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t > b, be a geodesic in M such that X a has no conjugate point along T = Xl for a < t < b. Let Y be a Jacobi field along T which vanishes at X a and is perpendicular to T. Let X be a piecewise differentiable vector field along T which vanishes at X a and is perpendicular to T. If XXb = Y Xb ' then we have PROPOSITION
and the equality holds only when X = Y. Proof. Let Jr,a be the space of all Jacobi fields along T which vanish at X a and is perpendicular to T. By Proposition 1.1, the space of all Jacobi fields along T which vanish at X a is of dimension n. It follows (cf. Proposition 2.1) that dim Jr,a = n  1. Let YI , ... , Y n 1 be a basis for Jr,a. Since Y E Jr,a' we may write Y = alYI
+ ... + anIYn l ,
where aI' ... , an  l are constants. Since there is no conjugate point of X a on T for a < t < b, YI , . . . , Yn  l are linearly independent at every point X t for a < t < b. There exist therefore functions fl (t), . . . ,fnl(t) such that
X = flYI We have
(~
standing for
g(X', X')
=
+ ... + fnIYnl.
~~l)
geL h'Yi ,
'L f:Yi) + 2g('L h'Yi , 'L hYD
+ g('Lfi Y;' I h Y;), where eachf; denotes rift. Idt. We also have
g(R(X, +)+, X) = Ifig(R(Yi , +)+, X) =
g('L fi Y;',
'L fiYi) ,
=
'Lhg(Y;', X)
VIII.
73
VARIATIONS OF THE LENGTH INTEGRAL
where the second equality is a consequence of the assumption that each Yi is a Jacobi field. On the other hand, we have
d
,
dt g(~ fiYi, ~ fiYi)
= g(~ f;Yi, ~ fiYD
+ g(! fiYi,
+ g(~ h Y ;, ~ hYD ~ h'YD + g(~ hYi, !
h Y;').
Combining these three equalities we obtain g(X', X')  g(R(X, =
+)+, X)
g(~ f:Yi, ~ f:Yi)
d
+ dt g(~ fiYi,
+ g(~ h'Yi, ~ fiyn
~ fiYD
 g(~ hYi, ~ h'Y~).
Since, by Proposition 2.5, we have g(~f/Yi' !fiyn  g(~ hYi' ~ f:Y~) =
we obtain
I:(X) =
r
g(JJ;Y"
I
~f:h(g(Yi' Y;) g(Yi , Y~))
i,i
j/Y,) dt
= 0,
+ g(I j,Yi , I j,Y;)t~b'
Similarly, we obtain
Since a~
=
daildt
=
0, we have
I~(Y)
By our assumption that ... , n  1. Hence,
I~(X) ~ I:(Y)
= g(~ aiYi , ~ aiYDt=b. XXb
=
= Y Xb ' we have
r
g(I f/ Y"
I
ai
= h(b) for i = 1,
f;Y,)
:2:
o.
Obviously, the equality holds if and only ifh ' = 0 for i = 1, ... , n  1. Since ai = f(b)'h' = 0 implies ai = f(t) for all t and hence X = Y. QED.
74
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
3.2. Let T = Xt, a < t :::;;; b, be a geodesic in M such that X a has no conjugate point along T = X t for a < t < b. If X is a piecewise differentiable vector field along T vanishing at X a and X b and perpendicular to T, then I~(X) :2 0, OOROLLARY
and the equality holds only when X = o. Proof. Set Y = 0 in Proposition 3.1.
QED.
As an application of this corollary we prove 3.3. Let M be a Riemannian manifold with sectional curvature > ko > o. Then, for every geodesic T of M, the distance between two consecutive conjugate points of T is at most 7T /~. Proof. Let T = Xt, a < t < c, be a geodesic such that Xc is the first conjugate point of xl on T. Let b be an arbitrary number such that a < b < c. Let Y be a parallel unit vector field along T which is perpendicular to T, and letf(t) be a nonzero function such that f(a) = f(b) = O. By Corollary 3.2 we have I~(fY) :2 O. On the other hand, since Y is parallel along T, we have THEOREM
I:CfY)
=
0 be such that x c a and x c+a are in U. Then since x c+a is not a conjugate point of x c a along the segment T' of T from X c a to x c+a' the linear mapping of the set of Jacobi fields on T' into T Xc _lJ (M) + T Xc +lJ (M) given by Z + (Zxc_lJ' Zxc+a) is onetoone and therefore onto, because both vector spaces have dimension 2n. Hence there is a Jacobi field on T' with prescribed values at both ends. We now choose a Jacobi field Z on T' such that Zxc_a = Y:2l ca and Zxc+a = O. We now define a vector field X along T as follows: PROPOSITION
X=Y =Z =0
from
xa
to
Xc~'
from
xc_~
to
xc+~'
from
xc+~
to
x b•
76
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Since 0 = have
I~( Y)
=
I~~( Y)
I:(X) = I:(X) 
+ I~_~( Y)
I~( Y)
I~~(Y)
+ I~~~(Z)
= IC+~(Z) c~
 Ic (Y).
=

I:~(Y)

I~_~(Y)
c~
Let Y be a vector field along
Y
by Proposition 2.3, we
T
from x c~ to xc+~ defined as follows:
=
Y from
xc_~
=
0
Xc
from
to to
xc, xc+~.
Applying Proposition 3.1 to vector fields Yand Z, we have I~~~( Z)
b, be a geodesic in M and X a non zero Jacobi field along (j perpendicular to (j. Let T = Yt, a < t < b, be a geodesic in Nand Y a nonzero Jacobi field along T perpendicular to T. Assume: (1) Both X and Y vanish at t = a; (2) X' and Y ' have the same length at t = a; (3) Xa has no conjugate point on (j = X t , a < t < b, and Ya has no conjugate point on T = Yt, a < t < b; (4) For each t, a < t < b, if P is a plane in the tangent space at X t and if q is a plane in the tangent space atYt, then THEOREM
K(p) > K(q), where K (P) and K (q) are the sectional curvatures for p and q respectively. Under these four assumptions, we have g(X:ll t ' X:llJ < h( YYt' YyJ Proof.
for every t, a < t < b.
We set for a < t < b.
VIII.
77
VARIATIONS OF THE LENGTH INTEGRAL
Since u(t) * 0 and v(t) * 0 for a < t < b by (1) and (3), we define functions fl(t) and v(t) by for a < t < b. Since du{dt = 2g(X, X') and dv{dt = 2h(Y, Y'), by Proposition 2.3 we have dv{dt = 2vv. du{dt = 2flu, Solving these differential equations, we have, for every 8, a
dt
hm e e
To complete the proof, it suffices to show that fl(t) < v(t) for a < t < b. We take an arbitrary nUInber C, a < c < b, and fix it in the rest of the proof. We set
x
Y = Y{V(c)l/2,
= X{U(c)l/2,
so that X and Yare Jacobi fields along (J and T respectively and Xx c and Yyc are unit vectors. We shall_ construct _ a vector field Z along such that g(Z, Z)x t = h(Y, Y)y t and g(Z', Z')x t = _ _ (J h(Y', Y')Yt for a < t < b. To this end, let Ut (resp. Vt ) be the normal space to (J at Xt (resp. the normal space to T atYt). Letfc be a metricpreserving linear isomorphism of Vc onto U c such that fc(YY ) = XXc • Let 01' (resp. be the parallel displacement along () (resp. along T) from x,. to Xt (resp.y,. tOYt). Letit be the linear isomorphism of Vt onto Ut defined by
Tn
it
0
T~
=
(J~
0
fc.
78
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
If we set ZXt = ft(YyJ for a ::;;: t < b, then we obtain the desired vector field Z along (j. We have I~(X) < I~( Z) < I~(Y). The first inequality is a consequence of Proposition 3.1 and the second inequality follows from our assumption (4) and the definition of I~. On the other hand, we have p(c) = I~(X)/u(c) = I~(X), v(c) = I~(Y)/v(c) = I~(Y). Hence, we have p(c) < v(c). Since c is arbitrary, this ·completes the proof. QED. 4.2. Let M and N be Riemannian manifolds of dimension n. Let (j = Xt, a < t < b, and T = Yt, a < t < b, be geodesics in M and N respectively. Assume that, for each plane p in the tangent space at Xt and for each plane q in the tangent space atYt, we have K(p) ~ K(q). If Xa has no conjugate point on (j = Xt, a < t < b, then Ya has no conjugate point on T = Yt, a < t < b. Proof. Assume that Yc, a < c < b, is the first conjugate point ofYa along T. Let Y be a nonzero Jacobi field along T vanishing atYa andy c. Let X be a J acobi field along (j vanishing at Xa such that the length of X' at Xa is the same as that of Y' atYa (cf. Proposition 1.1). Then, by Theorem 4.1, we have for a < t < c. g(XXt' XxJ < h( Y yt , YyJ Hence, we have g(Xxc' Xx) = lim g(XXt ' XxJ < lim h( Y yt , YyJ = h( YYc ' Y y) = O. tc tc This means that X is a Jacobi field vanishing at Xa and Xc and hence that Xc is conjugate to Xa along (j, thus contradicting our assumption. QED. COROLLARY
The following result is originally due to Bonnet [1]: 4.3. Let M be a Riemannian manifold whose sectional curvature K is bounded as follows: 0 < k o < K (p) < ki) where ko and k1 are positive constants. If T = Xt, a < t < b, is a geodesic such that Xb is the first conjugate point of Xa along T, then 7T/v!k;. < b  a < 7T/Vko• Proof. The second inequality has been already proved in Theorem 3.4. However, it follows also from Corollary 4.2 and COROLLARY
VIII.
VARIATIONS OF THE LENGTH INTEGRAL
79
Example 2.1 if we let N be a complete Riemannian manifold of constant curvature ko in Corollary 4.2. To obtain the first inequality, let N be a complete Riemannian manifold of constant curvature k1 and interchange the roles of M and N in Corollary 4.2. QED.
Remark. In Corollary 4.2, let M be a Euclidean space and N a Riemannian manifold with nonpositive curvature. Then we obtain an alternative proof for Corollary 2.4. For extensions of the comparison theorem, see Toponogov [lJ, Tsukamoto [2J, Berger [13J, and Warner [2J.
5. The first and second variations of the length integral Let M be a Riemannian manifold with metric tensor g. Throughout this section we fix two points y and z of M and denote by r the set of all piecewise differentiable curves T = Xt, a < t < b, from y to z parametrized proportionally to arc length. The tangent space of r at T, denoted by TT( r), is the vector space of all piecewise differentiable vector fields X along T vanishing at the end pointsy and z. As we shall soon see, r is similar to a manifold and TT(r) plays the role of the tangent space. We shall use this analogy to motivate certain definitions. The length function L on r assigns to each T the length of T. The main purpose of this section lies in the study of the Hessian of L at each critical point of L. The total differential dL of L assigns at each T a linear functional on TT( r) in the following way. Given X in TT( r), consider a Iparameter family of curves T S = x:, a < t < b and  8 < s < 8 such that (1) Each T is an element of r; (2) TO = T; (3) There exists a finite set of numbers ti E [a, bJ with a = to < t1 < ... < tk = b such that (t, s) + x~ is differentiable on each rectangle [ti) ti+1J X (  8, 8) ; (4) For each fixed t E [a, bJ, the vector i?t) tangent to the curve X~,  8 < S < 8, at the point Xt = x~ coincides with Xa:e. S
80
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
We then set
=
dL(X)
(~ L( T')
L.
We shall express dL(X) explicitly by means of X and its covariant derivatives; in particular, we shall see that the above definition of dL(X) is independent of the choice of the family T S•
5.1. Let T = x:, a ~ t < band 8 < s < 8, be a Iparameter family of curves such that (t, s) * x~ is a differentiable mapping of [a, b] X (  8, 8) into M and that each curve T is parametrized proportionally to arc length. Set X t = x~ and T = TO. Then we have S
THEOREM
S
where X is the vector field along T defined by XXt = i?n and r is the common length of tangent vectors to T. The proof will be given later. As a direct consequence of this theorem, we have
r
and X E Tr(r). Let a = Co < C1 < ... < Ch < Ch + 1 = b be a partition of [a, b] such that the restriction of T to each [c j , cj + 1 ] is differentiable. Then THEOREM
5.2.
Let
T
E
where + and ++ denote the left and right limits + at the points Xc •
of the tangent vector field
J
As we shall soon see, this implies 5.3. A curve T ofor all X E Tr(r).
THEOREM
dL(X)
=
E
r
is a geodesic
if and only if
This means that the geodesics belonging to r are precisely the critical points of L on r. We shall define the Hessian of L at a geodesic T E r. Following Morse we shall call it the index form and denote it by 1. It will be a real symmetric bilinear form on T r ( r). As in the definition of dL, given X E Tr(r), we consider a 1parameter family of curves T with properties (1), (2), (3), and (4). S
VIII.
VARIATIONS OF THE LENGTH INTEGRAL
81
We set
I(X, X) =
(~:L(T·)L.
By polarization, we define the index form I(X, Y) by
I(X, Y)
![I(X
=
+ Y, X + Y)
 I(X, X)  I(Y, Y)J.
Then we have THEOREM
If 7" E r
5.4.
I(X, Y)
=
~ Jb r
is a geodesic and if X, Y
E
TT( r), then
[g(X.1 /, y.1 /)  g(R(X.1, +)+, Y.1)] dt,
a
where x.1 = X  (1 /r)g( X, 7}r is the component of X perpendicular to 7" and X 1denotes the covariant derivative V TX.1 of X.1 along 7", and similarly for y.1 1
I.
This may be reformulated as follows: THEOREM
5.5.
Let
7",
X, and Y be as in Theorem 5.4. Then
where a = to < t1 < ... < th < th +1 = b is a partition of [a, b] such that X is differentiable in each interval [t;, t;+lJ, J = 0, 1, ... , h, and X.1 / (resp. X.1 /+) denotes the left (resp. right) limit of the covariant derivatives of X.1 with respect to + at the points xtJ • From this we shall obtain
5.6. Let 7" E r be a geodesic and XE TT(r). Then X.1 is a Jacobi field if and only if I(X, Y) = 0 for all Y E TT(r). THEOREM
The remainder of this section is devoted to the proofs of the above six theorems and to a few applications. Proof of Theorem 5.1. We lift the mapping (t, s) + x: to a mappingy: [a, b] X (8,8) + O(M) (where O(M) is the bundle of orthonormal frames over M) such that 7T y(t, s) = x: and that the curve yO defined by yO(t) = y(t, 0) is horizontal. 0
82
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Let Sand T be the vector fields in the rectangle [a, b] defined by
X
(8, 8)
a at
T=  . Let 0, w, and Q be the canonical form, the connection form, and the curvature form on O(M) respectively. We define forms 0*, w*, and Q* on the rectangle by
0*
=
y*(O),
w*
=
y*(w),
Q*
=
y*(Q).
Then we have the following formulas (cf. §1 ) : , (A) (B) (C) (D)
[S, T] = 0; w* (T) = 0 at the points (t, 0) ; S(O*(T)) = T(O*(S)) + w*(T)O*(S)  w*(S)O*(T); S(w*(T)) = T(w*(S)) + w*(T)w*(S)  w*(S)w*(T) + 2Q*(S, T).
Observe that (B) is a consequence of the assumption that yO is horizontal and that (C) and (D) follow from (A), the first and second structure equations and Proposition 3.11 of Chapter I. We define a function F on the rectangle by setting
F = (O*(T), O*(T))l/2, so that, at each (t, s), F(t, s) is the length of the vector x~s). The length L( T S ) of each curve T S is then given by the integral:
L( r) =
r
F(t, s) dt.
Since each T S is parametrized proportionally to arc length, the function F(t, s) actually depends only on s. In particular, we have
(E) r = F(t, 0). Now we prove the following formula: (F) S(F) =
!r (T(O*(S)),
O*(T))
at the points (t,O).
VIII.
83
VARIATIONS OF THE LENGTH INTEGRAL
In fact, by (C) we have 2F· S(F) = S(F2) = 2(S(0*(T)), O*(T)) = 2(T(B*(S)), O*(T)) + 2(w*(T)B*(S), B*(T))  2(w* (S) 0* (T), 0* (T)). Since w*(S) is in o(n) (i.e., skewsymmetric), the last term vanishes. Hence, (F) follows from (B) and (E). We are now in position to compute the first variation
r r r r
r
(dL( 7 8 ) / ds) 8=0.
(G)
~ L( TS),~O =
S(F) «,0) dt
=~
(T( IJ* (S)), IJ* (T)) It,O) dt
(T(IJ*(S), IJ*(T)))«,o) dt
~
=
=~
(IJ*(S), T(IJ*(T)))«,o) dt
~r (O*(S),
~
O*(T))(bO) 
~ (O*(S),
'r
O*(T))(ao) '
(IJ* (S), T( IJ* (T))) It,O) dt.
On the other hand, we have
(H) O*(S)(t,O) = O(y(S(t,O»)) = y0(t)I(7T y(S I(pX, pX) and the equality holds if and only if X (3) i(I IT;)
forXET;, E
J.
i(I I J), a(I I T;) = a(I I J), and n(I ITf) = a(I I J). =
(1) Suppose X EJ and (1. (X) = 0 so thatXo = Xl = ... = X h = o. By our choice of the positive number 0, xai+I is not conjugate to xai along 'T. Hence, X == 0 along 'T, proving that (1. is injective. To show that (1. is surjective, it suffices to prove that, given vectors Xi and x ai and X i+l at xai+I' there is a Jacobi field X along 'T I [ai' ai +l ] which extends Xi and X i + l (cf. Corollary 2.2). Since XaH1 and x ai are not conjugate along 'T, X + (Xi' X i + l ) defines a linear isomorphism of the space of J acobi fields along 'T I [ai' ai+IJ into the direct sum of the tangent spaces at xai and XaH1 • Since it is a linear isomorphism of a vector space of dimension Proof.
VIII.
VARIATIONS OF THE LENGTH INTEGRAL
91
2n into a vector space of the same dimension (cf. Proposition 1.1), it must be surjective. This completes the proof of (1). (2)
With the notations in §3, we have h1
h1
I(X, X) =
I
I~Hl(X),
I(pX, pX)
=
I
I:~Hl (pX) .
i=O
i=O
By Proposition 3.1, we have
I::+1(X) >
I:~i+1(pX)
and the equality holds if and only if X is a Jacobi field along 'T I [ai' ai +1]. (3) If U is a subspace of T j on which I is negative semidefinite, then I is negative semidefinite on pU by (2). Moreover, p: U + pU is a linear isomorphism. In fact, if X E U and p(X) 0, then (2) implies
o>
I(X, X) > I(pX, pX)
=
0,
and hence I(X, X) = I(pX, pX). Again by (2), we have X pX = o. Thus we have
=
a(I ITj) < a(I I J). The reverse inequality is obvious. The prooffor the index i(I ITj) is similar. Finally, to prove n(I IT;) = n(I I J), let X be an element of J such that I(X, Y) = 0 for all Y E J. Since X is a Jacobi field along T I [ai' ai +1] for all i, the formula in Theorem 5.5 reduces to the following
leX, Y) =
~ [:~ g(X'
 X'+, Y) ••,].
In the same way as we proved Theorem 5.6, we conclude that X' = X'+ at xai for all i so that X is a Jacobi field along T. This means that n(I ITj) ~ n(I I J). The reverse inequality is obvious. Lemma 1 and the next lemma imply (2) of Theorem 6.1. 2. If B is a symmetric bilinear form on a finitedimensional vector space V, then a(B) = i(B) + n(B). LEMMA
92
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Proof. Let VI' ••• , Vr be a basis for V with respect to which B is a diagonal matrix with diagonal entries d1 , • • • , dr. Set the space spanned by {Vi; di > O}; V_ = the space spanned by {Vi; di < O}; Vo = the space spanned by {Vi; di = O}, V+
=
so that V = V+ + V_ + Vo• We shall show that n(B) dim Vo, i(B) = dim V_ and a(B) = dim (Vo + V_). Clearly, Vo
=
{X
E
V; B(X, Y) =
°
for all Y
E
V}
and hence dim Vo = n(B). Let U be any subspace of Von which B is negative semidefinite. We claim that the projection p: U + Vo + V_ along V+ is injective. In fact, if X E U and p(X) = 0, then X E V+. Since B is negative semidefinite on U and positive definite on V+, X must be zero. Thus, dim U < dim (Vo + V_) and hence a(B) = dim (Vo + V_). Similarly, if U' is any subspace of V on which B is negativedefinite, then the projection p': U' + V_ is injective and hence i(B) = dim V_. This completes the proof of Lemma 2. Since dim J = (h  1) (n  1) < 00, Lemma 1 implies that both a(I T:j) and i(I T:j) are finite. The finiteness of conjugate points follows from the next lemma.
I
I
For any finite number of conjugate points xt1 ' • • • , xt , (a < t1 < . . . < ts < b) of Xa along 7' with multiplicity fll' ••• , fls' we have a(I IT:j) > fll + ... + fls· LEMMA
3.
Proof. For each i, let XI, ... , X~~ be a basis for the Jacobi fields along 7' [a, tiJ which vanish at t = a and t = ti and extend them to be zero beyond t = tie It suffices to prove that fll + ... + fls vector fields Xt, ... , X~i' i = 1, ... , s, along 7' are linearly independent and that I is negative semidefinite on the space spanned by them. Suppo~e
I
8
LXi i=1
where
=
0,
VIII.
VARIATIONS OF THE LENGTH INTEGRAL
I
93
Since Xl, ..• , Xs1 vanish on T [tS1, h], XS must vanish along T [tsl, t s]· Being a Jacobi field along T [a, ts], Xs must vanish identically along T. Thus, c~ = = c;, = O. Continuing this S 1 S 1 argument, we obtain c1 = = cJ = 0 and so on. To l,l' prove that I is negative semidefinite on the space spanned by Xt, ... , X~l' i = 1, ... , s, let
I
X
=
I
+ ... + Xs,
Xl
where each Xi is a linear combination of Then s
I(X, X)
=
~
Xl, ... , X:
+2 I
I(Xi, Xi)
t
as above.
I(Xi, Xi).
1;;;;;1 r, we have THEOREM
j.
Since bk is the distance between r
Xo
and xak ' we have
= lim bk • k+oo
Hence, the set of vectors bkXk is contained in some compact subset of Txo(M). By taking a subsequence if necessary, we may assume that the sequence b1X1, b2 X 2 , • • • converges to some vector of length r, say r Y, where Y is a unit tangent vector. Then, exp tY, 0 < t < r, is a geodesic from X o to Xr since exp r Y = limk+ 00 exp bkXk = limk>oo xak = X r • It is of length r and hence minimizing. If X =I= Y, then exp tX and exp tY, 0 < t < r, are two distinct minimizing geodesics j oining Xo and Xr and (2) holds. Assume X = Y. Assuming also that Xr is not conjugate to X o along T, we shall obtain a contradiction. Since the differential of exp: Txo(M) + M is nonsingular at r X (cf. §1), exp is a diffeomorphism of a neighborhood U of rX in Txo(M) onto an open neighborhood of Xr in M. Let k be a large integer such that both akX and bkXk are in U. Since exp akX = xak = exp bkXk, we can conclude that akX = bkXk, thus contradicting the fact that X =I= X k • We have shown that if X = Y, then X r is conjugate to X o along 'T. On the other hand, there is no conjugate point of Xo before X r along T. Indeed, if xs , where 0 < s < r, were conjugate to X o along 'T, then 'T would not be minimizing beyond X s by Theorem 5.7, which contradicts the definition of r. Hence, Xs is the first conjugate point of X o along 'T.
QED.
