follows also from a result of H. Hopf and H. Samelson, Emn Satz fiber die Wirkungsraume geschlossener Liescher Gruppen, Comm. Math. Helv., 13 (1940/41) pp. 240-251.

492 I. SATAKE

and that the fact Nnxn is an integer is trivial (hence containing no relationship between the Bernoulli numbers). He remarks also that the values for n=3 in our tables are false. The true values are as follows:

440 , N3x3 = 8, N3 =181440 x 2Y (0 s v 3). -6! = 360, x3 = _ 181440 6!

Bibliography

[1] C. B. Allendoerfer and A. Weil, The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. Amer. Math. Soc., 53 (1943), 101-129.

[2] S. S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifold, Ann. of Math., 45 (1944), 747-782.

[3] , On the curvatura integra in a Riemannian manifold, Ann. of Math., 46 (1945), 674-684.

[4] H. Hopf, Vektorfelder in n-dimensionalen Manigfaltigkeiten, Math. Ann., 96 (1926), 225-250.

[5] H. Minkowski, Uber den arithmetischen Begriff der Aquivalenz and uber die endlichen Gruppen linearer ganzzahliger Substitutionen, J. Reine Angew. Math.,

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[6] , Zur Theorie der positiven quadratischen Formen, J. Reine Angew. Math., 101 (1887), 196-202.

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Gauss Bonet Theorem