98
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
7.2. If X r is the cut point of Xo along T = X t , o < t < 00, then X o is the cut point of Xr along T (in the reversed direction). Proof. Extending the geodesic T in the negative direction, we may assume that X t is defined for  00 < t < 00. Let a be any negative number. We claim that T I [a, r] is not minimizing. Assume (1) of Theorem 7.1. Then X o is a conjugate point of Xr along T in the reversed direction and, by Theorem 5.7, T I [a, r] cannot be minimizing. Assume (2) of Theorem 7.1. Then the join of T I [a, 0] and a minimizing geodesic from X o to X r other than T I [0, r] gives us a nongeodesic curve of length a + r joining Xa and X r • Hence, T I [a, r] whose length is also a + r, cannot be minimizing. This proves our claim. There exists therefore a nonnegative number b < r such that X b is the cut point of X r along T in the reversed direction. Assuming that b is positive, we shall get a contradiction. We apply Theorem 7.1 to the geodesic T I [b, r] in the reversed direction. Then X b and X r are conjugate to each other along T or else there is another minimizing geodesic from X r to X b • By the reasoning we just used, we can prove that if c < b, then T I [c, r] cannot be minimizing. In particular, T I [0, r] cannot be minimizing, which contradicts the definition of r. Hence, b = O. QED. COROLLARY
Let Sre be the set of unit tangent vectors at a point x of M; it is a unit sphere in Tre(M). We define a function fl: Sre + R+ U 00, where R + denotes the set of. positive real numbers, as follows. For each unit vector X E Sre, consider the geodesic T = exp tX, o < t < 00. If exp rX is the cut point of x along T, then we set fl(X) = r. If there is no cut point of x along T, then we set fl(X) = 00. We introduce a topology in R+ U 00 by taking intervals (a, b) and (a, 00] = (a, 00) U 00 as a base for the open sets. Then we have
7.3. The mapping fl: Sre + R + U 00 is continuous. Proof. Suppose that fl is not continuous at a point X of Sre and let Xl' X 2 , • • • be a seqaence of points of Sre converging to X such that fl(X) * lim fl(Xk ). THEOREM
k..
00
In general, limk+oo fl(Xk ) may not even exist. However, by taking a subsequence if necessary, we may assume that limk..oo fl(Xk ) exists in R+ U 00.
VIII.
VARIATIONS OF THE LENGTH INTEGRAL
99
We consider first the case fl (X) > limk+ <Xl fl (Xk). Set ak
fl(Xk),
=
a
=
lim fl(Xk). k+<Xl
Since fl(X) > a, exp aX cannot be conjugate to x along the geodesic exp tX. By Theorem 1.4, exp: Txo(M) + M maps a neighborhood, say U, of aX diffeomorphically onto a neighborhood of exp aX. We may assume, by omitting a finite number of akXk if necessary, that all of akXk are in U. Since exp maps U diffeomorphically onto exp (U), exp akXk cannot be conjugate to X o along the geodesic exp tXk • By Theorem 7.1, there is another minimizing geodesic from X o to exp akXk. In other words, there exists, for each k, a unit vector Yk =I= X k at X o such that exp akXk = exp akYk. Since exp maps U onetoone into M, each akYk does not lie in U. By taking a subsequence if necessary, we may assume that Y1 , Y2 , • • • converges to some unit vector, say Y. Then aY, which is the limit vector of akYk, does not lie in U. We have exp aY = exp (lim akYk) = lim (exp akYk) = lim (exp akXk) k+ <Xl k+ <Xl k+ <Xl = exp (lim akXk) = exp aX. k+<Xl Hence, both exp tX and exp t Y, a < t < a, are minimizing geodesics from X o to exp aX = exp aYe This implies that if b is any number greater than a, then the geodesic exp tX, a < t < b, is no longer minimizing, contradicting the assumption fl (X) > a. We consider next the case fl(X) < limk+<Xl fl(Xk). Let b be a positive number such that limk+<Xl fl(Xk) > fl(X) + b. By omitting a finite number of X k if necessary, we may assume that for all k. By the very definition of fl( X), there exists a unit vector X' =I= X at Xo such that exp tX', a < t < fl(X) + b', is a minimizing geodesic from X o to exp (fl(X) + b)X, where b' < b. (Note that b' may be negative.) In particular, exp (fl(X)
+ b)X =
exp (fl(X)
We set
2c
=
b  b'.
+ b')X'.
100
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Since the sequence of points exp (p,(X) + b)Xk converges to exp (p,(X) + b)X, we may assume, by omitting a finite nurrLber of X k ifnecessary, that the distances between exp (p,(X) + b)X and exp (p,(X) + b)Xk are all less than c. For each fixed k, consider the curve from Xo to exp (p,(X) + b)Xk defined as follows. It consists of the geodesic exp tX', 0 < t < p,(X) + b', from X o to exp (p,(X) + b')X' = exp (p,(X) + b)X and a minimizing geodesic from exp (p,(X) + b)X to exp (p,(X) + b)Xk • Then the length of this curve is less' than p,(X) + b' + c = p,(X) + b  c. This means that the geodesic exp tXk , 0 < t < p,( X) + b, is not minimizing, which contradicts the inequality p,(Xk ) > p,(X) + b. QED. ~
Let C(x o) denote the set of all p,(X)X, where X are unit vectors at X o such that p,(X) are finite. Let C(x o) be the image of C(xo) under expo Obviously, C(xo) consists of all cut points of X o along all geodesics starting from X o. We shall call C(x o) the cut locus of '" X o and C(x o) the cut locus of X o in Txo(M). ~
THEOREM
7.4.
Let E
=
{tX; 0 < t < p,(X) and X unit vectors
at x o}. Then (1) E is an open cell in Txo(M); (2) exp maps E dijJeomorphically onto an open subset of M; (3) M is a disjoint union of exp (E) and the cut locus C(xo) of X o. Proof. (1) follows from Theorem 7.3. Obviously, exp maps E onetoone into M. By Theorem 5.7, there cannot be any conjugate point of X o in E. Hence, the differential of exp: E + M is nonsingular at every point of E. This implies (2). Since E and "" C(x o) are disjoint, exp (E) and C(x o) are also disjoint. To show that M is a union of exp (E) and C(x o), lety be an arbitrary point of M. Let exp tX, 0 < t < a, be a minimizing geodesic from X o to y, where X is a unit tangent vector at X o and a is the distance from X o toy. From the very definition of p,(X), it follows that a < p,(X). Hence, aX is either in E or in C(x o). The point y = exp aX is therefore either in exp (E) or in C(x o). QED. ~
Remark. The open subset exp (E) of M is the largest open subset of M in which a normal coordinate system around X o can be defined.
VIII.
VARIATIONS OF THE LENGTH INTEGRAL
101
7.5. Let M be a complete Riemannian manifold and X o a point of M. Then M is compact if and only if, for every unit tangent vector X at xo, fl(X) is finite. Proof. Suppose M is compact and let d be the diameter of M. If a > d, then exp tX, 0 < t < a, cannot be a minimizing geodesic from X o to exp aX. Hence, fl(X) s d. Conversely, assume that, for every unit vector X at xo, fl(X) is finite. Since fl is a continuous function defined on the unit sphere in Txo(M) (cf. Theorem 7.3), fl is bounded by a positive number, say b. Let B be the set of tangent vectors at X o whose length are less than or ..., equal to b. Then B is a compact set containing E and C(xo) of Theorem 7.4. By Theorem 7.4, exp maps B onto M. Hence, M is compact. QED. THEOREM
Remark. Theorems 7.3 and 7.5 imply that M is compact if and only if the cut locus C(x o) of X o in Txo(M) is homeomorphic with a sphere of dimension n  1, where n = dim M. Example 7.1. Let M be an ndimensional unit sphere and x its north pole. The cut locus of x in Tx(M) is the sphere of radius TT with center at the origin of Tx(M). The cut locus of x in M reduces to the north pole. Example 7.2. Let Sn be the unit sphere in Rn+l. Identifying each point of Sn with its antipodal point, we obtain the ndimensional real projective space, which will be denoted by M. The Riemannian metric of Sn induces a Riemannian metric on Min a natural manner so that the projection of Sn onto M is a local isometry. The cut locus of a point x EM in Tx(M) is the sphere of radius TT/2 in Tx(M) with center at the origin. If x corresponds to the north and south poles of Sn, then the cut locus of x in M is the image of the equator of Sn under the projection Sn + M. The cut locus C(x) is therefore a naturally imbedded (n  I)dimensional proj ective space. Example 7.3. In the Euclidean plane R2 with coordinate system (x,y), consider the closed square given by 0 < x, Y < 1. By identifying (x, 0) with (x, 1) for all 0 < x < 1 and (O,y) with (l,y) for all 0 II W* II under the assumption that W* is perpendicular to p. We define vector fields y* and Yalong p and T respectively as in the proof of Lemma 2. Since y* and Yare induced by Iparameter families of geodesics pS and T S = expx (pS) respectively, they are Jacobi fields along p and T respectively vanishing at t = O. We shall now apply the comparison theorem of Rauch (Theorem 4.1) to y* and Y. Assumptions (1) and (4) in Theorem 4.1 are obviously satisfied. By Corollary 2.4, assumption (3) is also satisfied. To complete the proof of Lemma 3, it is therefore sufficient to verify assumption (2) for the vector fields y* and Y. Take a Euclidean coordinate system in Tx(M) with origin at the origin of Tx(M) and the corresponding normal coordinate system in a neighborhood of x. From the construction of Y, it follows that y* and Y have the same components with respect to the coordinate systems chosen above. On the other hand, the Christoffel symbols vanish
VIII.
VARIATIONS OF THE LENGTH INTEGRAL
105
at x (cf. Proposition 8.4 of Chapter III) and hence the covariant differentiation and the ordinary differentiation coincide at x (cf. Corollary 8.5 of Chapter III). These two facts imply assumption (2) of Theorem 4.1. Combining Lemma 1 and Lemma 3, we obtain Theorem 8.1. QED.
Remark 1. If M has vanishing curvature in Lemma 3, then we may apply the comparison theorem of Rauch in both directions to obtain I W* I < I WII and I W* I > I WII. Hence we have: . If M. has vanishing curvature, then expx: Tx(M) * M is an isometric zmmerszon. Combining this with Theorem 8.1 we obtain
8.2.
If M
is a complete, simp(y connected Riemannian manifold with vanishing curvature and x EM, then expx: Tx(M) * M is an isometry. COROLLARY
Remark 2. Kobayashi [16] strengthened Theorem 8.1 as follows : If M is a connected complete Riemannian manifold and a point x of M has no conjugate point, then expx: Tx(M) * M is a covering map. The proof may be achieved by replacing Lemma 3 by the following: If a point x E M has no conjugate point, there is a complete Riemannian metric on Tx(M) which makes expx distanceincreasing. For the proof we refer the reader to Kobayashi [16]; it is more elementary than that of Lemma 3. Theorem 8.1 is due to Hadamard [1] and E. Cartan [8]. The proof presented here is based on Ambrose's lectures at MIT in 195758. Theorem 8.1 and its generalization as stated in Remark 2 may be proved by means of the theory of Morse (cf. Klingenberg [8] and Milnor [3]). 8.3. Let M be a connected homogeneous Riemannian manifold with nonpositive sectional curvature and negativedefinite Ricci tensor. Then Mis simp(y connected and, for every x EM, expx: Tx(M) * M is a diffeomorphism. Proof. An isometry of a Riemannian manifold is called a Clifford translation if the distance between a point and its image under the isometry is the same for every point. The following lemma is due to Wolf [1]. THEOREM
106
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
1. Let M be a homogeneous Riemannian manifold and M a covering manifold with the induced metric so that the covering projection p: M + M is an isometric immersion. Then a diffeomorphism f of M onto itself satisfying p f = P is a Clifford translation of M. Proof of Lemma 1. Let G be a connected Lie group of isometries acting transitively on M, and 9 the Lie algebra of G. Considering every element X E 9 as an infinitesimal isometry of M, let x* be the lift of X to £1. Then the set of these vector fields x* generates a transitive Lie group G* of isometries of £1 whose Lie algebra is isomorphic to g. Since f induces the identity transformation of M, it leaves every x* invariant. Hence f commutes with every element of G*. For any two points Y, Y' E £1, let 1p be an element of G* such thaty' = 1p(y). Then LEMMA
0
d(y',f(Y'))
=
d(1p(Y),f o 1p(y))
=
d(1p(Y),
1p
0
f(y))
=
d(Y,f(Y)),
where d denotes the distance between two points. This completes the proof of Lemma 1.
Let M, M, andf be as in Lemma 1. LetYo EM and let T* = Yt, 0 < t < a, be a minimizing geodesic from Yo to f (Yo) so that Ya =f(yo)· Set Xt = P(Yt) for 0 < t < a. Then T = Xt, 0 < t < a, is a smoothly closed geodesic, that is, the outgoing direction of T at Xo coincides with the incoming direction of T at Xa. Proof of Lemma 2. Let r be a small positive number such that the rneighborhoods V(Yo; r) and V(Ya; r) ofYo andYa are homeomorphic with the rneighborhood U(x o; r) of X o = Xa under the projection p. Assume that T is not smoothly closed at X o = Xa • Then there is a small positive number t5 such that the points xa  6 and X 6 can be joined by a curve (1 in U(x o; r) whose length is less than 2t5 (where 2t5 is equal to the length of T from xa _ 6 through X a = Xo to x6). Let (1* be the curve in V(Ya; r) such that P( (1*) = (1. Let y* be the end point of (1*. Then y* = f(Y6). Then LEMMA
2.
d(YMf(Y6))
d(YMY*) < d(YMYa6) + d(Ya6'Y*) < (a  2 t5) + (length of (1*) = (a  2t5) + (length of a) < (a  2t5) + 2t5 = a = d(Yo,f(Yo)). =
This contradicts Lemma 1, thus completing the proof of Leluma 2.
VIII.
VARIATIONS OF THE LENGTH INTEGRAL
107
We shall now complete the proof of Theorem 8.3. Assuming that M is not simply connected, let M be the universal covering manifold of M. Let f be a covering transformation of M different from the identity transformation. Let T = X t , 0 < t < a, be a smoothly closed geodesic of M given in Lemma 2. Let X be any infinitesimal isometry of M. We define a nonnegative function h(t), 00 < t < +00, as follows:
h(t)
=
g(X, X)Xt
for 0 < t < a,
and then extend h to a periodic function of period a. By Lemma 2, h(t) is differentiable at every point t,  00 < t < + 00. We have d
dt h(t) = 2g(X', X)Xt
d2
dt 2 h(t)
=
for 0 < t < a,
2g(X', X')Xt  2g(R(X, +)+, X)Xt
for 0 < t < a.
Since the sectional curvature is nonpositive, h"(t) > 0 for 0 < t < a. Since h(t) is periodic, h(t) is a constant function. Hence h"(t) = O. In particular, g(X', X') = 0 and g(R(X, +)+, X) = o. On the other hand, since M is a homogeneous Riemannian manifold with negativedefinite Ricci tensor, there exists an infinitesimal isometry X of M such that g(R(X, +)+, X):r: o < O. This contradiction comes from the assumption that M is not simply connected. The fact that exp:r:: Tx(M) + M is a diffeomorphism follows from Theorem 8.1. QED. Theorem 8.3 is due to Kobayashi [19].
8.4. Let M be a connected homogeneous Riemannian manifold with nonpositive sectional curvature and negativedefinite Ricci tensor. If a Lie subgroup G of the group I(M) of isometries of M is transitive on M, then G has trivial center. Proof. Let C be the center of G. Let G be the closure of G in I( M). Then every element of C commutes with every element of G and hence lies in the center of G. We may therefore assume that G is a closed subgroup of I(M). We first prove that C is discrete. Suppose that X is an infinitesimal isometry of M which generates a Iparameter group belonging to the center C. Since X is invariant under the action of G, THEOREM
108
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
the length g( X, X) 1/2 is constant on M. Since the Ricci tensor of M is negativedefinite, Theorem 5.3 of Chapter VI implies that X vanishes identically. Let cp be any element ofG. Then it commutes with each element of G and hence is a Clifford translation in the sense defined in the proof of Theorem 8.3. In fact, for x, x/ E M we choose an element 1p of G such that x/ = 1p(x). Then we have
d(x', cp(x/)) = d(1p(x), cp 1p(x)) = d(1p(x), 0
1p
0
cp(x)) = d(x, cp(x)).
It follows that the action of G on M is free. We shall show that G is properly discontinuous on M. Let H be the isotropy subgroup of G at a point of M so that M = GIH. Then H is compact (cf. Corollary 4.8 of Chapter I). By Proposition 4.5 of Chapter I, G is discontinuous on M. By Proposition 4.4 of Chapter I, G is properly discontinuous on M. Then the quotient space MIG is a manifold (cf. Proposition 4.3 of Chapter I) and M is a covering manifold of MIG with the natural projection p: M + MIG as a covering projection (cf. pp. 6162 of Chapter I). Since G is the center of G, the action of G on M induces an action of G on MIG. It follows that with respect to the induced Riemannian metric MIG is also a homogeneous Riemannian manifold with nonpositive sectional curvature and negativedefinite Ricci tensor. By Theorem 8.3, MIG is simply connected. Hence G reduces to the identity element. QED.
9. Center of gravity and fixed points of isometries Let A be a compact topological space and G(A) the algebra of realvalued continuous functions f on A. With the norm Ilfll defined by Ilfll
=
sup If(a) I, aEA
G(A) is a Banach algebra. A Radon measure (or simply, measure) on A is a continuous linear mapping fl: G(A) + R. For eachf E G(A), fl(f) is called the integral off with respect to the measure fl and will be denoted by Lf(a) dft(a) or simply by i f dfto A measure
fl is positive if fl(f) > 0 for all nonnegative f
E
G(A) and if fl
*" O.
VIII.
VARIATIONS OF THE LENGTH INTEGRAL
109
We need the following theorem to prove the existence of a fixed point of a compact group of isometries of a complete, simply connected Riemannian manifold with nonpositive curvature.
9.1. Let p be a positive measure on a compact topological space A. Let f be a continuous mapping from A into a complete, simply connected Riemannian manifold M with nonpositive curvature. We set THEOREM
J(x)
L
for x EM,
d(x,j(a))2 dp.(a)
=
where d(x, f(a)) is the distance between x andf(a). Then J attains its minimum at precisely one point. The point where J takes its minimum will be called the center of gravity off(A) with respect to p. Proof. By normalizing the measure if necessary, we may assume that the total measure of A, i.e., p( I), is 1. (I) The case where M = Rn. Let Xl, ••• , x n be a Euclidean coordinate system in Rn and letf: A + Rn be given by a
Then
J(x)
=
=
Lf
A, i = 1, ... , n.
(xi fi(a))2 dp(a)
~ (Xi)2L dp.(a)
2f x'f./i(a) dp.(a)

L
=
+ t (fi(a))2 dp.(a) 1 ((X~)2  2b ixi ci ), i
where
bi
=
L/i(a) dp.(a)
and
ci
L (fi(a))2 dp.(a).
Hence, we have
J(x)
=
1 ((Xi
 bi)2
ci  (b i)2),
i
showing that J takes its minimum at b at b.
=
(bI, ... , bn ) and only
110
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
(2) Existence of the center of gravity in the general case. Since J is continuous, it suffices to show the existence of a compact subset K of M and a positive number r such that for some Yo in K and
J(x) > r2
for all x not in K.
Choose an arbitrary point Yo of M and take a positive number r such that for all a EA. d(Yo,j(a)) < r We set K = {x EM; d(x,j(A)) < r}. Since j (A) is compact, K is bounded and closed. Being a bounded, closed subset of a complete Riemannian manifold M, K is compact by Theorem 4.1 of Chapter IV. Evidently,yo E K. We have
J(yo)
r" dft(a) = r".
> r for all a EA. Hence,
L
r" dft(a) = r"
for x
E
K.
(3) The uniqueness of the center of gravity in the general case. We shall reduce the problem to the Euclidean case. Let 0 be a point of M where J takes its minimum. Since expo: To(M) + M is a diffeomorphism by Theorem 8.1, there is a unique mapping F: A + To(M) such that j = expo F. We set 0
J'(X) =
L
d'(X, F(a))" dft(a),
where d' denotes the Euclidean distance in To(M). Let a be the origin of To(M). We shall prove the following relations which imply obviously that 0 is the only point where J takes its minimum:
J(o) = J'(O) < J'(X) < J(x)
if expo X
=
x
=I=
o.
VIII.
111
VARIATIONS OF THE LENGTH INTEGRAL
Since expo preserves the length of a ray emanating from the origin 0, we have d'(O, X) = d(o, x) for x expo X. Hence, d'(O,F(a)) = d(o,j(a)) for a A. This implies the equality J(o) = J'(O). Since To(M) is a Euclidean space, (1) implies that 0 is the only point where J' takes its .. . mInImum, I.e., J' (0) < J' (X)
for 0 ::j= X
E
To(M).
Since expo is a distanceincreasing mapping (cf. Lemma 3 in the proof of Theorem 8.1), we have
d'(X, F(a))
d(x,j(a))
for x
=
expo X
a
and
A.
This implies the inequality
J'(X)
J(x)
for x
=
expo X.
QED. As an important application of Theorem 9.1, we prove
9.2. Every compact group G oj isometries oj a complete, simply connected Riemannian manifold M with nonpositive curvature has a fixed point. Proof. We choose any point, say xo, of M and define a mapping j: G ~ M as follows: THEOREM
for a
j(a)
E
G.
Let p, be a left invariant measure on G. (It is known that every compact group admits a biinvariant measure; see for example, Nachbin [1; p. 81J.) By setting A G, we apply Theorem 9.1 and claim that the center of gravity ofj(G) with respect p, is a fixed point of G. Evidently, it suffices to show that J(b(x)) = J(x) for x E M and for bEG. Since the distance function d is invariant by G and p, is left invariant, we have
J(b(x)) =
r d(b(x), a(x ))2 dp,(a) = 1 d(x, b1a(x ))2 dp,(a) r d(x, b1a(x ))2 dp,(b1a) = J(x).
JaeG
=
JaeG
o
aeG
o
o
QED.
112
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
We owe the formulation of Theorem 9.1 to Iwahori [lJ. Theorem 9.2 was originally proved by E. Cartan [16J. (See also Borel [4].) As an immediate consequence of Theorem 9.2 we have COROLLARY 9.3. Let M be a connected, simply connected homogeneous Riemannian manifold of nonpositive sectional curvature and let G be a closed subgroup of the group I( M) of isometries of M. Assume that G is transitive so that M = GIH, where H is the isotropy subgroup of G at a point of M. Then (1) H is a maximal compact subgroup of G and every maximal compact subgroup of G is conJugate to H; (2) If G is connected, so is H. Proof. (1) Let K be any maximal compact subgroup of G. By Theorem 9.2, K is contained in the isotropy subgroup of G at a point say x, of M. Since the isotropy subgroup at x is compact (cf. Corollary 4.8 of Chapter I), it coincides with K. Since G is transitive, the isotropy subgroups are all conjugate to each other. Hence H is conjugate to K and is a maximal compact subgroup ofG. (2) Since M is simply connected, a simple homotopy argument shows that if G is connected, so is H. QED. The following theorem will not be used except in §ll of Chapter
XI. THEOREM 9.4. Let M be a connected, simply connected homogeneous Riemannian manifold with nonpositive sectional curvature. Let G be a closed subgroup of the group of isometries of M which is transitive on M so that M = GIH, where H is the isotropy subgroup ofG at a point, say 0, of M. Assume that the linear isotropy representation of H leaves no non zero vector of To(M) invariant. Then we have (1) M is in onetoone correspondence with the set of all maximal compact subgroups of G under the correspondence which assigns to each point x E M the isotropy subgroup Ga; of G at x; (2) If (X is an automorphism of G of prime period, then there is a maximal compact subgroup of G which is invariant by (X. Proof. (1) By Corollary 9.3, Ga; is a maximal compact subgroup of G and, conversely, every maximal compact subgroup of G coincides with Ga; for some x EM. It remains to prove that, for
VIII.
VARIATIONS OF THE LENGTH INTEGRAL
113
every x E M, x is the only fixed point ofGflJ. Because of homogeneity, it suffices to show that 0 is the only fixed point of H = Go. Assume that there is another fixed point x of H. By Theorem 8.1 there is a unique geodesic T from 0 to x. By uniqueness of T, H leaves T pointwise invariant. Hence the tangent vector to T at 0 is invariant by H, in contradiction to our assumption. (2) Let y be any automorphism of G. Then y permutes the maximal compact subgroups of G. Let y denote the corresponding transformation of M; if y(GflJ) = G,J' then y(x) = y by definition. Let r be the cyclic group generated by (X and assume that the order of r is prime. Then every element y =I= 1 of r is a generator of r, and a fixed point of y for any y =I= 1 is a fixed point of ~. If ~ has no fixed point on M, then r acts freely on M. But this is impossible since M is homeomorphic to a Euclidean space by Theorem 8.1 of Chapter VIII. If r acts freely on M, then
Hk(r; Z)
=
Hk(M/r; Z)
by a theorem of Eckmann, EilenbergMacLane and Hopf (see CartanEilenberg [1; p. 356]). On the other hand, Hk(Mjr, Z) = o for k > dim M. On the other hand, Hk (r; Z) = Z'J) for all even k (c£ CartanEilenberg [1; p. 251]). This shows that r cannot act freely on M. In other words, ~ has a fixed point, say Jf EM. Then (X leaves GfD invariant. QED.
CHAPTER IX
Complex Manifolds 1. Algebraic preliminaries The linearalgebraic results on real and complex vector spaces obtained in this section will be applied to tangent spaces of manifolds in subsequent sections. A complex structure on a real vector space V is a linear endomorphism J of V such that J2 = 1, where 1 stands for the identity transformation of V. A real vector space V with a complex structure J can be turned into a complex vector space by defining scalar multiplication by complex numbers as follows:
(a
+ ib)X =
aX
+ bJX
for X
E
V and a, b E R.
Clearly, the real dimension m of V must be even and lm is the complex dimension of V. Conversely, given a complex vector space V of complex dimension n, let J be the linear endomorphism of V defined by for X
JX= iX
E
V.
If we consider V as a real vector space of real dimension 2n, then J is a complex structure of V.
1.1. Let J be a complex structure on a 2ndimensional real vector space V. Then there exist elements Xl' ... , X n of V such that {Xl' ... , X n , JXI , • • • , JXn } is a basis for V. Proof. We turn V into an ndimensional complex vector space as above. Let Xl' ... , X n be a basis for V as a complex vector space. It is easy to see that {Xl' ... , X n , JXI , • • • , JXn } is a basis QED. for Vas a real vector space. PROPOSITION
114
IX.
115
COMPLEX MANIFOLDS
Let Cn be the complex vector space of ntuples of complex numbers Z = (Zl, ... , zn). If we set Zk
= xk
+ rye,
,
XkykER
,
k
=
1, ... , n,
then Cn can be identified with the real vector space R2n of2ntuples of real numbers (xl, ... , xn, y\ ... ,yn). In the following, unless otherwise stated, the identification of Cn with R2n will always be done by means of the correspondence (Zl, ... , zn) + (Xl, ... , xn, y\ ... ,yn). The complex structure of R2n induced from that of Cn maps (xl, ... , xn,y\ ... ,yn) into (y\ ... ,yn, Xl, ... , x n) and is called the canonical complex structure of R 2n. In terms of the natural basis for R2n, it is given by the matrix
!o
a ( = In
In) 0 '
where In denotes the identity matrix of degree n.
1.2. Let J and J' be complex structures on real vector spaces V and V', respectively. If we consider V and V' as complex vector spaces in a natural manner, then a real linear mapping f of V into V'is complex linear when and only when J' f = f J. Proof. This follows from the fact that J or J' is the multiplication by i when V or V'is considered as a complex vector space. QED. PROPOSITION
0
0
In particular, the complex general linear group GL(n; C) of degree n can be identified with the subgroup of GL(2n; R) consisting of matrices which commute with the matrix
JO = (
a 1n
In ). 0
A simple calculation shows that this representation of GL(n; C) into GL(2n; R), called the real representation of GL(n; C), is given by
A+ iB (AB +
B)
A
where both A and B are real n
+ iB E GL(n; C),
for
A
X n
matrices.
116
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
1.3. There is a natural onetoone correspondence between the set of complex structures on R2n and the homogeneous space GL(2n; R) /GL(n; C),. the coset represented by an element S E GL(2n; R) corresponds to the complex structure SJoSl, where J o is (the matrix representing) the canonical complex structure of R 2n. Proof. Every element S of GL(2n; R) sends every complex structure J ofR2n into a complex structure SJSl ofR2n; we consider GL(2n; R) as a transformation group acting on the set of complex structures of R2n. It suffices to prove that this action is transitive and that the isotropy subgroup of GL(2n; R) at J o is GL(n; C). To prove the transitivity of the action, let J and J' be two complex structures ofR2n. By Proposition 1.1, there are bases {e 1 , • • • , em Je 1 , • • • , Je n } and {ei, ... , e~, J'ei, . .. , J'e~} for R2n. If we define an element S of GL(2n; R) by PROPOSITION
SJe k
=
J'e'k
for k.
1,..., n,
then J' = SJSl, thus proving the transitivity. On the other hand, an element SofGL(2n; R) is in GL(n; C) if and only ifit commutes with J o, that is, J o = SJOSl (cf. the argument following Proposition 1.2). QED.
1.4. Let J be a complex structure on a real vector space V. Then a real vector subspace V' of V is invariant by J if and only if V' is a complex subspace of V when V is considered as a complex vector space. Proof. As in the case of Proposition 1.2, this also follows from the fact that J is the multiplication by i when V is considered as a complex vector space. QED. PROPOSITION
Let V be a real vector space and V* its dual space. A complex structure J on V induces a complex structure on V*, denoted also by J, as follows:
(JX, X*) = (X, JX*)
for X
E
V and X*
E
V*.
Let V be a real mdimensional vector space and Vc the complexification of V, i.e., Vc = V @R C. Then V is a real subspace of Vc in a natural manner. More generally, the tensor space T~(V) of type (r, s) over V can be considered as a real subspace of the tensor space T~( Vc) in a natural manner. The complex conjugation in Vc is the real linear endomorphism defined by Z
=
X
+ iY ~ Z
= X  iY
for X, Y
E
V.
IX.
117
COMPLEX MANIFOLDS
The complex conjugation of Vc extends in a natural manner to that ofT~(VC).
Assume now that V is a real2ndimensional vector space with a complex structure J. Then J can be uniquely extended to a complex linear endomorphism of Vc, and the extended endomorphism, 1. The eigendenoted also by J, satisfies the equation J2 values of J are therefore i and i. We set
VO.I = {Z E Vc,. JZ
VI,O = {Z E Vc ~. JZ = iZ},
iZ}.
The following proposition is evident. PROPOSITION
1.5. VL·
(1) Vl,O
=
{X  iJX; X
E
V} and VO,I =
{X iJX~· X E (2) Vc VI'o VO,I (complex vector space direct sum); (3) The complex conjugation in Vc defines a real linear isomorphism between VI'o and VO.I. Let V* be the dual space of V. Its complexification V*c is the dual space of Vc. With respect to the eigenvalues ± i ofthe complex structure Jon V*, we have a direct sum decomposition as above:
V*c = VI,o + VO,I. The proof of the following proposition is also trivial. PROPOSITION
VI,o
=
1.6. V*c~.
{X*
(X, X*) = 0
for all X
E
VO.I},
VO.I = {X* E V*o; (X, X*) 0 for all X E Vl,O}. The tensor space T;( vc) may be decomposed into a direct sum of tensor products of vector spaces each of which is identical with one of the spaces Vl,O, VO,\ VI,O, and VO,I. We shall study the decomposition of the exterior algebra A v*o more closely. The exterior algebras A VI.O and A VO,I can be considered as subalgebras of A v*c in a natural manner. We denote by Ap,q V*c the subspace of A v*c spanned by (J. A p, where (J. E APVI,o and PE AqVo,l. The following proposition is evident. PROPOSITION
1.7.
The exterior algebra A v*c may be decomposed
as fqllows:
A v*c
n
=
2
r=O
ArV*c
with
ArV*c
118
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
and the complex conjugation in V*c, extended to A v*c in a natural manner, gives a real linear isomorphism between Ap,qV*c and Aq,pV*c. If {e\ ... , en} is a basis for the complex vector space V1,O, then {P, ... , en}, where ek = ek , is a basis for VO,1 (cf. Proposition 1.5) and the set of elements eh A ••• A eJ1i A ek1 A ••• A ekq , 1 < il <  . . < i 'P < nand 1 < kl < . . . < kq < n, forms a basis for AP,fJV*c over the field of complex numbers. A Hermitian inner product on a real vector space V with a complex structure J is an inner product h such that h(JX, JY) = h(X, Y) for X, Y E V. It follows that h(JX, X) = 0 for every X E V. 1.8.  Let h be a Hermitian inner product in a 2ndimensional real vector space V with a complex structure J. Then there exist elements Xl' ... , X n of V such that {Xl' ... , X n, JXI , ••• , JXn} is an orthonormal basis for V with respect to the inner product h. Proof. We use induction on dim V. If Xl is a unit vector, then {Xl' JXI } is orthonormal. Let W be the subspace spanned by Xl and JXI and let W1.. be the orthogonal complement so that V = W + W 1... Then W 1.. is invariant by J. By the inductive assumption, W 1.. has an orthonormal basis of the form {X2 , • • • , X n, J X 2 , • • • , JXn}. QED. If ho is the canonical inner product in R2n, i.e., the inner product with respect to which the natural basis of R2n is orthonormal, then ho is a Hermitian inner product with respect to the canonical complex structure J o of R2n. PROPOSITION
1.9. There is a natural onetoone correspondence between the set of Hermitian inner products in R 2n with respect to the canonical complex structure J o and the homogeneous space GL(n; C) / U(n); the coset represented by an element S E GL(n; C) corresponds to the Hermitian inner product h defined by PROPOSITION
R2n, where ho is the canonical Jfermitian inner product in R 2n. h(X, Y) = ho(SX, SY)
for X, Y
E
Proof. The proof is similar to that of Proposition 1.3. An element S of GL(n; C) sends a Hermitian inner product h of R2n (with respect to J o) into a Hermitian inner product h' as follows:
h' (X, Y)
=
h(SX, SY)
for X, Y
E
R2n.
IX.
119
COMPLEX MANIFOLDS
(It should be remarked that, considered as a subgroup of GL (2n ; R) , the group GL(n; C) acts on R2n.) We consider GL(n; C) as a transformation group acting on the set ofHermitian inner products in R2n (with respect to J o) in the manner just described. It suffices to prove that this action is transitive and that the isotropy subgroup of GL(n; C) at ho is U(n). Given two Hermitian inner products hand h' ofR2n, by Proposition 1.7 there are orthonormal bases {el , • . . , em JOel, ... , Joe n } with respect to h and {e~, ... , e~, Joe~, ... , Joe~} with respect to h' for R2n. The element S of GL(2n; R) defined by for k = 1, ... , n, is an element of GL(n; 'C) and sends h into h', thus proving the transitivity of the action. On the other hand, the isotropy subgroup of GL(n; C) at J o is evidently the intersection GL(n; C) n O(2n), where both GL(n; C) and O(2n) are considered as subgroups of GL(2n; R). It is easy to see that U(n) consists of elements of GL(n; C) whose real representations are in O(2n). QED. The proof of the following proposition is straightforward. 1.10. Let h be a Hermitian inner product in a real vector space V with a complex structure J. Then h can be extended uniquely to a complex symmetric bilinear form, denoted also by h, of Va, and it satisfies the following conditions : PROPOSITION
(1) h(Z, W) = h(Z, W) for Z, WE Va; (2) h( Z, Z) > 0 for all nonzero Z E Va~. (3) h( Z, W) = 0 for Z E VL'o and W E
VO,l.
Conversely, every complex symmetric bilinearform h on Va satisfying (1), (2), (3) is the natural extension of a Hermitian inner product of V. To each Hermitian inner product h on V with respect to a complex structure J, we associate an element cp of A2 V* as follows: for X, Y E V. cp(X, Y) = h(X, JY) We have to verify that cp is skewsymmetric: cp(Y, X) = h(Y, JX) = h(JX, Y) = h(JX, J2Y) = h(X, JY) = cp(X, Y).
120
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
It can be easily seen that cp is also invariant by J. Since A2 v* can be considered as a subspace of A2 v*c, cp may be considered as an element of A2 V*c. In other words, cp may be uniquely extended to a skewsymmetric bilinear form on Vc, denoted also by cpo By Propositions 1.5, 1.6, and 1.10, we have
Let cp be the skewsymmetric bilinear form on Vc associated to a Hermitian inner product h of V. Then cp E A1,1 V*c. PROPOSITION
1.11.
We prove PROPOSITION
Z,,} a basis for We set
Let hand cp be as in Proposition 1.11, {ZI' ... , over C and {~\ ... , ~n} the dual basis jar V1 ,o.
1.12. Vl,O
for j, k = 1, ... , n. Then (1) hjl' = hkJ for j, k = 1, ... , n; h t j ~tic (2) cp   2z· ""n £...lj,k=l jkf; A 1\
•
Proof. (1) follows from (1) of Proposition 1.10. As for (2), given Z, W E Vc, we may write n
Z =
L (~j(Z)Zj + ~j(Z)Zj),
11
W
=
j=1
L (~j(W)Zj + ~j(W)Zj). j=1
A simple calculation then shows n
cp(Z, W) = i
L
hjk(~j(Z)~k(W)
 ~j(W)~k(Z)).
j,k=l
QED. Example 1.1.
Let 9 be a Lie algebra over C. Considering 9 as a real vector space we have a complex structure J defined by JX = iX. The complex structure J satisfies [JX, YJ = J[X, YJ = [X, JYJ for all X, Y E g, that is, J 0 ad (X) = ad (X) 0 Jfor every X E g. Conversely, suppose that 9 is a Lie algebra over R with a complex structure J satisfying J 0 ad (X) = ad (X) 0 J for every X E g. Then, defining (a + ib)X = aX + bJX, where a, b E R, we get a complex Lie algebra; we may verify complex bilinearity of the bracket operation as follows:
[(a
+ ib)X,
YJ = [aX, YJ
= a[X, YJ
+ [bJX, YJ + bJ[X, YJ
=
a[X, YJ
=
(a
+ b[JX, YJ
+ ib) [X, YJ.
IX.
COMPLEX MANIFOLDS
121
2. Almost complex manifolds and cOlnplex manifolds A definition of complex manifold was given in Chapter 1. For the better understanding of complex manifolds, we shall define the notion of almost complex manifolds and apply the results of §I to tangent spaces of almost complex manifolds. An almost complex structure on a real differentiable manifold M is a tensor field J which is, at every point x of M, an endomorphism of the tangent space Tx(M) such that J2 = 1, where 1 denotes the identity transformation of Tx(M). A manifold with a fixed almost complex structure is called an almost complex manifold.
2.1. Every almost complex manifold is of even dimensions and is orientable. Proof. An almost complex structure Jon M defines a complex structure in each tangent space Tx(M). As we have shown at the beginning of §1, dim Tx(M) is even. Let 2n = dim M. In each tangent space Tx(M) we fix a basis Xl' ... , X m JXI , ••• , JXn• The existence of such a basis was proved in Proposition 1.1 and it is easy to see that any two such bases differ from each other by a linear transformation with positive determinant. To give an orientation to M, we consider the family of all local coordinate systems xl, ... , x2n of M such that, at each point x where the coordinate system xl, ... , x2n is valid, the basis (oj oxl )x, •.• , (oj ox2n )x of T x ( M) differs from the above chosen basis Xl' ... , X m J Xl' ... , J X n by a linear transformation with positive determinant. It is a simple matter to verify that the family of local coordinate systems thus obtained gives a complete atlas compatible with the pseudogroup of orientationpreserving transformations ofR2n. QED. PROPOSITION
The orientation of an almost complex manifold M given in the proof above is called the natural orientation. To show that every complex manifold carries a natural almost complex structure, we consider the space C n of ntuples of complex numbers (zl, ... , zn) with Zi = xi + iyi, J = 1, ... , n. With respect to the coordinate system (xl, ... , xn,yl, .•. ,yn) we define an almost complex structure J on Cn by J( ojoyi) =  (ojox i ), J = 1, ... , n.
122
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
2.2. A mapping f of an open subset of Cn into Cm preserves the almost complex structures of Cn and cm, i.e., f* ,",o J = J f* , if and only iff is holomorphic. Proof. Let (WI, ••• , w m ) with w k = Uk + ivk , k = 1, ... , m, be the natural coordinate system in Cm. If we express f in terms of these coordinate systems in Cn and Cm = PROPOSITION
0
lr
Uk
=
Vk _
(xl, . . . , xn, yl, . . . ,yn), n yl , ••• , y n) , Vk(. 1 X, Uk
,x ,... ,
k
=
1, ... , m,
thenfis ho10morphic when and only when the following CauchyRiemann equations hold:
auk/aX]  avk/ayi auk/ayi + avk/axi
=
0,
=
o.
J=l, ... ,n;
k
=
1, ... , m.
On the other hand, we have always (whether f is holomorphic or not)
f* (aj axi ) = f*(a/ayi) =
m
~
(auk/ax i )(a/auk)
m
(auk/ayi)(a/au k)
k=l ~
k=l
for J =
m
+ ~ (avk/ax i)(a/av 'C ), k=l m
+ ~ (av k/ay 1) (ajavk),
k=l 1, ... , n.
From these formulas and the definition of J in Cn and Cm given above, we see that f* 0 J = J 0 f* if and only if f satisfies the CauchyRiemann equations. QED. To define an almost complex structure on a complex manifold M, we transfer the almost complex structure of Cn to M by means of charts. Proposition 2.2 implies that an almost complex structure can be thus defined on M independently of the choice of charts. An almost complex structure J on a manifold M is called a complex structure if M is an underlying differentiable manifold of a conlplex manifold which induces J in the way just described. Let M and M' be almost complex manifolds with almost complex structures J and J', respectively. A mapping f: M ~ M' is said to be almost complex if J' 0 f* = f* 0 J. From Proposition 2.2 we obtain
IX.
COMPLEX MANIFOLDS
123
2.3. Let M and M' be complex manifolds. A mapping f: M + M' is holomorphic if and only iff is almost complex with respect to the complex structures of M and M'. PROPOSITION
In particular, two complex manifolds with the same underlying differentiable manifold are identical if the corresponding almost complex structures coincide. Given an almost complex structure J on a manifold M, the tensor field J is also an almost complex structure which is said to be conjugate to J. If M is a complex manifold with atlas {( Uj, Pj)} then the family of charts (Uj, Pj), where Pj is the complex conjugate of Ph is an atlas of the topological space underlying M which is compatible with the pseudogroup of holomorphic transformations of Cn. The atlas {( Uj, Pj)} defines a complex manifold whose underlying differentiable manifold is the same as that of M; this new complex manifold is said to be conjugate to M and will be denoted by M. It is easy to verify that if J is the complex structure ofa complex manifold M, then J is the complex structure of M.
2.4. Let M be a 2ndimensional orientable manifold and L(M) the bundle of linear frames over M. Then the set of almost complex structures on M are in onetoone correspondence with the set of crosssections of the associated bundle B = L (M) jGL (n; C) with fibre GL(2n; R)jGL(n; C), where GL(n; C) is considered as a subgroup of GL(2n; R) by its real representation. PROPOSITION
Proof. This follows from Proposition 5.6 of Chapter I (see also the remark following it) and Proposition 1.3. QED. In general, given two tensor fields A and B of type (1, 1) on a manifold M, we can construct the torsion of A and B, which is a tensor field of type (1, 2) (cf. Proposition 3.12 of Chapter I). Specializing to the case where both A and B are an almost complex structure J, we define the torsion of J to be the tensor field N of type (1, 2) given by N(X, Y)
=
2{[JX, JY]  [X, Y]  J[X, JY]  J[JX, Y]} for X, Y E X(M).
Let xl, ... , x 2n be a local coordinate system in M. By setting X = a/ax; and Y = a/axk in the equation defining N, we see that the components NJk of N with respect to xl, ... , x 2n may be
124
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
expressed in terms of the components J; of J and its partial derivatives as follows: 2n
N;k
=
2 I (J:
0hJ~  J~ Oh J ;  JJ~ 0iJ~
h=l
+ J~ 0k J ;£),
o;ox
h. An almost where 0h denotes the partial differentiation complex structure is said to be integrable if it has no torsion. THEOREM 2.5. An almost complex structure is a complex structure if and only if it has no torsion. Proof. We shall only prove here that a complex structure has no torsion. The converse will be proved in Appendix 8 only in the case where the manifold and its almost complex structure are real analytic. Let Zl, ••• , zn, Zi = Xi + ~i, be a complex local coordinate system in acornplex manifold M. From the construction of the complex structure J given before Proposition 2.2, it is clear that the components of J with respect to the local coordinate n c . . .In t h e cosystem x1 , . . . , x n ,y1 , . . . ,yare constant . lunctlons ordinate neighborhood and hence have vanishing partial derivatives. By the expression above for NJk it is clear that the torsion N is zero. QED. The complex tangent space T~(M) of a manifold M at x is the complexification of the tangent space Tx(M) and its elements are called complex tangent vectors at x. Ifwe denote by !)r(M) the space of rforms on M, then an element of the complexification [r(M) of !)r(M) is called a complex rform on M. Every complex rform OJ may be written uniquely as OJ' + iOJ", where OJ' and OJ" are (real) rforms. If we denote by T:c the complexification of the dual space of Tx(M), then a complex rform OJ on M gives an element of At' T:c at each point x of M; in other words, a skewsymmetric rlinear mappings T~(M) X ••• X T~(M) ~ C at each point x of M. More generally, we can define the space of complex tensor fields on M as the complexification of the space of (real) tensor fields. Such operations as contractions, brackets, exterior differentiation, Lie differentiations, interior products, etc. (cf. §3 of Chapter I) can be extended by linearity to complex tensor fields or complex differential forms. If M is an almost complex manifold with almost complex structure J, then by Proposition 1.5 T~(M) =
T~'o
+ T2'\
IX.
125
COMPLEX MANIFOLDS
where ~,o and T~,l are the eigenspaces of J corresponding to the eigenvalues i and i respectively. A complex tangent vector (field) is of type (1, 0) (resp. (0, 1)) if it belongs to T~'o (resp. T~,l). By Proposition 1.5 we have
2.6. A complex tangent vector Z of an almost complex manifold M is of type (1, 0) (resp. (0, 1)) if and only if Z = X  iJX (resp. Z = X + iJX) for some real tangent vector X. PROPOSITION
By Proposition 1.7, the space [ = [(M) of complex differential forms on an almost complex manifold M of dimension 2n may be bigraded as follows: n
L
[ =
[p,q.
p,q=O
An element of [p,q is called a (complex) form of degree (p, q). By Proposition 1.6, a complex Iform w is of degree (1, 0) (resp. (0, 1)) if and only if w( Z) = 0 for all complex vector fields Z of type (0, 1) (resp. (1, 0)). If wI, ... , w n is a local basis for [1,0, then its complex conjugate WI, ••• , wn is a local basis for [0,1 (cf. Proposition ~.5). It follows that the set of forms w il 1\ ••• 1\ w ip 1\ Wk1 1\ ••• 1\ {jjk a, 1 0 for every nonzero X E C n ; (3) E(X, Y) is an integer if X, Y E D.
For n = 1, every complex torus is an abelian variety. As a quotient group of cn, cnlD is a connected compact complex Lie group. Conversely, every connected compact complex Lie group G is a complex torus. In fact, the adjoint representation of G is a holomorphic mapping of G into GL(n; C) c C n \ where n = dim G. Since a holomorphic function on a compact complex manifold must be constant, the adjoint representation of G is trivial, that is, G is abelian. Thus G is an evendimensional torus (Pontrjagin [1]) and hence a complex torus.
132
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
If Zl, ••• , zn is the natural coordinate system in cn, then the holomorphic Iforms dz 1 , • • • , dz n can be considered as forms on a complex torus CnjD. Every holomorphic Iform on CnjD is a linear combination of dz 1 , • • • , dz n with constant coefficients. In fact, every holomorphic Iform on CnjD is a linear combination of dzI, ... , dz n with holomorphic functions as coefficients and since CnjD is a compact complex manifold, these coefficient functions are constant functions. Let CmjD' be another complex torus and wI, ... , wm the natural coordinate system in Cm. A homomorphismCnjD + CmjD' is induced by a complex linear transformation of C n into C m which sends D into D'. If f: CnjD + Cm/D' is a holomorphic rnapping, then n
f*(dw k ) =
L a~ dzi ,
;=1
k
=
1, ... , m,
a~ E
C,
showing that f is induced by a mapping C n + C m of the form wk = ~;=1 a;z3 + bk, k = 1, ... , m, with bk E C. Thus, every holomorphic mapping of cn/D into CmjD' is a homomorphism modulo a translation in Cm/D'. Example 2.3. An ndimensional complex manifold M is said to be complex parallelizable if there exist n holomorphic vector fields Zl' ... , Zn which are linearly independent at every point of M. Every complex torus is complex parallelizable. More generally, let G be a complex Lie group of complex dimension n. Taking n linearly independent complex vectors of type (1, 0) at the identity element of G and extending them by left translations, we obtain n left invariant holomorphic vector fields Zl' ... , Zn on G which are linearly independent at every point of G. If D is a discrete subgroup of G, then Zl' ... , Zn induce n holomorphic vector fields on the quotient complex manifold G/D which are linearly independent at every point of G/D, showing that GjD is complex parallelizable. H. C. Wang [5] proved that, conversely, every compact complex parallelizable manifold may be written as a quotient space GjD of a complex Lie group G by a discrete subgroup D. In fact, if Zl' ... , Zn are everywhere linearly independent holomorphic vector fields on a compact complex manifold M, then [Z;, Zk] = ~/7=1 C~kZh' where, being holomorphic functions on a compact complex manifold M, C~k are all constant functions on M. Let
IX.
COMPLEX MANIFOLDS
133
Xl' ... , X n be the corresponding infinitesimal automorphisms of the complex structure of M, i.e., X j = Zj + Z; for J = 1, ... , n (c£ Proposition 2.11). Let G be the universal covering manifold of M and Xi the natural lift of Xj to G. Then, (i) Xi, ... , X: are infinitesimal automorphisms of the complex structure of G; (ii) [X;*, X:J = L~=1 CJk X: for j, k = 1, ... , n; (iii) Xi, ... , X: are complete vector fields on G. We point out that (iii) follows from the fact that all vector fields, in particular Xl' ... , X m on a compact manifold M are complete (cf. Proposition 1.6 of Chapter I). From (i), (ii), (iii), and the simple connectedness of G, it follows that G can be given a complex Lie group structure such that Xi, ... , X: are left invariant infinitesimal automorphisms of the complex structure of G. Let D be the group of covering transformations of G which gives M. Each element of D leaves Xi, ... , X: invariant and hence is a left translation of G. Thus D can be identified with a discrete subgroup of G acting on the left. Example 2.4. The complex Grassmann manifold Gp,q(C) ofpplanes in Cp+q is the set ofpdimensional complex subspaces in Cp+q with the structure ofa complex manifold defined as follows. Let Zl, ••• , zp+q be the natural coordinate system in Cp+q, each Zi being considered as a complex linear mapping Cp+q "'" C. For each set cx = {CXI' ••• , cx p } of integers such that 1 < CX I < ... < CX p P + q, let Ua be the subset of GfI.q(C) consisting of pdimensional subspaces 8 such that za.l 18, ... , zap I 8 are linearly independent. We shall define a mapping CPa of Ua into the space M(q, p; C) of P X q complex matrices, which may be identified with Cpq. Let {cxp +l , ••• , CX P +q } be the complement of {cx H ••• , cx p } in {I, ... , p + q} in the increasing order. Since, for each 8 E Ua , Z al 18, . . . , za.:ol 8 form a basis of the dual space of 8, we may write p
za.P+k 1
8
=
2 s:(za/
I
8),
k
=
1, ... , q.
j=1
We set
CPa(8) (J1) E M(q, p; C). It is easy to see that CPa maps Ua onetoone onto M(q, p; C) and that the family of
(P P q)
charts (U., 'P.) turns G•. q(C) into a
complex manifold of complex dimension pq.
134
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
The group GL(p + q; C) acting in Cp+q sends every pdimensional subspace into a pdimensional subspace and hence can be considered as a transformation group acting on Gp,q(C). The action is holomorphic and transitive. If So denote the pdimensional subspace spanned by the first p elements of the natural basis of Cp+q, then the isotropy subgroup at So is given by '.
H = {(:
:)
E
GL(p
+ q; C) ),
where 0 denotes the zero matrix with p columns and q rows. Thus Gp,q(C) is a quotient space GL(p + q; C)/H of a complex Lie group GL(p + q; C) by a closed complex Lie subgrotlp Hand the natural projection GL(p + q; C) + G'P,q(C) is holomorphic. A similar argument applied to the unitary group U(p + q) acting in Cp+q shows that G'P,q(C) may be written also as a quotient space U(p + q)/U(p) X U(q), where U(p) X U(q)
= {(:
~); A
E
U(p), B
E
U(q)).
Since U(p + q) is compact, G'P,q(C) is also compact. The Grassmann manifold Gn,1 (C) is called an ndimensional complex projective space, denoted by Pn(C). The multiplicative group C* of nonzero complex numbers acts freely on cn+1  {O} by (c, z) E c* X (Cn+1  {O}) ~ cz E Cn+1  {O}. Let zo, z1, , zn be the natural coordinate system in Cn+1. For each j,j = 0, 1, , n, let Ur be the set of points of Cn+1  {O} where Z1 =1= 0 and let U; be the image of under the natural projection Cn+1  {O} + (cn+1  {O})/C*. It is easy to see that considering ZO/Z1, ••• , Z11/Z1, Z1+1/Z1, ... , zn/ Z1 as functions defined on Uj we may identify Pn(C) with (cn+1  {O})/C* whose complex manifold structure is defined by the family of coordinate neighborhoods U; with local coordinate system ZO/Z3, ••• , z11/ z 1, Z1+1/ Z;, ... , zn/ z 1, called the inhomogeneous coordinate system of Pn(C) in Uj" The coordinate system zO, z1, ... , zn of Cn+1 is called a homogeneous coordinate system of Pn(C); homogeneous coordinates of a point of Pn(C) ~ (cn+1  {O})/C* is by definition the coordinates of a point of cn+1  {O} representing it. Thus, homogeneous coordinates are defined up to a nonzero constant factor.
ut
IX.
COMPLEX MANIFOLDS
135
What we have just said may be rephrased more geometrically as follows. Cn+l  {O} is a principal fibre bundle over Pn(C) = (cn+l  {O})/C* with group C*. If we denote by 7T the projection Cn+l  {O} + Pn(C), then local triviality "Pj: 7T1 ( U j) ~ U j X C* is given by "Pj(z) = (7T(Z), Zi) E U j X C* for z
=
(ZO, ••• ,
zn)
The transition functions "Pkj: Uj n Uk
"Pkj
=
E
+
Cn+l  {O}.
C* are given by
Zk/Zi,
where zO, ... , zn is considered as a homogeneous coordinate system in Pn(C). Let S2n+l be the unit sphere in Cn+l defined by Izol2 + ... +Iznl2 = 1 and Sl the multiplicative group of complex numbers of absolute value 1. Then S2n+l is a principal fibre bundle over Pn(C) with group Sl; indeed, it is a subbundle of Cn+l  {O} in a natural manner. If we denote by 7T the projection S2n+l + Pn(C), then local triviality "Pi: 7T1 ( Ui) ~ Ui X Sl is given by
"Pi (z) =
z), Zi II zi I)
Ui X Sl for z = (ZO, ••• , zn) E S2n+l. The transition functions "Pkj: Ui n Uk + Sl are given by (7T (
E
"Pki = Zk Izil/Zi Izkl· It is sometimes necessary to identify the group Sl with the additive group R/Z (the real numbers modulo 1); the isomorphism is given by A = e21ri9 E Sl + (1 /27Ti) log A = () E R/Z. If we consider S2n+l as a principal fibre bundle over Pn(C) with group R/Z, then the transition functions are given by 1 . log "Pki 27TZ
=
1 2 . (log Zk/ Zi log 7TZ
Izkl/lzil).
Example 2.5. Let M be a complex manifold with an open covering {Uj } and G a complex Lie group. Given a family of holomorphic mappings "Pjk: Uj n Uk + G such that "Pz;(x) = "PZk(X)"Pki(X),
x
E
Uj n Uk nUL'
we can construct a principal fibre bundle P over M with group
136
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
G and transition functions 'lfJkj (cf. Proposition 5.2 of Chapter I). From the proof of Proposition 5.2 of Chapter I we see that P has a natural complex manifold structure such that the projection 7T is holomorphie and 7T1 ( U j ) ~ U j X G holomorphically. We shall now apply this construction to the case where . M . = P'P(C) X Pq(C) and G is a Idimensional complex torus CjD. Let zo, ... , z'P and wo, ..• , wq be the homogeneous coordinate systems for P'P(C) and Pq(C), respectively. For each cx, cx = 0, ... ,p (resp. A, A = 0, ... , q), let Ua (resp. V;.) be the open subset of P'P(C) (resp. Pq(C)) defined by za =I 0 (resp. w), =I 0). We take {Ua x V;,} as an open covering of P'P(C) x Pq(C). Let T 1 and T2 generate D. We then define holomorphic mappings 'lfJPI!,a),: (Ua X V;.) n (Up X VI!) + CjD by 1 'lfJPI!,a), = 27Ti (T1 log zP j za + T2 log wl!jw),) mod D.
Then (Ua
on
X
V;J
n
(Up
X
VI!)
n
(lly
X
Vv!.
We denote by M~;~ the bundle over P'P(C) X Pq(C) with group CjD constructed by the transition functions {'lfJpl!,a),}. We shall show that the complex manifold M~;T~ is diffeomorphic with the product S2'P+l X S2q+l of a (2p + 1 )sphere S2'P+l and a (2q + 1)sphere S2q+l. Define a family of mappings CPa),: Ua X V), + CjD by
and set 'lfJPI!,a),
=
CPPI!
+ 'lfJpl!,a),
 CPa),·
The transition functions {'lfJpl!,a),} define a bundle equivalent to M~;~. Define a family of mappings ha: Ua n Up + RjZ by fpa
=
1
2. (log zPfza  log IzPljlzal) 7TZ
and a family of mappings gl!),: V),
n
VI!
+
RjZ by
1 gl!), = 2. (log wU/w),  log IWl!ljlw"'I). 7TZ
IX.
COMPLEX MANIFOLDS
137
The principal bundle over P'P(C) with group RjZ defined by the transition functions {fpr) is a (2p + 1) sphere S2'n+l (cf. ExampIe 2.4). Similarly, the principal bundle over Pq(C) with group RjZ defined by {gJL)) is a (2q + 1) sphere S2 q +l. The mapping (a, b) E R X R + aTl + bT2 E CjD induces a group isomorphism of (RjZ) X (RjZ) onto CjD. Since
"PPJL,a;,
=
1 27Ti {Tl (log zP j za  log IzP Ijl zal)
+
T2
(lOg wJLjw;'  log
IwJLljlw;'I)},
the principal bundle over P'P(C) X Pq(C) with group CjD E (RjZ) X (RjZ) defined by {"PPJL,a;'} is isomorphic with the product S2'P+l X S2q+l of the two bundles S2'P+l (P 'P (C), RfZ)
and
S2q+l
(P q (C), RjZ).
The fact that S2'P+l X Sl admits a complex structure was discovered by Hopf [5J, and M~;~2 is called a Hopf manifold. Calabi and Eckmann [1] later discovered a complex structure on S2'P+l X S2q+l for all p, q > O. (M~;~2 is nothing but CfD.) In Hopf's paper, M~;~2 was described as follows. Let A be a nonzero complex number with IAI =F 1 and Ll;, the cyclic group of linear transformations of C'P+l generated by the transformation
Since Ll;, is a properly discontinuous group acting freely on C'P+l  {O}, (C'P+l  {O})/Ll;, is a complex manifold in a natural way • We shall show that M11,O = (C'P+l  O)jLl;, with A = e21TiT2/Tl • T1T2 For each ex, ex = 0, ... , p, let ha be the mapping from the set U: = {z = (ZO, ••• , z'P) E C'P+l  {O}; za =F O} into Ua x CjD defined by
h.(z) = (17(Z),
2~ 7"1 log z.)
where 7T: (C'P+l  {O}) + P'P(C) is the projection described in Example 2.4 and (lj27Ti)T1 log za defines an element of CjD. From the way M~;~2 was constructed from Ua({ex} X Ua X CjD) (cf. Proposition 5.2 of Chapter I), we see that the family of mappings {h a } defines a holomorphic mapping h of C'P+l  {O} into M~;~2. It is easy to verify that h is a covering projection and induces a
138
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
holomorphic diffeomorphism of (CP+1  0) j il A onto .Ii.
=
M~;~,
where
e2rriT2/Tl.
We now define an action of GL(p + 1; C) X GL(q + 1; C) on Mr2 q • Let A E GL (p + 1; C) and B E GL(q 1; C). The group GL(P + 1; C) (resp. GL(q + 1; C)) acts on Pp(C) (resp. Pq(C)) in a natural way with respect to the homogeneous coordinate system). Let (z, w, t) E Ua X VA X CjD, where z and ware represented by homogeneous coordinates (zo, ... , zP) and (WO, ... , w q ) respectively so that za =F 0 and w A =F O. Let
+
z'
=
Az
with homogeneous coordinates (z'O,
, z'p),
w'
=
Bw
with homogeneous coordinates (w'O,
, w'q).
Assume that (z', w') E Up X VJL , i.e., z'P =F 0 and w'JL =F O. Then we set 1 t' = t + 2. (71 log z'Pjza + 7 2 log w'JLjw A ) mod D. 7T'l
It is straightforward to verify that the point x' of M~;~ represented by (z', w', t') E Ua X VA X CjD depends only on the point x of M~;i2 represented by (z, w, t) E Ua X VA X CjD and on the element (A, B) of the group GL(p + 1; C) X GL(q + 1; C). It is easy to see that the action of GL(p + 1; C) X GL(q + 1; C) on M~;i2 defined by ((A, B), x) + x' is holomorphic and transitive. It should be pointed out that the action is not effective but is fibrepreserving with respect to the fibring M~;i2 + Pp(C) X PQ(C). The class of complex manifolds M~;i2 (p, q > 0) constructed above is contained in the class of Cspaces of H. C. Wang [4] consisting of simply connected, compact homogeneous complex manifolds (see Note 24). (See also Ise [2] for Hopf manifolds.) Example 2.6. Let Sn denote the unit sphere in Rn+1. Kirchhoff [1, 2] has shown that if Sn admits an almost complex structure, then Sn+1 admits an absolute parallelism. On the other hand, Borel and Serre [1] have proved that sn, for n =F 2, 6, does not admit almost complex structures. Later, Adams [1] proved that Sn+1 admits an absolute parallelism only for n 1 = 1, 3, and 7. The result of Adams combined with that of Kirchhoff implies that of Borel and Serre. The proofs of the theorems of Adams and BorelSerre are beyond the scope of this book. We shall prove only the elementary result of Kirchhoff. Let J be an almost complex structure on Sn.
+
IX.
139
COMPLEX MANIFOLDS
We fix Rn+l as a subspace of Rn+2 and a unit vector e of Rn+2 perpendicular to R n+l. We shall construct a field (J of linear frames on Sn+l. Let x E Sn+l. If x =I= e, then we may write uniquely as follows:
x = ae
+ by,
a, b
E
R,
b> 0
and y
E
Sn.
Let Vy be the ndimensional vector subspace ofRn+2 parallel tothe tangent space Ty(sn) in Rn+2 and J y the linear endomorphism of Vy corresponding to the linear endomorphism of Ty(sn) given by J. We define a linear frame (J a:: R n+l ~ Ta: (sn+l) as follows:
(Ja:(Y) = ay  be, (Ja:( z) = az
+ bJy( z)
for z
E
Vy•
(Note that Rn+l is spanned by y and Vy and that both (Ja:(Y) and (Ja:(z) are perpendicular to x and hence can be considered as elements of Ta:(sn+l).) We define (Je to be the identity transformation: Rn+l ~ Rn+l = T e(sn+l). It is easy to see that (J is a continuous field. Since the underlying differentiable manifold of Pl(C) is S2, S2 admits a complex structure. We shall construct an almost complex structure on S6 using Cayley numbers. A Cayley number x = (ql' q2) is an ordered pair of quaternions. The set of Cayley numbers forms an 8dimensional nonassociative algebra over R with the addition and the multiplication defined as follows:
(ql' q2) ± (q~, q;) = (ql ± q~, q2 ± q~) (ql' q2) (q~, q;) = (qlq~  1j~q2' q;ql
+ q21j~),
where 1j denotes the quaternion conjugate of q. Define the conjugate of a Cayley number x = (ql' q2) to be x = (1jl' q2). Then xx = (q/h + 1j2q2, 0) and we set Ixl 2 = ql1jl + q2q2. Evidently Ixl > 0 unless x = O. It can be verified by direct calculation that Ixx'i = Ixllx'i. Therefore xx' = 0 implies x = 0 or x' = o. Although the associative law does not hold, the socalled alternative law is valid: x(xx') = (xx)x', (x'x)x = x' (xx). A Cayley number x = (ql' q2) is real if ql is real and q2 = O. It is called purely imaginary if ql is a purely imaginary quaternion. Let
140
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
U7 be the 7dimensional real vector space formed by the purely imaginary Cayley numbers. We define an inner product ( , ) and a vector product X in U7 as follows:
(x, x') = the real part of xx', X X
x, x'
x' = the purely imaginary part of xx',
E
U7 ,
x, x'
.
E
U7 •
(This generalizes the inner product and the vector product in R3 which are defined by considering R3 as the space of purely imaginary quaternions.) It can be verified that if x, x', x" E U7 , then (i) xx = (x, x) lxI 2, (ii) x X x' = x' X x, (iii) (x X x', x") = (x, x' X x"). (For more details and references on Cayley numbers, see Jacobson [2].) Let 8 6 be the unit sphere in U7 defined by 8 6 = {x E U7 ; Ixl = I}. We identify the tangent space Ta:(8 6 ) with the subspace Va: = {y E U7 ; (x,y) = O} of U7 parallel to it. Define a linear endomorphism Ja: of Vx such that J a: 0 J a: = 1 by Ja:(y) = x X y, Y EVa:' 1=
By (ii) and (iii), y E Va: implies x X y E Va: and hence Ja: is an endomorphism of Va:' It follows also that Ja:(Ja:(Y)) = x X
(x xy) = x(x xy)  (x,x xy) = x(x xy) = x(xy)  x(x,y) = x(xy) = (xx)y = lxl7 = :Y, showing that Ja: Ja: = 1. The family of endomorphisms Ja:' x E 8 6 , defines an almost complex 0
structure on 8 6 • It may be verified by direct calculation that this almost complex structure has nonvanishing torsion (cf. Frolicher [1]). The group of automorphisms of the algebra of Cayley numbers is an exceptional simple Lie group G2 (see Jacobson [1, 2]) and the group G2 acts transitively on the unit sphere 8 6 in U7 , leaving the almost complex structure defined above invariant. It is not known whether 8 6 admits any complex structure. Invariant (almost) complex structures on homogeneous spaces will be discussed later in §6 of Chapter X. Example 2.7. Let M be a 6dimensional orientable manifold immersed in R7. From each almost complex structure J on 8 6 we shall induce an almost complex structure on M. (When J is the almost complex structure on 8 6 defined from Cayley numbers as in Example 2.6, the induced almost complex structure on M coincides with the one constructed by Calabi [5].) Let g: M ~ 8 6 be the spherical map of Gauss (cf. §2 of Chapter VII). The
IX.
141
COMPLEX MANIFOLDS
tangent spaces Tx(M) and Tg(x) (86 ) are parallel in R7 and can be identified in a natural manner. Hence every almost complex structure on 8 6 induces an almost complex structure on M6 in a natural manner. Example 2.8. We show that every almost complex structure J on a 2dimensional orientable manifold M has vanishing torsion. For any vector field X on M, we have
N(X, JX) = 2([JX, XJ  [X, JXJ 0 .
+ J[X, XJ
 J[JX, JXJ)
On the other hand, in a neighborhood of a point where X =1= 0, every vector field Y is a linear combination of X and J X. Hence N = 0, proving our assertion. We show also that every Riemannian metric on a 2dimensional oriented manifold M defines an almost complex structure J in a natural manner. In fact, every Riemannian metric on M defines a reduction to 80(2) of the structure group GL(2; R) of the bundle L(M) of linear frames. Since 80(2) is contained in the real representation of GL(l; C), i.e., 80(2) c GL(l; C) c GL(2; R), our assertion follows from Proposition 2.4.
3. Connections in almost cOTnplex manifolds On each almost complex manifold M, we shall construct the bundle C( M) of complex linear frames and study connections in C(M) and their torsions. Let M be an almost complex manifold of dimension 2n with almost complex structure J and let J o be the canonical complex structure of the vector space R2n defined in §1. Then a complex linear frame at a point x of M is a nonsingular linear mapping u: R2n + Tx(M) such that u 0 J o = J 0 u. In §l we showed that J defines the structure of a complex vector space in Tx(M) and, by Proposition 1.2, u: R2n + Tx(M) is a complex linear frame at x if and only if it is a nonsingular complex linear mapping of Cn = R2n onto Tx(M). The set of complex linear frames forms a principal fibre bundle over M with group GL(n; C) ; it is called the bundle of complex linear frames and is denoted by C( M). The proof of this fact is almost identical with the case of bundles of linear frames (see Example 5.2 of Chapter I), except perhaps local triviality ofC( M). To prove local triviality of C( M), let xl, ... , x 2n
142
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
be a local coordinate system valid in a neighborhood of a point o of M. By a change of ordering if necessary, we may assume that ajax!, ... , oj oxn form a basis for To (M) as a complex vector space and hence for T",,(M) for all x in a neighborhood U of o. Let e1, • • • , en be the natural basis for R2n Cn as a complex vector space. To each complex linear frame u at x, we assign an element (x, a) E U x GL(n; C), where a = (aD is defined by n
u(ek ) =
I
a{( oj oxi ) ,
k
1, . . . , n.
i=1
I t is an easy matter to verify that u (x, a) gives the local triviality ofC(M). We remark here that, although we may obtain an open covering {Ua} of M and transition functions "Pap: Ua n Up GL(n; C) as explained in §5 of Chapter I, these transition functions are in general not almost complex.. Since the bundle C( M) is a subbundle of the bundle L(M) of linear frames, each almost complex structure gives rise to a reduction of the structure group GL(2n; R) of L(M) to GL(n; C). ')00
')00
3.1. Given a 2ndimensional manifold M, there is a natural onetoone correspondence between the almost complex structures and the reductions of the structure group of L( M) to GL(n; C). Proof. Since we have already defined a mapping from the set of almost complex structures of M into the set of reductions of the structure group of L(M) to GL(n; C), we shall just construct its inverse mapping. Let P be a subbundle of L(M) with structure group GL(n; C). For each point x of M, choose a linear frame u E P at x and then transfer the canonical complex structure of R2n onto T",,( M) by the linear transformation u: R2n T",,( M) to obtain a complex structure on the vector space T",,(M). Since any other frame u' E P at x differs from u by right multiplication by an element of GL(n; C), the complex structure defined on T",,( M) is independent of choice of u. QED. PROPOSITION
')00
From Proposition 3.1 above and Proposition 5.6 of Chapter I, we obtain
3.2. Given a 2ndimensional manifold M, there is a natural onetoone correspondence between the almost complex structures of M and the crosssections ofthe associated bundle L(M)jGL(n; C) over M. PROPOSITION
IX.
COMPLEX MANIFOLDS
143
We know that, given a Riemannian manifold M with metric tensor g, a linear connection r of M is a metric connection, i.e., r comes from a connection in the bundle 0 (M) of orthonormal frames if and only if g is parallel with respect to r (cf. Proposition 2.1 of Chapter IV as well as Proposition 1.5 of Chapter III). The proof of the following proposition is analogous to that of Proposition 1.5 of Chapter III and is left to the reader.
3.3. For a linear connection r on an almost complex manifold M, the following conditions are equivalent: (a) r is a connection in the bundle C(M) of complex linear frames)· (b) The almost complex structure J is parallel with respect to r. PROPOSITION
A linear (or affine) connection on M is said to be almost complex if it satisfies anyone (and hence both) of the conditions above. From the general theory of connections (cf. Theorem 2.1 of Chapter II) we know that every almost complex manifold admits an almost complex affine connection (provided it is paracompact) . We now prove the existence of a connection of more special type.
3.4. Every almost complex manifold M admits an almost complex affine connection such that its torsion T is given by THEOREM
N
8T,
=
where N is the torsion of the almost complex structure J of M. Proof. Consider an arbitrary torsionfree affine connection on M with covariant differentiation V', and let Q be the tensor field of type (1, 2) defined by 4Q(X, Y) = (V'JyJ)X
+ J((V'yJ)X) + 2J((V'xJ)Y),
where X and Y are vector fields. Consider an affine connection whose covariant differentiation V' is defined by ~
,..,
V'x Y = V'xY  Q(X, Y).

By Proposition 7.5 of Chapter III, V' is really covariant differentiation of an affine connection. We shall show that this is a desired connection.
144
FOUNDATIONS OF DIFFERENTIAL GEOMETRY ~
To pro~e that the conn:,ction given by V is almost complex, we compare Vx(JY) with J(V xY). Then ~
V x(JY) = V x(JY)  Q(X, JY)
= (V xJ) Y
,..,
+ J(V xY)
 Q(X, JY),
J(VxY) = J(V xY)  J(Q(X, Y)). ~
~
To prove that Vx(JY) = J(V xY), we shall establish the equality Q(X, JY)  J(Q(X, Y)) = (V xJ) Y.
We have 4Q(X, JY) = (VyJ)X + J((V JyJ)X) + 2J((V x J) 4J(Q(X, Y)) = J((VJyJ)X)  (VyJ)X  2(V x J)Y.
On the other hand, from 0 = Vx (.]2) = (V xJ)J obtain 2J( (VxJ)
0
0
JY)
+ J(V xJ), we
JY) = 2J(J 0 (VxJ) Y) = 2(VxJ) Y. ~
This establishes the desired equality, thus showing that V commutes with J, i.e., J is parallel with respect to the connection ,.., given by V. ,.., The torsion T of V is given by (cf. Theorem 5.1 of Chapter III) ~
T(X, Y)
= =
,..,
VxY  VyX  [X, Y] VxY  VyX  [X, Y]  Q(X, Y)
+ Q(Y, X).
Since V has no torsion, we obtain T(X, Y)
=
Q(X, Y)
+ Q(Y, X).
From the defining equation of Q, we obtain 4(Q(Y, X)  Q(X, Y)) = (VJxJ)Y + J((VyJ)X) (V JyJ)X  J((V xJ)Y).
The four terms on the right may be written as follows:
(V JXJ) Y = VJx(JY)  J(V JXY), J((VyJ)X) = J(Vy(JX)) + VyX, (V JyJ)X = VJy(JX)  J(VJyX), J((VxJ)Y) =J(Vx(JY))
+ VxY
IX.
145
COMPLEX MANIFOLDS
Hence, 4(Q(Y, X)  Q(X, Y)) =
(VJx(JY)  VJy(JX))  (VxY  VyX)  J(VJxY  Vy(JX))  J(Vx(JY)  VJyX).
Since V has no torsion, the four terms on the right may be replaced by the ordinary brackets and we have 4(Q(Y, X)  Q(X, Y))
= [JX, JY]  [X, Y]  J[JX, Y]  J[X, JY]
= tN(X, Y). QED.
3.5. An almost complex manifold M admits a torsionfree almost complex affine connection if and only if the almost complex structure has no torsion. Proof. Assume that M admits a torsionfree almost complex affine connection and denote its covariant differentiation by V. If we use this V in the proof of Theorem 3.4, then from VJ = a we obtain Q = 0 and hence N = O. The converse is a special case of Theorem 3.4. QED. COROLLARY
3.6. Let M be an almost complex manifold with almost complex structure J. Then the torsion T and the curvature R of an almost complex affine connection satisfy the following identities: PROPOSITION
(1)
T(JX, JY)  J(T(JX, Y))  J(T(X, JY)) 
T(X, Y)
= tN(X, Y) for any vector fields X and Y, where N is the torsion of J,. (2) R(X, Y)
0
J = J
0
R(X, Y)
for any vector fields X and Y. Proof. This is an immediate consequence of the two formulas T(X, Y) = VxY  VyX  [X, Y]
and
R(X, Y)
=
[V x, V y
]
in Theorem 5.1 of Chapter III and VJ

V[X,y]
=
O.
QED.
146
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
We shall conclude this section by a simple remark on the structure equations of an almost complex affine connection. Let C( M) be the bundle of complex linear frames on an almost complex manifold M, () the canonical form on C(M), i.e., the restriction of the canonical form of L(M) to C(M), and w the connection form of an almost complex affine connection with torsion form 0 and curvature form Q. Taking the natural bases in R2n and gI(2n; R), we write the structure equations as follows (cf. §2 of Chapter III) : 2n
d()i = 
I
w~ A Ok
+ 0 i,
j = 1, ... , 2n,
k=1 2n
dw ki
=  ~ '"
11=1
Wih
A whk
+ Q k' j
j, k
=
1, . . . , 2n.
Since the connection is almost complex, wand Q are gI (n; C)valued on C(M), where gI(n; C) is considered as a subalgebra of gl (2n; R) as explained in §1. If we therefore set cpa = Oa + iOn+a, a = 0 a + i0 n+a, cx. = 1, ... , n, '
'ljJp = wp + iw~+p,
'Y p
=
Qp
+ iQ~+p,
cx., ~ = 1, . . . , n,
then cp = (cpa) and = (a) are Cnvalued and 'ljJ = ('ljJp) and 'Y = ('Y p) are gl(n; C)valued, where gI(n: C) is now considered as the Lie algebra of n X n complex matrices. The structure equations on C(M) may be now written as follows: 11
dcpa = 
I
'ljJp A cpP
+ a,
'ljJ~ A 'ljJ~
+ 'Yp,
cx. = 1, ... , n,
P=1
n
d'ljJp = 
I
y=1
cx., ~
= 1, ! ... , n.
4. Hertnitian metrics and J(aehler metrics A Hermitian metric on an almost complex manifold M is a Riemannian metric g invariant by the almost complex structure J, i.e., g(JX, JY) = g(X, Y)
for any vector fields X and Y.
A Hermitian metric thus defines a Hermitian inner product (cf. §l) on each tangent space Tz(M) with respect to the complex structure defined by J. An almost complex manifold (resp. a
IX.
147
COMPLEX MANIFOLDS
complex manifold) with a Hermitian metric is called an almost Hermitian manifold (resp. a Hermitian manifold).
4.1. Every almost complex manifold admits a Hermitian metric provided it is paracompact. Proof. Given an almost complex manifold M, take any Riemannian metric g (which exists provided M is paracompact; see Example 5.7 of Chapter I). We obtain a Hermitian metric h by setting PROPOSITION
h(X, Y) = g(X, Y)
+ g(JX, JY)
for any vector fields X and y.
QED. Remark. By Proposition 1.10, every Hermitian metric g on an almost complex manifold M can be extended uniquely to a complex symmetric tensor field of covariant degree 2, also denoted by g, such that (1) g(Z, W) = g( Z, W) for any complex vector fields Z and W; (2) g( Z, Z) > 0 for any nonzero complex vector Z; (3) g( Z, W) = 0 for any vector field Z of type (1, 0) and any vector field W of type (0, 1). Conversely, every complex symmetric tensor field g with the properties (1), (2), and (3) is the natural extension of a Hermitian metric on M. In §l we associated to each Hermitian inner product on a vector space V a skewsymmetric bilinear form on V. Applying the same construction to a Hermitian metric of an almost complex manifold M we obtain a 2form on M. More explicitly, the fundamental 2form of an almost Hermitian manifold M with almost complex structure J and metric g is defined by (X, Y)
=
g(X, JY)
for all vector fields X and Y.
Since g is invariant by J, so is , i.e., (JX, JY) = (X, Y)
for all vector fields X and Y.
The almost complex structure J is not, in general, parallel with respect to the Riemannian connection defined by the Hermitian metric g. Indeed we have the following formula.
148
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Let M be an almost Hermitian manifold with almost complex structure J and metric g. Let 0, where I p is the p X P identity matrix and the symbol ">0" means "positive definite". Then Dp,q is a
162
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
bounded domain in Cpq. Let U(q, p) be the group of (q (q + p) matrices
(A: q
X
q matrix; D:
p
+ p)
X
P matrix)
X
such that
( tA tc )( l
q
tB
a
tD
0) (A B) l G D
=
(
l
p
If we set
W
=
(AZ
q
0
a). lp
+ B)(GZ + D)I,
then which shows that W is in Dp,q if Z is in Dp,q' It is a simple exercise to show that the group U(q, p) thus acts transitively on Dp,q' We set = 4i log det(lp  tZZ).
aa
From the preceding equation it follows that is invariant by the action of U(q, p). In the same way as in Example 6.4 we see that gives rise to a Kaehler metric g which coincides at Z = 0 with 4 trace (dtZ dZ). In particular, D1,n is the interior of the unit ball in Cn and the metric g may be written in terms of the coordinate system Z1, ••• , zn of Cn as follows:
Example 6.6. We shall sketch briefly how the construction of the metric in Example 6.5 may be generalized to the case of an arbitrary bounded domain in cn. Let M be an ndimensional complex manifold and H the space of holomorphic nforms cp (i.e., forms of degree (n, 0) such that acp = 0) such that
L
in'cp
II
if
0 everywhere, we set ds 2 = 2 I grxP dz a dz P with grxP = 02 log kjoza ozp. Then ds 2 is in general positive semidefinite. For any bounded domain in en, K =I= 0 everywhere and ds 2 is positive definite. Since every holomorphic transformation of M induces a linear transformation of H preserving the inner product (i.e., a unitary transformation of H), it preserves K and ds 2 • The Kaehler metric of a bounded domain in en thus constructed is called the Bergman metric of M. Let i n2 Gdz l A ••. A dz n A dz 1 A ••• A dz n be the volume element of M with respect to the Kaehler metric constructed above (where G is given by G = det (grxp); cf. Appendix 6). Since both K and the volume element are forms of degree (n, n) which are invariant by the group ofholomorphic (and hence isometric) transformations, they coincide up to a constant multiple if M is homogeneous, i.e., if the group of holomorphic transformations is transitive on M. It follows from (24) of §5 that K rxp = grxP, i.e., S = g (where S denotes the Ricci tensor and g denotes the metric tensor) if M is homogeneous. For the domain D p ,q in Example 6.5, the function k (with respect to the natural coordinate system) is given by c{det (lp  tZZ) }(p+q), where
c=
1 ! 2 ! . . . (p + q  1) ! 1 ! 2! ... (p  1) ! 1 ! 2! ... (q  l)!
7T PQ
.
For the detail of the construction of Bergman metrics, we refer
164
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
the reader to Bergman [1], Helgason [2; pp. 293300], Kobayashi [14] and Weil [2; pp. 5565]. For the domain Dp,q, see for instance Hua [1; p. 83]. Example 6.7. Let M be a complex analytic submanifold of a Kaehler manifold M', i.e., the immersion M + M' is holomorphic. Then the Riemannian metric induced on M is Hermitian. It is easy to see that the fundamental 2form associated with this Hermitian metric is the restriction of the fundamental 2form of M' and hence is closed. This shows that every complex analytic submanifold M of a Kaehler manifold is also a Kaehler manifold with the induced metric. In particular, every closed complex analytic submanifold of the complex projective space PN(C) is a Kaehler manifold. It is a celebrated theorem of Chow that such a manifold is always algebraic, i.e., is given as the zeros of homogeneous polynomials in the homogeneous coordinate system zo, z1, ••• , ZN of PN( C). (See Chow [1] and also the anonymous letter in Amer. ]. Math. 78 (1956), p. 898 for a simpler proof in the nonsingular case.) Example 6.8. Let M be a Idimensional complex manifold. Then every Hermitian metric is a Kaehler metric since every 2form and, in particular, the fundamental 2form on a manifold of real dimension 2 is closed. From Theorem 2.5 and Example 2.8 we see that every Riemannian metric on an oriented 2dimensional manifold is a Kaehler metric with respect to the naturally induced complex structure. Example 6.9. We shall now give examples of complex manifolds which do not admit any Kaehler metric. For this purpose we shall first show that the evendimensional Betti numbers of a compact Kaehler manifold M are all positive. If is the fundamental 2form, then k = A . . . A (k times) is a closed 2kform and defines an element of the 2kth cohomology group H2k(M; R) in the de Rham theory. Since n, n = dim M, coincides with the volume element up to a nonzero constant multiple, its integral over M is different from zero and hence it gives a nonzero element of H2n (M; R). It follows that each k, 1 < k < n, defines a nonzero element of H2k(M; R), thus proving our assertion. The complex manifold M~;i2 defined in Example 2.5 is diffeomorphic with S2P+l X S2q+l but cannot carry any Kaehler metric except for p = q = O.
IX.
165
COMPLEX MANIFOLDS
Example 6.10. Let M be an arbitrary manifold of dimension nand T*(M) the space of covectors of M. We define a Iform w on T*(M) as follows. If X is a tangent vector of T*(M) at ~ E T*(M), then we set w(X) = 2, deprived of the origin. The largest group of affine transformations of M is easily seen to be GL(n; R). Let K be the identity component of GL(n; R) and H the isotropy subgroup of K at a point, say (1, 0, ... , 0) E M. A simple calculation using matrices shows that K/H is not reductive. But many a homogeneous space is reductive. In any of the following cases, a homogeneous space K/H is reductive: (a) H is compact; (b) H is connected and I) is reductive in f in the sense that ad (h) in f is completely reducible. This is the case if H is connected and semisimple; (c) H is a discrete subgroup of K. To prove that K/H is reductive in case (a), let ( , )' be an arbitrary inner product on f. Define a new inner product ( , ) on f by
(X, Y) = t(ad h(X), ad h(Y))' dh, where dh denotes the Haar measure on H. The new inner product is invariant by ad (H). If we denote by m the orthogonal complement of I) with respect to the inner product ( , ), then we have an ad (H)invariant decomposition f = I) + m. Most of the results in this section are due to Nomizu [2]. (We note that the canonical connection in Proposition 2.4 was called the canonical connection of the second kind, and the natural
200
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
torsionfree connection in Theorem 2.10 was called the canonical connection of the first kind.) Theorem 2.8 is a technical improvement of Kobayashi [3]. Invariant connections on reductive v homogeneous spaces have been studied independently by Rasevskii [1], [2], Kurita [1], and Vinberg [1], [2].
3. Invariant indefinite Riemannian metrics Let M KfBbe a homogeneous space, where K is a connected Lie group and, unless otherwise stated, acts effectively on M. We shall often identify the tangent space To(M) at the origin 0 with the quotient space ffI) in a natural manner. If KfB is reductive with an ad (H)invariant decomposition f = I) + m, then both To(M) and f/I) will be identified with m also in a natural manner. Since, for each h E H, ad h: f + f maps the subalgebra I) into itself, it induces a linear transformation of f/I) which will be also denoted by ad h. As in the preceding sections, we shall identify each element X f with the vector field on M induced by X. To express some of the basic properties of an invariant metric on KfH in the Lie algebraic language, we first prove
3.1. There is a natural onetoone correspondence between the Kinvariant indefinite Riemannian metrics g on M = KIH and the ad (H)invariant nondegenerate symmetric bilinear forms B on f/I). The correspondence is given by PROPOSITION
B(X, Y)
=
g(X, Y)o
for X, Y
E
f,
where X and Yare the elements of f/I) represented by X and Y respectively. A form B is positive definite if and only if the corresponding metric g is positive definite. Proof. We shall only show how to construct g from Band shall leave the rest of the proof to the reader. Any point of M is of the formf(o) for somefE K and any vector atf(o) EM is of the formf(Xo) for some X E f. It is straightforward to verify that the following equality defines a Kinvariant metric g on M:
g(f(Xo), f(Yo)) = B(X, Y) COROLLARY
3.2.
If
M
invariant decomposition f = I)
=
+
for X, Y
f.
QED.
KfH is reductive with an ad (H)m, then there is a natural onetoone
X.
201
HOMOGENEOUS SPACES
correspondence between the Kinvariant indefinite Riemannian metrics g on M = K/H and the ad (H) invariant nondegenerate symmetric bilinear forms B on m. The correspondence is given by B(X, Y)
=
g(X, Y)o
for
x,
Y
E
m.
The invariance of B by ad (H) implies
B([Z, XJ, Y)
+ B(X,
for X, Y
E
m
[Z, YJ) and
=
Z
E
0 I),
and the converse holds if H is connected. Remark. Proposition 3.1 is a special case of a natural onetoone correspondence between the Kinvariant tensor fields L of type (r, s) on KjH and the ad (H)invariant tensors of type (r, s) on f/I). The same remark applies to Corollary 3.2. 3.3. Let M = K/ H be a reductive homogeneous space with an ad (H) invariant decomposition f = I) + m and an ad (H)invariant nondegenerate symmetric bilinear form B on m. Let g be the KTHEOREM
invariant metric corresponding to B. Then (1) The Riemannian connection for g is given by Am(X) Y = (1/2) [X, YJm
+ U(X,
where U(X, Y) is the ,symmetric bilinear mapping defined by 2B(U(X, Y), Z)
=
B(X, [Z, YJm)
Y),
of m
X
minto m
+ B([Z, XJm,
Y)
for all X, Y, Z E m. (2) The Riemannian connection for g coincides with the natural torsionfree connection if and only if B satisfies B(X, [Z, YJm)
+ B([Z, XJm,
Y) = 0
for X, Y, Z
E
m.
Proof. Identifying m and To(M) we have Am(X) = (Ax)o by Corollary 2.2. Since Ax is skewsymmetric with respect to g (cf. Proposition 3.2 of Chapter VI), Am(X) is skewsymmetric with respect to B, that is, B(Am(X) Y, Z) + B(Y, Am(X)Z) = 0 for Y, Z E m. We have also Am(X)Y  Am(Y)X = [X, YJm by Proposition 2.3. If we set
U(X, Y) = Am(X) Y  (1/2) [X, YJm,
202
FOUNDATIONS OF DIFFERENTIAL GEOMETRY I
then U( X, Y) is symmetric in X and Y and satisfies
+ B( Y,
B( U(X, Y), Z) =
U(X, Z)) (1/2){B([Y, X]m, Z)
+ B(Y,
[Z, X]m)}.
From this and from the two identities resulting by cyclic permutations of X, Y, Z, we obtain by using symmetry of U
2B( U(X, Y), Z) = B(X, [Z, Y]m)
+ B([Z, X]no
Y)
for X, Y, Z E m, proving the first part. The second part follows immediately. QED. Theorem 3.3 was proved by Nomizu [2], where the formula (13.1), p. 51, has an error of the sign (as was pointed out by H. Wu). A homogeneous space M = KIH with a Kinvariant indefinite Riemannian metric g is said to be naturally reductive if it admits an ad (H)invariant decomposition f = h + m satisfying the condition
B(X, [Z, Y]m)
+ B([Z, X]m,
for X, Y, Z Em
Y) = 0
in Theorem 3.3. 3.4. Let M = Kill be a naturally reductive homogeneous space with an ad (H) invariant decomposition f = 1) + m and a Kinvariant indefinite Riemannian metric g. Let B be the bilinear form on m which corresponds to g. Then the curvature tensor R of the Riemannian connection satisfies PROPOSITION
g(R(X, Y) Y, X)o = lB([X, Y]no [X, Y]m)  B([[X, YJr" Y], X) for X, YEm. Proof.
From Proposition 2.3 and Theorem 2.10 we obtain
(R(X, Y)Z)o =
leX,
[Y, Z]m]m 
ley,
[X, Z]m]m
 i[[X, Y]no Z]m  [[X, Y]I)' Z] for X, Y, Z
E
m.
From this formula and Theorem 3.3 we obtain Proposition 3.4 easily. QED. The following theorem furnishes a very simple case where Theorem 3.3 and Proposition 3.4 may be applied.
x.
HOMOGENEOUS SPACES
203
3.5. Let K/H be a homogeneous space. Assume that the Lie algebra f of K admits an ad (K) invariant nondegenerate symmetric bilinear form B such that its restriction Bf) to 1) is nondegenerate. Then (1) The decomposition f = 1) + m defined by THEOREM
= {X E f,. B (X, Y) = 0
for all Y E 1)} is ad (H)invariant and the restriction B m ofB to m is also nondegenerate and ad (H) invariant,. (2) The homogeneous space KjH is naturally reductive with respect to the decomposition f = 1) + m defined above and the Kinvariant metric g defined by B m,. (3) The curvature tensor R defined by the metric g satisfie~ m
g(R(X, Y) Y, Xh = iBm([X, Y]m, [X, Y]m) + Bf)([X, Yh, [X, Yh) for X, Y Em. Proof. The proof of (1) is trivial. To prove (2), let X, Y, Z E m. Since B is ad (K)invariant, we have B([Z, X], Y) + B(X, [Z, Y]) = o. Since 1) and m are perpendicular with respect to B, we obtain B([Z, X]m, Y)
+ B(X, [Z,
thus proving (2). To prove (3), let X, Y
Bm([[X, Yh, Y], X)
Y]m) = 0, E
m. Then
B([[X, Yh, Y], X) =  B ([X, Y] f)' [X, Y]) = B( [X, Y]f)' [X, Y]f)) = Bf)( [X, Y]f)' [X, Y]f)). =
Now (3) follows from Proposition 3.4.
QED.
3.6. Let KjH be a homogeneous space such that the Lie algebra f of K admits an ad (K) invariant positive definite symmetric bilinear form B. Then KjH is naturally reductive with respect to the decomposition f = 1) + m and the Kinvariant Riemannian metric g defined in Theorem 3.5. The sectional curvature of g is nonnegative. Remark. Samelson [2] gave a more direct proof of the last assertion of Corollary 3.6. Example 3.1. As a special case of Corollary 3.6 we take a compact Lie group K, considered as a homogeneous space Kj{e}. For an ad (K)invariant positive definite symmetric bilinear form B on COROLLARY
204
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
the Lie algebra f, we get a left invariant Riemannian metric g on K, which is also right invariant. The curvature tensor R is given by
R(X, Y)
=
i ad
([X, Y]).
In fact, this Riemannian connection coincides with the (O)connection of K in Proposition 2.12. Example 3.2. The KillingCartan form cp on the Lie algebra f of a Lie group K is negative semidefinite if K is compact (cf. Appendix 9). shall prove that, conversely, if cp is negative definite, then K is compact (a theorem of Weyl). Starting from B = cp, which is an ad (K)invariant positive definite symmetric bilinear form on f, we obtain a biinvariant Riemannian metric g. on K in the manner of Example 3.1. Since R(X, Y) = (  i) ad ([X, Y]), the Ricci tensor S is given by
We
S(X, Y) = trace of {Z ~ R(Z, X) Y} =
trace of {Z ~ 1[[Z, X], X]}
=
1 trace ad (X)
ad (Y)
=
lcp(X, Y)
= i g(X, Y). Thus the metric g is Einstein, and Theorem 3.4 of Chapter VIII implies that K is compact. The theorem of Weyl implies the following. If K is a compact semisimple Lie group (so that cp is negative definite; cf. Appendix 9), then the universal covering group K is also compact and the fundamental group 7T1 ( K) is finite.
4. Holonomy groups of invariant connections For the sake of convenience we shall restate the theorem of Wang on the holonomy algebra of an invariant connection specialized to the case of an invariant affine connection (cf. Theorem 11.8 of Chapter II). 4.1. Let P be a Kinvariant Gstructure on M = KjH and A: f ~ 9 a linear mapping defining a Kinvariant connection in P as in Theorem 1.2. Then the Lie algebra of the holonomy group 'Y(uo) of the invariant connection defined by A is given by THEOREM
mo
+ [A(f),
mo]
+ [A(f),
[A(f), mo]]
+ ... ,
x.
205
HOMOGENEOUS SPACES
where m o is the subspace of 9 spanned by {[A(X), A(Y)]  A([X, Y]),. X, Y
E
f}.
In the reductive case, we obtain 4.2. In Theorem 4.1, assume further that K/H is reductive with an ad (H) invariant decomposition f = 1) + m and let Am: m ~ 9 be the restriction of A to m. Then the Lie algebra of the holonomy group 'Y (u o) is given by COROLLARY
mo
+ [Am ( m),
mo]
+ [Am ( m), [Am ( m),
mo]]
+ . . .,
where mo is the subspace of 9 spanned by
Proof. We shall first show that m o in Corollary 4.2 coincides with mo in Theorem 4.1. Let X, Y E f and decompose them according to the decomposition f = 1) + m:
X = XI}
+ Xm,
Y = YI}
+ Ym.
Using the property of Am stated in Theorem 2.1 and the fact that A is a homomorphism of 1) into 9, we obtain
[A(X), A(Y)]  A([X, Y])
+ Am(Xm), A(YI}) + Am(Ym)] A( [XI}' YI}] + [Xm, Ym]I}) Am([XI}, Ym] + [Xm, YI}] + [Xm, Ym]m)
= [A(XI}) 

= [Am(Xm), Am(Ym)]  Am([Xm, Ym]m)  A([Xm, Ym]g), thus proving our assertion. Let X, Y E m and set
A(X, Y) = [Am(X), Am(Y)]  Am([X, Y]m)  A([X, Y]I}).
+ Zm E f. By a simple calculation, we obtain [A(Z), A(X, Y)] = A([Zg, X], Y) + A(X, [ZtP Y]) + [Am(Zm), A(X, Y)],
Let Z = ZI}
thus proving
206
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Hence we obtain also
[A(Z), [Am(m), mo]] C [A(Zl)) , [A m(m), mo]] + [Am(Zm), [A m(m), mo]] C [[A(Zl))' Am(m)], mo] + [Am(m), [A(Z{)), mo]] + [Am(Zm), [Am(m), mo]] C [Am([Zl)' m]), mo]+ [Am(m), mo + [Am(m), mo]] + [Am(Zm), [Am(m), mo]] c [Am(m), mo] + [Am(m), [Am(m), mo]], thus proving
[A(f), [A m(m), mo]]
c
[Am(m), mo]
+ [Am(m),
[A m( m), mo]].
Continuing this way, we obtain finally
mo] + [A(f), [A(f), mo]] + ... c mo + [Am(m), mo] + [Am(m), [Am(m), mo]] + .... Since the reversed inclusion holds trivially, Corollary 4.2 follows mo
+ [A(f),
now from Theorem 4.1. Setting Am
=
QED.
0 in Corollary 4.2, we obtain (Nomizu [2])
4.3. In Theorem 4.1, assume that KjH is reductive with an ad (H)invariant decomposition f = ~ + m. Then the Lie algebra of the holonomy group 'F (u o) of the canonical connection is spanned by {A([X, Y]l))" X, Y Em}. COROLLARY
We remark here that {[X, Y]l); X, Y E m} spans an ideal of ~ and, consequently, the restricted linear holonomy group of the canonical connection is a normal subgroup of the linear isotropy group H = 1(H). As we have already remarked (cf. Remark after Corollary 2.5), the canonical connection is related to the invariant connection in K(K/H, H) defined in Theorem 11.1 of Chapter II. Corollary 4.3 corresponds to the statement (4) in Theorem 11.1 there. In spite of the fact that, for the natural torsionfree connection, Am is explicitly written by means of the bracket, Corollary 4.2 does not give a particularly simple expression for the holonomy algebra in this case.
X.
HOMOGENEOUS SPACES
207
The following reformulation of Theorem 4.1 and Corollary 4.2 is sometimes more useful for applications. 4.4 In Theorem 4.1, the Lie algebra of the holonomy group \f(uo) is equal to the smallest subalgebra 1)* of 9 such that (1) R(X, Y)o E 1)* for all X, Y E f and (2) [A(X), 1)*] E h* for all X E f. As usual To(M) is identified with Rn and hence R(X, Y)o above really means U;;l (R(X, Y)o) Uo' Proof. By Theorem 4.1, the Lie algebra of'Y(uo) is equal to the smallest subalgebra 1)* of 9 such that (1) mo c 1)*, where mo is defined in Theorem 4.1 and (2) [A(X), 1)*] C 1)* for all X E fOn the other hand, mo is generated by R (X, Y) 0' X, Y E f, by Proposition 1.4. QED. THEOREM
0
0
The same reasoning using Corollary 4.2 and Proposition 2.3 yields 4.5. In Corollary 4.2, the Lie algebra of the holonomy group 'Y(uo) is equal to the smallest subalgebra 1)* of 9 such that (1) R(X, Y)o E 1)* for all X, Y E m and (2) [Am(X), 1)*] C 1)* for all COROLLARY
XEm.
In Nomizu [3], Corollary 4.5 was derived more directly from a result of Nijenhuis (cf. Theorem 9.2 of Chapter III). For an invariant connection on a homogeneous space, we shall sharpen results in §4 of Chapter VI. Let P be a Kinvariant Gstructure on a homogeneous space K/H and A: f ~ 9 a linear mapping defining a Kinvariant connection in P (cf. Theorem 1.2). We set
at = the subalgebra of 9 generated by {A(X); X E fl. By Corollary 1.3, at may be considered also as the Lie algebra of linear transformations of To(M) generated by {(Ax)o; X E f}, where Ax = Lx  V x' Originally, at was introduced as such in the Riemannian case by Kostant [1], [2], and has been used by Lichnerowicz [3] and Wang [1] under more general circumstances. The basic properties of at are given by PROPOSITION 4.6. Let P be a Kinvariant Gstructure on M = K/H and A: f ~ 9 a linear mapping defining a Kinvariant connection in P. Let 1)* be the Lie algebra of the holonomy group \f(uo)' Then (1) 1)* C at C ng (1)*),
208
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
where n g (1)*) is the normalizer
ng ( 1)*) = {A
of 1)* E
in g, i.eo,
g.; [A, 1)*]
C
1)*};
(2) A(1») C Of Proof. (1) By Theorem 4.1, Of clearly contains 1)*. Also by Theorem 4.1, [A(X), 1)*] C 1)* and hence [Or, 1)*] C 1)*. (2) Since A(X) = A(X) for X E 1) by definition, Of contains A(1») = {A(X); X E 1)}. QED. 0
We shall give several cases in which the holonomy algebra 1)* coincides with Of. We shall say that a Kinvariant connection in P is normal if 1) * = Or. Let M = KIH be a homogeneous space with a Kinvariant Riemannian metric. Then its Riemannian connection is normal in any of the following cases: (a) M is compact; (b) M does not admit a nonzero parallel 2form. THEOREM
4.7.
Proof. As we have shown in the proof of Theorem 4.5 of Chapter VI, every infinitesimal isometry X of M gives rise to a parallel tensor field B x of type (1, 1) which is skewsymmetric with respect to the Riemannian metric and the vanishing of B x implies (Ax)o E 1)*. If (a) is satisfied, then Of C 1)* by Theorem 4.5 of Chapter VI. Let f3 x be the 2form corresponding to B x by the duality defined by the metric tensor g, i.e.,
f3 x( Y, Z)
=
g(BxY, Z)
for all vector fields Y, Z on M.
If (b) is satisfied, then f3 x = 0 for all X all X E f. It follows that Of C 1)*.
E
f and hence B x = 0 for QED.
The proof of Theorem 4.5 of Chapter VI actually yields the following slightly more general result: Remark.
(a')
If M
KIH is a compact homogeneous space with a Kinvariant indefinite Riemannian metric, then its Riemannian connection is normal; (b') For a homogeneous space M = KIH with a Kinvariant indefinite Riemannian metric, every Kinvariant metric connection (i.e., every Kinvariant connection for which the metric tensor is parallel) which does not admit a nonzero parallel 2form is normal. =
Theorem 4.7 is due to Kostant [1]. As we see from Proposition 4.6, if 1)* = ng (1)*) then 1)* = Of.
X.
HOMOGENEOUS SPACES
209
The following theorem gives a geometric interpretation to the notion of normal connection. 4.8. Let P be a Kinvariant Gstructure on a homogeneous space M = K/H. Fixing a Kinvariant connection in P, let P(uo) be the holonomy bundle through aframe Uo E P. Then the connection is normal if and only if every element of K maps P(uo) into itself. Proof. Let OJ be the connection form for the given Kinvariant connection. For each f E K, the induced transformation f of P maps every horizontal curve into a horizontal curve. It follows thatf maps P(uo) into itself if and only if f(u o ) E P(uo). Since K is connected by assumption, f(u o ) E P(uo) for allf E K ifand only if g is tangent to P(uo) at Uo for all X E f. (Here, g denotes as before the natural lift of X to P.) Since the horizontal component of g at Uo is always tangent to P(u o), X is tangent to P(u o ) at U o if and only if its vertical component is. But the latter holds if and only if OJ(X)uo is in the Lie algebra 1)* of the holonomy group '¥(u o). Hence, jmaps P(uo) into itself for allf E K if and only if OJ(X)uo E 1)* for all X E f. On the other hand, Corollary 1.3 implies that OJ(X)u o E 1)* if and only if X E Uf. QED. THEOREM
~
We have already seen in Chapter II that, by virtue of the reduction theorem (cf. Theorem 7.1 of Chapter II), for certain types of problems concerning a connection in a principal bundle P we can assume that P is the holonomy bundle. If an automorphism group K of P is involved, such a simplification is not in general available unless K maps the holonomy bundle into itself. Theorem 4.8 means that if an invariant connection on a homogeneous space is normal, then the reduction theorem can be still used advantageously. The proof of the following corollary will illustrate the point.
4.9. Let P be a Kinvariant Gstructure on a homogeneous space M = K/H. If a Kinvariant connection in P is normal, then every parallel tensor field on M is invariant by K. Proof. By Theorem 4.8 we may assume that P itself is the holonomy bundle. Let S be a parallel tensor field of type (r, s) on M and S the corresponding mapping of P into the tensor space • T rs' I.e., S(u) = u 1 (S7T(u») ' COROLLARY
210
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
where 7T: P
~
M is the projection and u1 denotes the isomorphism T~(7T(U)) ~ T; induced by u1 : Ta:(M) ~ Rn. (More intuitively, S(u) may be considered as the components of S with respect to the frame u.) From the definition of covariant differentiation given in §1 of Chapter III, the assumption that S is parallel means that S is a constant map of P into T~. Hence, S is invariant by for all f E K. This means that S is invariant by K. QED.
J
Remark. Corollary 4.9 has been proved by Lichnerowicz [3] along the following line of argument. For each x EM = KIH, let Uf(X) be the Lie algebra of linear transformations of Ta:(M) generated by {(Ax)a:; X E fl. Then a tensor field Son M satisfying any two of the following three conditions satisfies necessarily the third: (a) S is invariant by K; (b) S is parallel; (c) Sa: is invariant by the linear transformation group generated by Of (X) for all x E M. (This fact easily follows from Ax = Lx  V x without the assumption that the given invariant connection is normal.) If the connection is normal, then (b) implies (c) and hence (a) also. From Theorem 4.7 and Corollary 4.9 it follows that on a compact homogeneous Riemannian manifold M every parallel tensor field S on M is invariant by the largest connected group of isometries of M. This is a special case of the result of Wang proved in Volume I (cf. Theorem 4.6 of Chapter VI) and was originally obtained by Kostant [2].
5. The de Rham decomposition and irreducibility Let M be a simply connected complete Riemannian manifold. Then (cf. Theorem 6.2 of Chapter IV), M is isometric to the direct product M o X M 1 X ••• X M r , where M o is a Euclidean space (possibly of dimension 0) and M 1 , ••• , M r are all simply connected, complete, irreducible Riemannian manifolds, and such a decomposition is unique up to an order. By Theorem 3.5 of Chapter VI, the largest connected group IO(M) of isometries of M is naturally isomorphic to the direct product of the largest
X.
211
HOMOGENEOUS SPACES
connected groups ]o( M i ) of isometries of the factors M i :
]O(M)
~
]O(Mo) X ]O(M1 ) X ... X ]O(Mr ).
From this, it is evident that M is a homogeneous Riemannian manifold if and only if all factors M o, M 1, • • • , M r are. The following theorem (due to Nomizu [3]) gives a more precise result. 5.1. Let M = K/H be a simply connected homogeneous space with an invariant Riemannian metric. Then there exist connected closed subgroups K o, K 1, ... , Kr of K, all containing H, such that THEOREM
K/H = Ko/H
X
K1/H
X ••• X
Kr/H
is the de Rham decomposition, where each factor K i / H is provided with an invariant Riemannian metric. (K i may not be effective on KdH.) Proof. We first remark that, being homogeneous, M is complete (cf. Theorem 4.5 of Chapter IV) and hence the de Rham decomposition theorem stated above may be applied to M. Let M = M o X M 1 X ••• X M r be the de Rham decomposition of M. We may identify each M i with a totally geodesic submanifold M i of M through the origin 0 of M = K /H (cf. §6 of Chapter IV). Since K is assumed to be connected, it is contained in ]O(M). For each fixed i, set that is, K~ =
K
n ]O(Mi ).
Since, for each x E M i , there exists an element f E K such that f(o) = x and since such an elementf is necessarily in ]O(Mi ), K i is transitive on M i and the isotropy subgroup of K i at 0 coincides with H. Hence, M i = KdH. QED. Although Theorem 5.1 holds for all simply connected, homogeneous Riemannian manifolds, it is more desirable to have, even under a stronger assumption, the decomposition of the following type: with
K = Ko
X
K1
X ••• X
Kr,
H = Ho
X
HI
X ••• X
Hr.
The following theorem ofKostant [2] gives results in this direction.
212
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
5.2. Let M = K/H be a simply connected naturally reductive homogeneous space with an ad (H) invariant decomposition I = 1) + m and a Kinvariant Riemannian metric g. Let THEOREM
T~O)
To(M) =
+ T~l) + ... + T~r)
be the de Rham decomposition of the tangent space To(M) and 
m = mo
+ m1 + . .. + m
r
the corresponding decomposition of m under the natural identification To(M) = m. If we set
I(m) Ii = mi
=
+ [mi, mi],
m
+ Em, m],
1)i = Ii n 1)
for i = 0, 1, . . . , r,
then I(m), 10' II, ... , Ir are ideals qf I and
I( m) = 10 and
1) = 1)0 Proof.
+ II +
0
+ 1)1 +
0
0
+t
0
0
0
r
+ 1)r
(direct sum of Lie algebras) (direct sum of Lie algebras).
We first prove the following general fact.
1. Let T x (M) = ~r=o T~i) be the de Rham decomposition of the tangent space Tx(M) at x of a simply connected Riemannian manifold M. Let N be the normalizer ofthe (restricted) linear holonomy group o/(x) (it consists of linear transformations s of T x (M) su~h that s l o/(x)s c o/(x)). Then the subspaces T~) are all invariant by the identity component N° of N. Proof of Lemma 1. Let sEN. Since for each a E 0/ (x) there exists an element a' E o/(x) such that LEMMA
as( T~))
=
sa' (T~))
=
s( T~i))
for i
0, 1, ... , r,
=
each s( T~)) is invariant by o/(x). On the other hand, Tx(M) ~r=o s( T~)). From the uniqueness of the de Rham decomposition (cf. Theorem 5.4 of Chapter IV), it follows that each s permutes CO) ' TCI) ThoIS gIves rIse to a h omox,... , TCr) x. t h e su b spaces T x morphism of N into the group of permutations of the set {O, 1, ... , r}. Evidently, the kernel of this homomorphism contains the identity component N°. This completes the proof of Lemma 1. 0
0
x. LEMMA
213
HOMOGENEOUS SPACES
The following relations hold:
2.
(1) af(m i) c mi, (2) [mi, mi] c mi (3) [mi,m j ] =0
[1), mi]
for i = 0, 1, ... , r; + 1) for i = 0, 1, , r; fori=f=.j,i,j=O,l, ,r. c
mi
Proof of Lemma 2. Since af is contained in the normalizer of the holonomy algebra 1)* by Proposition 4.6, Lemma 1 implies the first relation in (1). Since [1), mi ] c af( mi ) also by Proposition 4.6, the second relation in (1) holds. Since KjH is naturally reductive, the Riemannian connection coincides with the natural torsionfree connection. Hence (cf. Theorem 2.10),
[X, Y]m
2(A x Y)o
=
for X, Y
E
m.
I t follows that that is,
(4) em, mi]m C mi, which i;mplies (2) and also (5) [mi , mj]m
C
mi n m j
=
°
for i =f=.j.
By Proposition 2.3, the curvature tensor R is given by
(R(X, Y)Z)o = leX, [Y, Z]m]m  [Y, [X, Z]m]m  l[[X, Y]m' Z]m  [[X, Y]I)' ZJ for X, Y, Z E m. If X E mi , Y E m j and i =f=.j, then (4) and (5) imply (R(X, Y)Z)o
=
[[X, Y]I)' Z]
for Z
m.
E
But the left hand side vanishes since X and Y belong to different factors in the de Rham decomposition. Since the linear isotropy representation is faithful, from [[ X, Y]I)' m] = we obtain [X, Yh = O. This together with (5) implies (3), thus completing the proof of Lemma 2. We shall now complete the proof of Theorem 5.2 with the aid of Lemmas 1 and 2. Using the Jacobi identity we obtain from Lemma 2
°
(6) [1, mi]
C
[1), mi]
C
m;
+ [m o, mi] + ... + [mr , m
i]
+ [mi, m
i ].
214
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
By the Jacobi identity and (6), we obtain
(7) [I,[m i, mi]]
c c
+ [mi, mi], mi ] [I, mi] c mi + [mi, mi].
[m i
By (6) and (7), each Ii is an ideal of 1. By Lemma 2, (8) [Ii, Ij] = 0
for i =1= J.
By Lemma 2,
I(m)
=
10
+ II + . . . + I
(not necessarily a direct sum).
r
To show that the right hand side is actually a direct sum, let X E Ii n Ij for some i =1= J. By (8),
(9) [X, m]
c
+ [X,
[X, 10 ]
On the other hand, Ii
:::>
+ ... + [X, I = mi and I = 1) + m imply II]
r]
O.
Since Ii n Ij c 1)i n 1)j c 1) for i =1= J, X is in 1). From (9) and the faithfulness of the linear isotropy representation, it follows that X = O. The rest of the theorem is now evident. QED. Remark. If K is connected and K/H is simply connected, a simple homotopy argument shows that H is connected. By taking the universal covering group of K, we may assume in Theorem 5.2 that K is simply connected; K remains to be almost effective on K/H although it may no longer be effective. Since K is simply connected, the normal subgroups K( m), Ko, K 1 , • • • , K r of K generated by I( m), 10, II,"" Ir , respectively, are closed and simply connected, and moreover
K( m)
=
K o x K1
X ...
x Kr.
IfwesetH(m) =K(m) nH,Hi=Ki nHfori=O,I, ... ,r, then K(m)/H(m), KdHi, i = 0, 1, ... , r, are naturally reductive, and
KfH = K(m)fH(m) = KofHo
X
K1 /H1 x ...
X
Kr/Hr
coincides with the de Rham decomposition of M = K/H. I t is now quite natural to look for cases in which f = I( m).
x.
HOMOGENEOUS SPACES
215
5.3. In Theorem 5.2, if the ad (H)invariant inner product B m on m corresponding to the metric g can be extended to an ad (K)invariant nondegenerate symmetric bilinear form B on f such that B( m, 1)) = 0, then f = f o + f 1 + ... + fro COROLLARY
Proof. It suffices to prove f = f( m). But this holds more generally as follows. LEMMA.
Under the same assumption as in Theorem 3.5, we have
+ em, m]. Since f ( m) = m + [m, m] f = m
Proof of Lemma. is an ideal of f, its orthogonal complement n (in f with respect to B) is also an ideal of f. Since n is perpendicular to m, it is contained in 1). Since K is (almost) effective on KIH, n reduces to O. QED.
5.4. Let M = KIH be a simply connected, naturally reductive homogeneous space with a Kinvariant Riemannian metric g. If K is simple, then M is irreducible (as a Riemannian manifold). Proof. Since f has no nontrivial ideal, f = f( m). By Theorem 5.2, M is either irreducible or a Euclidean space. The following lemma whose proof is taken from Lichnerowicz [1; p. 113] completes the proof. COROLLARY
If a connected Lie group K acts transitively on a Euclidean as a group of Euclidean motions, then K is not semisimple.
LEMMA.
space Rn
Proof of Lemma. Assume that K is semisimple. Every element of K can be uniquely written as rt, where r is a rotation at the origin and t is a translation. Let p: K + SO (n) be the homomorphism which sends rt into r. Its kernel N is an abelian normal subgroup of K and hence must be discrete. On the other hand, the image of p, being a connected semisimple subgroup of SO(n), must be closed (cf. p. 279 of Volume I) and hence compact. Hence, KI N (which is isomorphic to the image of p) is compact and semisimple. Its covering group K is also compact by the theorem of Weyl (cf. Example 3.2). Being compact, K has a fixed point in Rn (cf. the last 6 lines on p. 193 of Volume I or Theorem 9.2 of Chapter VIII), which is a contradiction. QED.
216
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
For further results on the de Rham decomposition and irreducibility of a homogeneous Riemannian manifold, see Note 25.
6. Invariant almost complex structures Let M = K/H be a homogeneous space of a Lie group K. In this section we shall first give an algebraic characterization of an invariant almost complex structure on M and its integrability. We denote by 0 the origin of M. We choose once and for all a vector space direct sum decomposition of the Lie algebra f = 1) + m, where we are not assuming ad (H) m c m as in the reductive case. Accordingly we write X = XI) + X m for X E f. We identify m with the tangent space To(M) by the restriction of 7T *: Te(K) = f + To(M) to the subspace m. Each element X of f generates a Iparameter subgroup exp tLY and this induces a vector field on M, which we denote by X again. Let J be an almost complex structure on M invariant by K. Restricting J to To(M) = m we obtain a linear endomorphism I: m + m such that 12X = X for X E m. Since J is invariant by K, J o: T o(M) + T o(M) conlmutes with the linear isotropy representation 1 of H. But under the identification To(M) = m, 1 is given by
1(a)X
for a
(ad (a)X)m
=
E
and
H
X
E
m,
where ad (a) denotes the adjoint action of a in f. Thus we obtain I(ad (a)X)m
=
(ad (a)IX)m
for a
E
H
and
X
Em.
6.1. The invariant almost complex structures on M = K /H are in a natural onetoone correspondence with the set of linear endomorphisms I of m such that PROPOSITION
(1) 12X = X
.for X
E
m;
(2) I(ad (a)X)m = (ad (a)IX)m
for a
E
H
and X
E
m.
The proof is straightforward and is left to the reader. The following proposition is equally easy; (2)' below is an infinitesimal version of (2) above.
6.2. If H is connected, then the invariant almost complex structures on M = K/H are in a natural onetoone correspondence with PROPOSITION
X.
217
HOMOGENEOUS SPACES
the set of linear endomorph isms I of m such that (1) 12X = X for X E m; (2)' I[Y, X]m = [Y,IX]m for X E m and
Y
E
1).
We extend I: m + m to a linear endomorphism I: f + f by setting IX = 0 for X E 1). Then conditions (1), (2), and (2)' imply (a) ZX = 0 for X E 1); (b) 12X = X mod 1) for X E f; (c) I(ad (a)X) = ad (a)IX mod 1) for a E H and X E f; (c)' I[Y, X] = [Y, iX] mod 1) for X E f and Y E 1). ,.. It is obvious that the almost cbmplex structure J on KIH and I are rdated by (d) 7T* (IX) = J o(7T*X) for X E 1. ~
~
~
~
~
~
Identifying I and I' such that I(X)  I' (X)
E
1) for X
E
f we have
6.3. The invariant almost complex structures J on M = KIH are in a natural onetoone correspondence with the linear endomorphisms I of f (mod 1)) satisfying (a), (b), and (c). When H is connected, (c) may be replaced by (c)'. The correspondence is given by (d). Proo£ Given J on KIH, we have indicated how to obtain f on 1. Conversely, given I on f satisfying (a), (b), and (c) (or (c')), we may choose a subspace m such that 9 = 1) + m and define Ion m by IX = (IX)m for X E m. We may easily verify that I satisfies (1) and (2) (or (2) ') and thus determines an invariant almost complex structure Jon KIH satisfying (d). It remains to show that the correspondence J * I is onetoone. If J and J' correspond to the same 1, then (d) implies J o(71*X) = J~( 7T*X) for every X E 1. Since 7T * (f) = To(KIH), this means that J o and J~ coincide. Since J and J' are invariant, they coincide everywhere on KIH. PROPOSITION
~
QED. As for the integrability we prove THEOREM 6.4. An invariant almost complex structure J on M = KIH has no torsion if and only if the corresponding linear endomorphism i qf f satisfies #"'I>J
#"'I>J
#"'I>J
#"'I>J
[IX, IY]  [X, Y]  I[X, IY]  I[IX, Y] for all X, Y E 1.
E
1)
218
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Proof. Since f is the Lie algebra of left invariant vector fields "" on K, I defines in a natural manner a left invariant tensor field "" of type (1, 1) on K, which we shall denote by 1. We define a tensor field S of type (1, 2) on K by
S( U, V) = [Iu, IV]
+ I2[U,
V]  I[u, Iv]  I[Iu, V]
for all vector fields U and V on K. (The tensor field 2S is what we called the torsion of two tensor fields A and B of type (1, 1) when A = B = I (pp. 3738 of Volume I).) S is also a left invariant tensor field. Our problem is to show that J is integrable if and only if S(X, Y) is tangent to H at e for all X, Y E f. We prepare the following terminology. We say that a vector field U on K is projectable if there is a vector field U' on K / H such that U iS7Trelated to U' (cf. p. 10 of Volume I), namely, (7T*) Ux = U;Cx) for every x E K. In this case we shall denote U' by 7T* U. If U and V are projectable, U + V, cU (c constant), and [U, V] are proj ectable and ~
and Thus the set of all projectable vector fields on K is a subalgebra, say, ~(K), of the Lie algebra X(M) of all vector fields on K and 7T* gives a Lie algebra homomorphism of ~(K) onto the Lie algebra X(K/H). We claim that if U is projectable, so is IU and 7T*(IU) = J(7T*U). It suffices to show that 7T*(IU) = J(7T*U) when U is a vector at a point a of K. This equality is obvious from the relationship (d) of I at e and J at o. If we denote by La the left translation of K by a as well as the transformation of K/H by a, then U is of the form U = La V for some vector V at e and ~
7T*(I U)
=
=
7T*(1 LaV) L aJ(7T* V)
= =
7T*(L aiV)
=
~
L a7T*(IV)
JL a7T*(V) = J(7T*(L aV)) = J(7T* U),
proving our assertion. If U and V are projectable vector fields on
x.
HOMOGENEOUS SPACES
219
K, then S( U, V) is also projectable and
7T*(S(U, V))
=
[J7T*U,J7T*V] +J'2[7T*U,7T*V]  J[7T* U, J7T* V]  J[J7T* U, 7T* V]
= [J7T* U, J7T* V]  [7T* U, 7T* V]
 J[7T* U, J7T* V]  J[J7T* U, 7T* V], which is half of the value of the torsion tensor of J for the vector fields 7T* U and 7T* V on K/H. It follows that J is integrable if and only if 7T * (S( U, V)) = 0 for all projectable vector fields, since U E ~(K) + 7T* U E X(K/H) is onto. Since S is a left invariant tensor field, 7T * (S( U, V)) is equivalent to the condition that S( U, V) is tangent to H at e for all vectors X, Y E Te(K), that is, S( U, V) E 1) for all X, Y E f. QED. Proposition 6.3 and Theorem 6.4 are due to Koszul [2]. (See also Frolicher [1].) Specializing to reductive homogeneous spaces we obtain
6.5. Let M = K/H be a reductive homogeneous space with a decomposition f = 1) + m, where ad (H) m c m. Then (i) There is a natural onetoone correspondence between the set of all invariant almost complex structures J on K/H and the set of linear endomorphisms I of m satisfying (1) 12 = 1,(2) load (a) = ad (a) I for every a E H. When H is connected, (2) can be replaced by a weaker condition (2)' load (Y) = ad (Y) I for every Y E 1). (ii) An invariant almost complex structure J is integrable if and only if the corresponding endomorphism I of m satisfies PROPOSITION
0
0
[IX, IY]m  [X, Y]m  I[X, IY]m  I[IX, Y]m = 0 for all X, Y E m. Proof. The first part follows from Propositions 6.1 and 6.2 if we note that ad (a)m c m and ad (Y)m c m for a EH and Y E 1). To prove the second assertion we check the condition given in Theorem 6.4. The condition is trivially satisfied if X, Y E 1). It is satisfied for X E m and Y E 1), since it reduces to [X, Y] + I[IX, Y] E 1), which is satisfied by virtue of load (Y) = ad (Y) 01.
220
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Thus J is integrable if and only if
[IX,IY]  [X, Y]  I[X, IY]m  I[X, IY]m
E
1)
[IX, IY]m  [X, Y]m  I[X, IY]m  I[X, IY]m
=
for all X, Y
for all X, Y
E
E
m, that is,
m.
0 QED.
A homogeneous space M = K/H with an invariant complex structure is called a complex homogeneous space. If an invariant almost complex structure J on K/H is integrable, then it is an invariant complex structure (by the real analytic version of Theorem 2.5 of Chapter IX; see Appendix 8). Example 6.1. Let K be an evendimensional Lie group. Considering K as a homogeneous space K/{e} with the left action of K, we may obtain a left invariant almost complex structure on K by simply taking an arbitrary complex structure on f = Te(K), namely, a linear endomorphism I of f such that 1 2 = 1. This does not mean, however, that K is a complex Lie group (cf. Example 2.1 of Chapter IX). Example 6.2. Let K be a complex Lie group and let H be a closed complex Lie subgroup. In terms of Lie algebras, this means that there is a complex structure J on f such that ad (X) 0 J = J 0 ad (X) for every X E f and that the subalgebra 1) is stable by J. In the homogeneous space M = K/H, we have an invariant complex structure, as can be proved by following the analogy of the proof for the existence of a differentiable structure on an arbitrary homogeneous space of a Lie group (cf. Chevalley [1, p. 110]). Here we indicate how our results can be applied to this situation. In f we may take a subspace m such that f = 1) + m and J m c m (cf. the proof of Proposition 1.1 of Chapter IX). Define Ion f by IX = JXfor X Em and IX = 0 for X E 1). Then i satisfies conditions (a), (b), and (c) of Proposition 6.3. We may also verify the integrability condition in Theorem 6.4. For example, for X, Y E m, we have ~
~
~
 
[IX,IY] = [JX, JY] = F[X, Y] = [X, Y]. Also, [X, JY] = J[X, Y] implies [X, JY]m = J[X, Y]m so that
x.
221
HOMOGENEOUS SPACES
....
"'"
I[X, IY] = J[X, JY]m = P[ X, Y]m =  [X, Y]m and, similarly, I[IX, Y] = [X, Y]m. Thus the mcomponent of [IX,IY] I'ftt.I
f'IY
,..,
f'IY
[X, Y]  j[ X, IY] I[iX, Y] is O. Remark. In Chapter XI we shall reexamine mapy examples given in Chapter IX from the point of view of complex homogeneous spaces (in fact, Hermitian symmetric spaces).
CHAPTER XI
Symmetric Spaces
1. Affine locally symmetric spaces Let M be an ndimensional manifold with an affine connection. The symmetry Sa: at a point x E M is a diffeomorphism ofa. neighborhood U onto itself which sends exp X, X E Ta:(M), into exp (X). Since the symmetry at x defined in one neighborhood U of x and the symmetry at x defined in another neighborhood V of x coincide in U n V, we can legitimately speak of the symmetry at x. If {xl, ... , xn } is a normal coordinate system with origin at x, then n Sa: sends (xl, ... , x ) into (Xl, ... , xn ). The differential of Sa: at x is equal to la:, where Ia: is the identity transformation of Ta:(M). The symmetry Sa: is involutive in the sense that Sa: 0 Sa: is the identity transformation of a neighborhood of x. If Sa: is an affine transformation for every x E M, then M is said to be affine locally symmetric.
1.1. A manifold M with an affine connection is qffine locally symmetric if and only if T = 0 and VR = 0, where T and Rare the torsion and the curvature tensors respectively. Proof. Assume that M is affine locally symmetric. Since Sa: is an affine transformation, it preserves T and VR. On the other hand, T is a tensor field of degree 3 and VR is a tensor field of degree 5. (The degree of a tensor field oftype (r, s) is, by definition, r + s.) THEOREM
On an affine locally symmetric space M, a tensor field oj odd degree which is invariant by Sa: vanishes at x. Proof of Lemma. Since the differential of Sa: at x is la:, Sa: sends a tensor K of degree p at x into (1) P K. From the lemma it follows that T = and VR = O. LEMMA.
°
222
XI.
223
SYMMETRIC SPACES
Conversely, assume T = 0 and VR = O. Since R is a tensor field of degree 4, 1(t preserves R(t. By Theorem 7.4 of Chapter VI, for each fixed x there exists a local affine transformationfsuch that f(x) = x and the differential off at x coincides with 1(t. Being an affine transformation,fcommutes with exp (cf. Proposition 1.1 of Chapter VI), i.e., f(exp X) = exp (1(tX)
for X
E
T(t(M).
Hence,f = S(t.
QED.
Theorem 1.1 enables us to apply a number of results obtained in §7 of Chapter VI to affine locally symmetric spaces. In particular, every affine locally symmetric space is a real analytic manifold with a real analytic connection with respect to the atlas consisting of normal coordinate systems (cf. Theorem 7.7 of Chapter VI). A manifold M with an affine connection is said to be affine ~mmetric if, for each x E .N!, the symmetry S(t can be extended to a global affine transformation of M. From Corollary 7.9 of Chapter VI we obtain
1.2. A complete, simply connected, affine locally space is affine symmetric. THEOREM
~mmetric
The following theorem shows that the assumption of completeness in Theorem 1.2 is necessary.
1.3. Every affine ~mmetric space is complete. Proof. Let T = xt , 0 t s a, be a geodesic from x toy. Using the symmetry Sy, we can extend T beyondy as follows. Set THEOREM
for 0 < t < a.
QED. 1.4. On every affine symmetric space, the group of affine transformations is transitive. Proof. Given any two points x and y, there exists a finite sequence of convex neighborhoods U1, U2 , • • • ,Uk such that x E U1,y E Uk' and Ui n Ui +1 ep for i = 1, ... , n  1. Hence, x and y can be joined by a finite number of geodesic segments. (For the existence and the property of a convex neighborhood, see THEOREM
224
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Theorem 8.7 of Chapter III.) It suffices therefore to prove that, for a pair of points x andy which can bejoined by a single geodesic segment, there exists an affine transformation sending x into y. Let z be the midpoint on the geodesic segment from x to y (with respect to the affine parameter). Then the symmetry Sz maps x into y. QED. The group m(M) of affine transformations of M is known to be a Lie group (cf. Theorem 1.5 of Chapter VI). Let G denote the identity component mO( M) of m( M) for the sake of simplicity. Since, as is easily seen, the identity component of a Lie group acting transitively on a manifold M is itself transitive on M, an affine symmetric space M may be written as a homogeneous space G/H. We shall later see that M = G/H is reductive and that the given connection on M coincides with the canonical connection as well as the natural torsionfree connection defined in §2 of Chapter X. As a preparation, we prove
1.5. Let G be the largest connected group of affine transformations of an affine symmetric space M and H the isotropy subgroup of G at a fixed point 0 of M so that M = G/H. Let So be the symmetry of M at 0 and a the automorphism of G defined by THEOREM
a (g)
=
So 0 g 0
s~1
for g
E
G.
Let Ga be the closed subgroup of G consisting of elements fixed by a. Then H lies between Ga and the identity component of Ga' Proof. Let h E H and consider a(h) = So 0 h 0 S;1. Since the differential of So at 0 coincides with 10 , the differential of a(h) at 0 coincides with that of h at o. Since two affine transformations with the same differential at one point coincide with each other (see the end of §l of Chapter X), a(h) is equal to h, thus proving H c Ga' Let gt be an arbitrary Iparameter subgroup of Ga' Then So 0 gt(o) = gt 0 so(o) = gt(o), which shows that the orbit gt (0) is left fixed pointwise by so' Since o is an isolated fixed point of so, the orbit gt( 0) must reduce to the point 0, that is gt(o) = o. Hence gt E H. Since a connected Lie group is generated by its Iparameter subgroups, we may conclude that the identity component of Ga is contained in H. QED.
XI.
SYMMETRIC SPACES
225
2. Symmetric spaces Theorem 1.5 suggests the following definition. A symmetric space is a triple (G, H, a) consisting of a connected Lie group G, a closed subgroup H of G and an involutive automorphism a of G such that H lies between G(1 and the identity component of G(1' where G(1 denotes the closed subgroup of G consisting of all the elements left fixed by a. A symmetric space (G, H, a) is said to be effective (resp. almost effective) if the largest normal subgroup N of G contained in H reduces to the identity element only (resp. is discrete). If (G, H, a) is a symmetric space, then (GIN, H(N, a*) is an effective symmetric space, where a* is the involutive automorphism of GI N induced by a. Here it is more convenient not to assume that (G, H, a) is effective, i.e., G acts effectively on GIH. The results in Chapter X we need here are all valid without the assumption that G be effective on GIH (cf. the remark following Corollary 2.11 of Chapter X). Given a symmetric space (G, H, 0'), we shall construct for each point x of the quotient space M = GIH an involutive diffeomorphism Sa:, called the symmetry at x, which has x as an isolated fixed point. For the origin 0 ofGIH, So is defined to be the involutive diffeomorphism of GIH onto itself induced by the automorphism 0' of G. To show that 0 is an isolated fixed point of So, let g( 0) be a fixed point of So, where g E G. This means o'(g) E gH. Set h = gla(g) E H. Since a(h) = h, we have
h2 = ha(tz) = gla(g)a(gla(g)) = gla(g)a(gl)g = 1, thus showing that h2 is the identity element. If g is sufficiently close to the identity element so that h is also near the identity element, then h itself must be the identity element and hence a(g) = g. Being invariant by a and near the identity element, g lies in the identity component of G(1 and hence in H. This implies g(o) = 0, thus proving our assertion that 0 is an isolated fixed point of So' For x = g(o), we set Sa: = g 0 So 0 gl. Then Sa: is independent of the choice of g such that x = g(o). We define now an infinitesimal version of a symmetric space. A symmetric Lie algebra (sometimes called an involutive Lie algebra) is a triple (g, 1), a) consisting of a Lie algebra g, a subalgebra 1) of g, and an involutive automorphism a of 9 such that 1) consists
226
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
of all elements of 9 which are left fixed by G. A symmetric Lie algebra (9, 1), 0') is said to be effective if 1) contains no nonzero ideal of 9. Every symmetric space (G, H, G) gives rise to a symmetric Lie algebra (9, 1), G) in a natural manner; 9 and 1) are the Lie algebras of G and H, respectively, and the automorphism G of 9 is the one induced by the automorphism G of G. Conversely, if (9, 1), G) is a symmetric Lie algebra and if G is a connected, simply connected Lie group with Lie algebra 9, then the automorphism G of 9 induces an automorphism G of G (cf. Chevalley [1; p. 113]) and, for any subgroup H lying between G(J and the identity component of G(J' the triple (G, H, G) is a symmetric space. Note that since G(J is closed, so is H. If G is not simply connected, the Lie algebra automorphism G may not induce an automorphism of G. It is clear that (G, H, G) is almost effective if and only if (9, 1), 0') is effective. It is also easy to see that there is a natural onetoone correspondence between the (effective) symmetric Lie algebras (9, 1), G) and the (almost effective) symmetric spaces (G, H, G) with G simply connected and H connected. Let (9, 1), G) be a symmetric Lie algebra. Since G is involutive, its eigenvalues as a linear transformation of 9 are 1 and 1, and 1) is the eigenspace for 1. Let m be the eigenspace for 1. The decomposition 9=1)+m is called the canonical decomposition of (9, 1), (J).
2.1. If 9 = 1) + m is the canonical decomposition ofa symmetric Lie algebra (9, 1), G), then PROPOSITION
[1), 1)]
C
1),
[1), m]
c
m,
em, m]
c
1).
Proof. The first relation just expresses the fact that 1) is a subalgebra. If X E 1) and Y E m, then
G([X, Y]) = [G(X), G(Y)] = [X, Y] = [X, Y], which proves the second relation. If X, Y
E
m, then
G([X, Y]) = [G(X), G(Y)] = [X, Y] = [X, Y], which proves the third relation.
QED.
XI.
227
SYMMETRIC SPACES
Remark. The inclusion relations given in Proposition 2.1 characterize a symmetric Lie algebra. Given a Lie algebra 9 and a decomposition 9 = 1) m (vector space direct sum) satisfying the relations in Proposition 2.1, let a be the linear transformation of 9 defined by a(X)
=
X
for X
E
1)
a(Y)
and
=
Y
for Y Em.
I t is easy to verify that a is an involutive automorphism of 9 and (g, 1), a) is a symmetric Lie algebra.
2.2. Let (G, H, a) be a symmetric space and (g, 1), a) its symmetric Lie algebra. If 9 = 1) + m is the canonical decomposition of (g, 1), a), then ad (H)m c m. PROPOSITION
Proof.
Let X
a(ad h . X)
=
E
m and h E H. Then
ad a(h) . a(X)
=
ad h . (X)
ad h· X.
QED. A homomorphism of a symmetric space (G', H', a') into a symmetric space (G, H, a) is a homomorphism ex. of G' into G such that ex.(H') cHand that a ex. = ex. a'; it will be denoted by ex.: (G', H', a') ~ (G, H, a). A homomorphism ex.: (G', H', a') ~ (G, H, a) is called a mono or epimorphism according as ex.: G' ~ G is a monoor epimorphism. It is called an isomorphism if ex.: G' ~ G is an isomorphism and if ex.(H') = H. A triple (G', H', a') is called a symmetric subspace (resp. symmetric closed subspace or symmetric normal subspace) of a symmetric space (G, H, a) if G' is a Lie subgroup (resp. closed subgroup or normal subgroup) of G invariant by a, if H' = G' () H and if a' is the restriction of a to G'. Unless otherwise stated we shall assume that G' is also connected, although the definition above makes sense for a nonconnected G'. Given a symmetric closed normal subspace (G', H', a') of a symmetric space (G, H, a), we obtain the symmetric quotient space (GIG', HIH', a"), where (1" is the involutive automorphism of GIG' induced bya. For symmetric Lie algebras, we define the notions of homomorphism, monomorphism, epimorphism, isomorphism, symmetric subalgebra, and symmetric ideal in the same way. It should 0
0
228
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
be remarked that, in the definition of a homomorphism rl: (g', ~', a') + (g, ~, a), the condition rl( ~') C ~ follows from a rl = rl a'. Each homo, mono, epi, or isomorphism rl: (G', H', a') + (G, H, a) induces in a natural manner a homo, mono, epi, or isomorphism rl: (g', ~', a') + (g,~, a), respectively. The converse holds under suitable assumptions. For instance, each homomorphism rl: (g', ~', a') + (g, ~,a) generates a homomorphism rl: (G', H', a') + (G, H, a) if G' is simply connected and H' is connected. Each homomorphism rl: (G', H', a') + (G, H, a) induces a mapping 1i: G' jH' + GjH which commutes with the symmetries, I.e., for x' E G' jH', 1i 0 sx ' = s( (Xx '} 0 1i 0
0
where Sx' denotes the symmetry of G' j H' at x' and sa(x') the symmetry of GjH at 1i(x'). Each homomorphism rl: (g', ~', a') + (g, ~,a) is compatible with canonical decompositions: if g' = ~' + m' and 9 = ~ + m are the canonical decompositions, then tt: g' + 9 sends m' into m. Given two symmetric spaces (G, H, a) and (G', H', a'), their direct product is a symmetric space given by (G X G', H X H', a X a'). Similarly, the direct sum of two symmetric Lie algebras (g, ~, a) and (g', ~', (J") is given by (g + g', ~ + ~', a + a'). The following example will show that the concept of symmetric space is a generalization of that of Lie group. We shall see more examples in due course. Example 2.1. Let G be a connected Lie group and D,.,G the diagonal of G X G, i.e., D,.,G = {(g, g) E G X G; g E G}. Define a: G X G+G X G by a(g,g') = (g', g). Then (G X G, D,.,G,a) is a symmetric space. The quotient space (G X G) / D,.,G is diffeomorphic with G, the diffeomorphism being induced from g E G + (g, e) E G X G, where e is the identity element of G. Given two connected Lie groups G and G' with a homomorphism rl: G' + G, the homomorphism rl X rl: G' X G' + G X G gives rise to a homomorphism ex X rl: (G' X G', D,.,G', a') + (G X G, D,.,G, a). If G' is a (closed or normal) subgroup ofG, then (G' X G', D,.,G', a') is a symmetric (closed or normal) subspace of (G X G, D,.,G, a). Similarly, every Lie algebra 9 gives rise to a symmetric Lie algebra (g + g, D,.,g, a).
XI.
SYMMETRIC SPACES
229
Example 2.2. Let M be an ndimensional affine locally symmetric space. At a point x of M, consider the tangent space m = Tx(M) and the curvature tensor R x. We denote by 1) the set of all linear endomorphisms U of Tx(M) which, when extended to a derivation of the tensor algebra at x, map R x into 0, that is, (writing R for R x ) (U· R) (X, Y)  UR(X, Y)  R( UX, Y)  R(X, UY)  R(X, Y) U
=0 for all X, Y E m. Then 1) is a Lie algebra under the usual bracket operation (cf. Proposition 2.13 of Chapter I). Note that R(X, Y) E 1) for any X, Y E m, si,nce if we extend X and Y to vector fields, then R(~, Y) = [V x, V y]  V[X,y] maps R x into 0 by virtue of VR = O. In the direct sum 9 = m + 1), we define [X, Y]
=
R(X, Y)
[U, X]
=
UX
[U, V] = [U, V]
for X, Y E m,
for U
E
1), X
for U, V
m,
E E
1), as already defined in 1).
We may verify that the Jacobi identity is satisfied in 9 as follows: If X, Y, Z E m, then [[X, Y], Z] = R(X, Y)Z so that 6[[X, Y], Z] = 0
by Bianchi's first identity (Theorem 5.3 of Chapter III). If X, Y E m and U E 1), then [[X, Y], U] = [R(X, Y), U], [[Y, U], X] = [UY, X]
= R(UY, X)
and [[U, X], Y] = [UX, Y]
= R( UX, Y)
so that 6[[X, Y], U] = 0 by virtue of the fact that U· R = O. Finally, for U, V E 1) and X E m, 6[[U, V], X] = 0 follows immediately from [U, V] = UV  VU. Hence 9 = m + 1) is a Lie algebra, and obviously [m, m] c 1), [m, 1)] c m, and [1), 1)] c 1). If we define (j(X) = X for X E m and 0'( U) = U for U E 1), then (j is an involutive automorphism of g. Thus (g, 1), (j) is a symmetric Lie algebra. It is effective, for if U is an element of an
230
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
ideal n of 9 contained in 9, then for X E m we have [U, XJ E n and also [U, XJ = UX E m, showing UX = 0 for every X E m, that is, U = o.
3. The canonical connection on a symmetric space We have shown in §1 that a simply connected manifold with a complete affine connection such that T = 0 and VR = 0 gives rise to a symmetric space (G, H, a) such that M = GIH. We shall show now that, conversely, if (G, H, a) is a symmetric space, then the homogeneous space GIH admits an invariant affine connection with T = 0 and VR = O. Let
9=9+ m be the canonical decomposition of the symmetric Lie algebra (g, 9, a). According to Proposition 2.2, GIH is reductive with respect to the canonical decomposition in the sense defined in §2 of Chapter X. Theorem 2.1 of Chapter X states that there is a onetoone correspondence between the set of Ginvariant affine connections on GIH and the set of linear mappings Am: m+ gI(n; R) such that Am(ad h(X))
=
ad (A(h)) (Am (X))
for X
E
m
and
h
E
H,
where A denotes the linear isotropy representation H ~ GL(n; R). (We recall that both m and the tangent space To(M) at the origin oEM = GIH are identified with Rn.) The invariant connection corresponding to Am = 0 is called the canonical connection of (G, H, a) or GIH. Since [m, mJ c 9 for a symmetric space, the canonical connection coincides with the natural torsionfree connection defined in §2 of Chapter X. We have T = 0 and VR = O.
3.1. Let (G, H, a) be a symmetric space. The canonical connection is the only affine connection on M = GIH which is invariant by the symmetries of M. Proof. We shall first prove that the canonical connection is invariant by the symmetries SX. Since Sx = g So gl if x = g( 0) as was shown in §2 and since the canonical connection is invariant by G, it suffices to show that the canonical connection is invariant THEOREM
0
0
XI.
by
SO.
From the way
g
0
231
SYMMETRIC SPACES
So
was constructed by a in §2, it follows that
So
=
So
0
a(g)
for g
E
G,
considered as transformations on M = GIH. This implies that So maps every Ginvariant connection into a Ginvariant connection. Let So map a Ginvariant connection with Am into a Ginvariant connection with A~. From Theorems 1.2 and 2.1 of Chapter X it follows that A~ = Am. In particular, So maps the canonical connection (defined by Am = 0) into itself. We claim that the symmetry Sx constructed from a coincides with the symmetry of M at x with respect to the canonical connection in the sense of§l. In fact, since a maps X Em into X E m, the differential of Sx at x coincides with Ix. On the other hand, since Sx is an affine transformation (with respect to the canonical connection), it commutes with the exponential maps, i.e.,
sx( exp X) = exp (X) (cf. Proposition 1.1 of Chapter VI), which proves our claim. By the lemma in the proof of Theorem 1.1, every tensor field of odd degree invariant by all Sx must vanish identically. We use this fact to prove the uniqueness of a connection invariant by Sx. We know (cf. Proposition 7.10 of Chapter III) that the difference of two connections on a manifold is a tensor field of type (1, 2). Considering a connection on M = GIH invariant by all Sx' let S be its difference with the canonical connection. Being the difference of two connectiqns invariant by all sx, S is invariant by all Sx and hence vanishes identically. QED. Specializing results in §2 of Chapter X to the canonical connection of a symmetric space, we state 3.2. With respect to the canonical connection of a symmetric space (G, H, a), the homogeneous space M = GIH is a (complete) affine symmetric space with symmetries Sx and possesses the following properties: (1) T = 0, VR = 0, and R(X, Y)Z = [[X, Y], Z] for X, Y, Z E m, where m is identified with To(M), 0 being the origin of M; (2) For each X E m, the parallel displacement along 1T( exp tX) coincides with the differential of the transformation exp tX on M,. THEOREM
232
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
(3) For each X E m, 7T( exp tX) = (exp tX) . 0 is a geodesic starting from 0 and, conversely, every geodesic from 0 is of this form; (4) Every Ginvariant tensor field on M is parallel; (5) The Lie algebra of the linear holonomy group (with reference point 0) is spanned by {R(X, Y) = adm ([X, Y]), X, Y Em}. We note that (5) follows from Corollary 4.3 of Chapter X or more directly from Theorem 9.2 of Chapter III. Remark. If a tensor field F on M = G/H is invariant by G, then each symmetry Sx maps F into F or F according as the degree ofF is even or odd. Since Sx = g 0 So 0 gl for some g E G, it suffices to prove the assertion above for so. Since g 0 So = So 0 a(g) for every g E G as in the proof of Theorem 3.1, so(F) is also invariant by G. But clearly so(F) = ±F at 0 depending on the parity of the degree of F. Since both F and so(F) are invariant by G, we have so(F) = ±F according as the degree of F is even or odd. 3.3. Let (G, H, a) be a symmetric space. A Ginvariant (indefinite) Riemannian metric on M = G/H, if there exists any, induces the canonical connection on M. Proof. Such a metric is parallel with respect to the canonical connection by (4) of Theorem 3.2. Since the canonical connection is also torsionfree by Theorem 3.2, it must be the Riemannian connection (cf. Theorem 2.2 of Chapter IV). This theorem may be also derived from Theorem 3.3 of Chapter X. QED. THEOREM
We also remark here that if (G, H, a) is a symmetric space, then G/H is naturally reductive (in the sense of §3 of Chapter X) with respect to an arbitrary Ginvariant indefinite Riemannian metric (if there exists any). There is a large class of symmetric spaces which admit invariant indefinite Riemannian metrics.
Let (G, H, a) be a symmetric space with G semisimple and let 9 = 1) + m be the canonical decomposition. Then the restriction of the KillingCartan form of 9 to m defines a Ginvariant (indefinite) Riemannian metric on G/H in the manner described in Corollary 3.2 of Chapter X. Proof. Since the KillingCartan form of a semisimple Lie algebra is nondegenerate and invariant by all automorphisms (cf. Appendix 9), it suffices to prove the following THEOREM
3.4.
XI.
233
SYMMETRIC SPACES
Let (g, 1), a) be a symmetric Lie algebra and 9 = 1) + m the canonical decomposition. If B is a symmetric bilinear Jorm on 9 invariant by a, then B(1), m) = O. If B is moreover nondegenerate, so are its restriction Bl) to 1) and its restriction B m to m. Proof of Lemma. If X E 1) and Y E m, then LEMMA.
B(X, Y)
=
B(a(X), O'(Y))
=
B(X, Y)
=
B(X, Y),
which proves our assertion. It is clear that B is nondegenerate if and only if both Bl) and B m are so. QED. Let M be an affine symmetric space and G the largest connected group of affine transformations of M. Choosing an origin 0 E M, we obtain a symmetric space (G, H, a) such that M = GjH (cf. Theorem 1.5). In general, there might exist a symmetric subspace (G', H', a') of (G, H, a) such that M = GjH = G' jH' in a natural manner. To see this, let 9 = 1) + m be the canonical decomposition. It is a simple matter to verify that 1)(m) = em, m] is an ideal of 1) and g( m) = 1)( m) + m is an ideal of g. For each subalgebra g' of 9 containing g( m), we obtain a symmetric subspace (G', H', a') such that M = GjH = G' jH' by taking for G' the connected Lie subgroup of G generated by g' and setting H' = G' () Hand a' = a I G'. Conversely, if (G', H', a') is a symmetric space with M = G' jH' (with the same origin as M = GjH) and if G' is effective on M, then (G', H', a') is a symmetric subspace of (G, H, a) and g' contains g(m). If M = Rn with the natural flat affine connection, then 1)( m) = 0 and g( m) = m. Consequently, every connected Lie subgroup G' of G containing the group of translations of Rn yields a symmetric subspace (G', H', a') of (G, H, a) such that Rn = G'jH'. ' When M is an affine locally symmetric space, we may construct, for each point x of M, a symmetric Lie algebra (g, 1), a) as in Example 2.2. Let (G, H, a) be a symmetric space corresponding to (g, 1), a), where G and GjH may be assumed simply connected. Let 0 be the origin of Gj H. Then we have a linear isomorphism F of Ta/M) onto T o(GjH) which maps the curvature tensor Rg; of M upon the curvature tensor of the canonical connection of GjH. By Theorem 7.4 of ChapterVI we see thatFis the differential of a certain affine isomorphismJ of a neighborhood U of x in M onto a neighborhood Vof 0 in GjH. We have thus shown that each
234
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
point of an affine locally symmetric space M has a neighborhood on which the affine connection is isomorphic to the canonical connection of a certain symmetric space GIH restricted to a neighborhood of the origin. The proof above is a special case of a method which works for a manifold with an affine connection invariant by parallelism (V T = 0 and VR = 0) and which gives a local version of (2) of Theorem 2.8 of Chapter X.
4. Totally geodesic submanifolds Let (G, H, a) be a symmetric space. With respect to the canonical connection, M = GIH is an affine symmetric space (cf. Theorem 3.2). Let (G', H', a') be a symmetric subspace of (G, H, a), i.e., let G' be a connected Lie subgroup of G invariant by (1, H' = G' n H and a' = (1 I G'. Then there is a natural imbedding of M' = G'IH' into M = GIH. From the way the symmetries of M and M' are constructed from a and a' (see §2), we easily see that the symmetry Sx of M at a point x of M' restricts to the symmetry of M' at x. We may therefore denote the symmetry of M' at x by sX'
4.1. If (G', H', a') is a symmetric subspace of a symmetric space (G, H, a), then M' = G' IH' is a totally geodesic submanifold of M = GIH (with respect to the canonical connection of M). The canonical connection of M restricted to M' coincides with the canonical connection of M. Proof. Let 9 = 1) + m and g' = 1)' + m' be the canonical decompositions of 9 and g', respectively. For the moment, we consider only the canonical connection of M. A geodesic of M tangent to M' at the origin 0 is of the form h (0 ), where h = exp tX with X E m' c m. Obviously, the geodesic ft(o) lies in M'. Given any point x = g'(o) of M' with g' E G', a geodesic of M tangent to M' at x is of the form g' (ft( 0)), which is obviously contained in M', thus proving that M' is a totally geodesic submanifold of M. Since the canonical connection of M is invariant by the symmetries of M, its restriction to M' is also invariant by the symmetries of M'. By Theorem 3.1, the connection induced on M' by the canonical connection of M coincides with the canonical connection of M'. QED. THEOREM
XI.
235
SYMMETRIC SPACES
Conversely, we have
4.2. Let (G, H, 0') be a symmetric space, M = GfH the affine symmetric space with the canonical connection and M' a complete totally geodesic submanifold of M through the origin o. Let G' be the largest connected Lie subgroup of G leaving M' invariant, H' the intersection G' n H and a' the restriction of 0' to G'. Then (G', H', a') is asymmetric subspace of (G, H, 0') and M' = G' fH'. THEOREM
By a complete totally geodesic submanifold, we mean a totally geodesic submanifold of M which is complete with respect to the connection induced from the connection of M. The assumption that M' contains 0 is not restrictive because every totally geodesic submanifold can be translated by an element of G so as to contain o. Although not explicitly stated, the theorem asserts that G' is invariant by 0'. Proof. We need a few lemmas.
1. The product of two symmetries formation belonging to G. Proof of Lemma 1. Write LEMMA
Sx = Sy =
g g,
So
0
0
So
0
gl , 1 0 g 
where x wherey
= =
Sx
and Sy of M is a trans
g(o), g'(o),
g E G, g' E G.
Then On the other hand (cf. the proof of Theorem 3.1), we have So
0
gl 0 g'
0
So =
O'(glg').
Hence, Sx
2.
0
Sy
=
gO'(glg')g'l.
Lit M' and Mil be two complete totally geodesic submanifolds of M. If at one point of M' n Mil the tangent space of M' coincides with that of Mil, then M' = Mil. Proof of Lemma 2. Let x E M' n Mil and T x ( M') = T x ( Mil). Any pointy of M' can bejoined to x by a broken geodesic (that is, a curve composed of a finite number of geodesic segments) of M'. This broken geodesic may be considered as a broken geodesic of M because M' is totally geodesic in M. Since Mil is totally geodesic and complete in M, Mil contains this broken geodesic. Hence, y E Mil, thus proving M' c Mil. Similarly, Mil eM'. LEMMA
236
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
LEMMA 3. For eve~ x EM', the symmetry Sx of M at x sends M' into itself Proof of LEMMA 3. Since Sx is an affine transformation of M, sx(M') is also totally geodesic in M. Applying Lemma 2 to M' and sx(M'), we obtain Lemma 3. We are now in a position to complete the proof of Theorem 4.2. We shall first prove that G' is transitive on M'. Given two points x andy of M', we have to show the existence of an element of G' which maps x into y. Since any two points of M' can be joined by a broken geodesic of M', we may assume that x and y can be joined by a geodesic segment T of M'. Write T
=
Xt,
with x =
0 < t < 4a,
For each fixed t, consider the symmetries
Xo SXt
and y =
and
S~3t
X4a
and set
Thenfo is the identity element andfa(x) = y. Eachh is a transformation belonging to G by Lemma 1 and sends M' into itself by Lemma 3. Hence, eachh lies in G', proving that G' is transitive on M'. . We have shown earlier (cf. the proof of Theorem 3.1) that
g
0
So
=
So
0
a(g)
for g
E
G.
Since So leaves M' invariant by Lemma 3, a(g) = S~l 0 g 0 So leaves M' invariant for g E G'. Let gt be a curve in G' starting from the identity element. Then a(gt) leaves M' invariant. Since a(go) is the identity element of G', the curve a(gt) lies in G'. This proves that a sends G' into itself. If we set H' = G' n H and a' = a G', then (G', H', a') is a symmetric subspace of (G, H, a). The transitivity of G' on M' implies M' = G' /H'. QED.
I
Let T be a geodesic from x to y in an affine symmetric space M. The product of two symmetries Sx and Sy is known as a transvection (along T). Lemma 1 says that every transvection is contained in the connected group G although the symmetries may not be. The proof above shows that the set oftransvections generates a transitive group.
XI.
237
SYMMETRIC SPACES
Theorems 4.1 and 4.2 do not give a onetoone correspondence between the complete totally geodesic submanifolds M' through 0 of M and the symmetric subspaces (G', H', a') of (G, H, a). 'fwo different symmetric subspaces (G', H', a') and (G", H", a") may give rise to the same totally geodesic submanifold (c£ Remark at the end of §3).
4.3. Let (G, H, a) be a symmetric space and 9 = 1) + m the canonical decomposition. Then there is a natural onetoone correspondence between the set oflinear subspaces m' of m such that [[ m', m'], m'] c m' and the set of complete totally geodesic submanifolds M' through the origin 0 of the ajjine symmetric space M = GIH, the correspondence being given by m' = To(M') (under the identification m = To(M)). Proo£ Let M' be a complete totally geodesic submanifold through 0 of M and (G', H', a') a symmetric subspace of (G, H, a) such that M' = G'IH'. Let g' = 1)' + m' be the canonical decomposition and m' = To(M'). Since em', m'] c 1)', we have [em', m'], m'] em'. Conversely, let m' be a subspace of m such that [em', m'], m'] c m'. Then set THEOREM
1)' = em', m'],
g' = 1)'
.+
m',
I
a' = a g'.
Then (g', 1)', a') is a symmetric subalgebra of (g, 1), a). Let G' be the connected Lie subgroup of G generated by g' and set H' = G' ('\ H. Since a leaves g' invariant, it leaves G' invariant. By setting a' = a I G', we obtain a symmetric subspace (G', H', a') of (G, H, a). The totally geodesic submanifold M' = G'IH' has the property To(M') = m'. Now, from Lemma 2 in the proof of Theorem 4'.2, we may conclude that the correspondence M' ~ m' is onetoone. QED. It would be of some interest to note that the symmetric subalgebra (g', 1)', a') constructed in the proof above by setting g' = em', m'] + m' is the smallest symmetric subalgebra of (g, 1), a) with the desired property. In general, a subspace t of a Lie algebra 9 such that [[ t, t], t] c t is called a Lie triple system. 4.4. Let (G, H, a) be a symmetric space and M' a complete totally geodesic submanifold through the origin 0 of M = GIH. PROPOSITION
238
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Then M' isfiat (i.e., has vanishing curvature) if and only if the corresponding subspace m' of m in Theorem 4.3 satisfies
[[ m', m'], m'] =
o.
Proof. Apply (1) of Theorem 3.2 to a symmetric subspace (G', H', 0") such that M' = G'IH'. QED.
5. Structure of symmetric Lie algebras Let 9 be a Lie algebra and t its radical, i.e., the largest solvable ideal of g, so that g/t is semisimple. There exists a semisimple subalgebra 5 of t which is complementary to t (Levi's theorem; cf. for instance, Jacobson [1; p. 91]). In other words; the short exact sequence
o + t
+
9
+
g/t
+
0
splits. It is also a well known theorem (cf. Appendix 9 or Jacobson [1; p. 71]) that a semisimple Lie algebra 5 is the direct sum of its simple ideals. We shall extend these results to symmetric Lie algebras. Since the radical t is the largest solvable ideal of g, it is invariant by every automorphism of g. Hence (cf. §2), given a symmetric Lie algebra (g, 1), 0') we have a symmetric ideal (t, 1) () t, 0"), where 0" = 0' I t. We obtain also a symmetric Lie algebra (g/t, 1)11) () t, 0'*) together with a natural epimorphism (g, 1), 0') + (g/t, 1)11) () t, 0'*). 5.1. Let (g, 1), 0') be a symmetric Lie algebra and t the radical of g. Then there exists a symmetric subalgebra (5, 1) () 5, a") of (g, 1), 0') such that 5 is a semisimple subalgebra oj' 9 which is complementary to t. In other words, the short exact sequence THEOREM
o + (t, 1)
() t, 0")
+
(g, 1), 0')
+
(g/t, 1)11) () t, 0'*)
+
0
splits. Proof. All we have to know is that in Levi's theorem there exists a semisimple subalgebra 5 which is invariant by 0'. This fact is proved in Appendix 9. QED.
XI.
239
SYMMETRIC SPACES
5.2. Let (g, 1), 0') be a symmetric Lie algebra with 9 semisimple. Then (g, 1), 0') can be decomposed into the direct sum THEOREM
(gl
+ g~,
1)1' 0'1)
+ ... + (gk + g~, 1)k' O'k) + (gk+1' 1)k+l' O'k+1) + ... + (gr'
1)r' O'r)' where (1) gl' g~, , gk' g~, gk+1' ••• , gr are the simple ideals of g; (2) gl + g~, , gk + g~, Ak+1' ••• , gr are invariant by 0' and O'i = 0' I gi + g; for i = 1, ... , k and O'j = 0' I gj for J = k + 1, ... \, r; (3) For i = 1, ... , k, each gi is isomorphic with g; under 0' and 1)i = {(X, O'(X)) E gi + g;; XE gi}. For J = k + 1, ... , r, 1)j = gj () 1). Proof. Let g1' ... , gp be the simple ideals of g. Being an automorphism of g, 0' permutes these simple ideals. Since 0' is involutive, we have either or (2) 0'( gi) = gi'
and
0'( gi') = gi·
Changing notations we may write 9 = (gl
+ g~) + . . . + (gk + g~) + gk+1 + . . . + gr'
where O'(gi) = g~ and O'(g~) = gi for i = 1, ... , k and O'(gj) = gj for J = k + 1, ... , r. Theorem 5.2 follows now immediately.
QED. A short exact sequence of symmetric Lie algebras may be considered as an infinitesimal version of the following situation. 5.3. Let (G, H, 0') be a symmetric space and (G', H', 0") a closed normal symmetric subspace of (G, H, 0'). If we set G* = GIG', H* = HIH', and define 0'* to be the involutive automorphism of G* induced by 0', then (G*, H*, 0'*) is a symmetric space and the homogeneous space GIH is a fibre bundle over G* IH* with fibre G' IH' in a natural manner. PROPOSITION
The proof is trivial. We may apply Proposition 5.3 to the case where G' is the radical of G, i.e., the connected Lie subgroup of G generated by the
240
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
radical of g. As we shall see below, the radical G' is closed in G. Thus, GIH is a fibre bundle over G* IH* with fibre G'IH' where G* is semisimple and G' is solvable. By Theorem 4.1 the fibres are all totally geodesic with respect to the canonical connection of GIH. We shall now prove that the radical G' of G is closed in G. Let G' be the closure of G' in G. Since G' is a connected normal subgroup of G, so is G'. It suffices therefore to prove that G' is solvable. But, in general, a connected dense Lie subgroup G' of a connected Lie group G' is normal in G' and there is a connected abelian Lie subgroup A of G' such that G' = G'A (for the proof of this fact, see Hochschild [1; p. 190J). Hence, G' is soJvable. Assume, in addition, that G is simply connected. Let 9 = g' + g" be a semidirect sum, where g' is the radical of 9 and g" is a semisimple subalgebra of 9 invariant by a (cf. Theorem 5.1). Let G" be the connected Lie subgroup of G generated by g". Since G is simply connected, G is written as the semidirect product G"G'. Indeed, the projection G + GIG' induces a homomorphism p: G" + GIG' which is obviously a local isomorphism. Hence, p is a covering map. On the other hand, since G is simply connected, so is GIG' (c£ Hochschild [1; pp. 135, 136J). It follows that p is an isomorphism of G" onto GIG' and that G" may be considered as (the image of) a global cross section of the fibring G + GIG'. Our assertion follows now immediately. Setting H" = G" (\ H and a" = a I G", we obtain a closed symmetric subspace (G", H", a") of (G, H, a). In general, we have only H"H' c H. But 1) is a semidirect sum of 1)' and 1)". Hence, if H is connected, then H = H"H'. We may now conclude that if G is simply connected and H is connected, then the
fibre bundle GIH over G* IH* with fibre G'IH' admits a global cross section G" IH" which is totally geodesic with respect to the canonical connection of GIH. Let (G, H, a) be a symmetric space where G is simply connected and semisimple and H is connected. Then corresponding to the direct sum decomposition of (g, 1), a) in Theorem 5.2, we have the following direct product decomposition of (G, H, a): (GI
X G~,
HI' a l )
X ••• X X
(Gk X G~, Hk , ak ) (Gk +l , Hk +l , ak +l ) X
• • • X
(Gr , Hr , ar ),
and consequently the direct product decomposition of the affine
XI.
241
SYMMETRIC SPACES
symmetric space G/H:
(Gl x Gn/Hl x ...
X
(Gk x Gn/Hk
x Gk +l / Hk +l x···
X
Gr/Hr.
The proof is straightforward and is left to the reader. Theorem 5.1 and Theorem 5.2 show that a symmetric Lie algebra is built of symmetric Lie algebras of the following three types: (1) (g g, 8g, a), where 9 is simple, 8g = {(X, X); XE g}, and a(X, Y) (Y, X) for X, Y E g; (2) (g, 1), a), where 9 is simple; (3) (g, 1), a), where 9 is solvable. The symmetric Lie algebras of type (1) are in onetoone correspondence with the simple Lie algebras in a natural manner. The symmetric Lie algebras of type (2) have been classified by Berger [2J. Example 5.1. Given a symmetric Lie algebra (g, 1), a), we shall construct a new symmetric Lie algebra which will be denoted by (T(g), T(1)), T(a)). Let T(g) be the Lie algebra obtained by defining a new bracket operation [ , J' in 9 + 9 as follows: [(X, X'), (Y, Y')]'  ([X, Y], [X, Y']
[X', Y]) for (X, Y)
E
9
+ g.
Then T(g) is a semidirect sum of an abelian ideal g' = {(O, X'); X' E g} and a subalgebra g" = {(X, 0); X E g}, which is naturally isomorphic with g. If 9 is semisimple, then g' is the radical of T(g) and g" is a semisimple subalgebra of T(g). Let T(1)) be the subalgebra of T(g) consisting of elements (X, X') E 1) + 1). We define an involutive automorphism T( a) of T(g) by setting T (a)(X, X')
=
(a(X), a(X')).
Setting 1)'
=
g' (\ T(1)),
1)"
=
g" (\ T(1)),
a' a"
= =
I I
T(a) g', T (a) g",
We obtain a semidirect sum decomposition of a symmetric Lie algebra (T(g), T(1)), T(a)) into a symmetric ideal (g', 1)', a') and a symmetric subalgebra (g", 1)", a"). The corresponding construction for a symmetric space (G, H, a) is more geometric. Let
242
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
T(G) and T(H) be the tangent bundles of G and H respectively; the differential of the multiplication G X G + G defines a multiplication T(G) X T(G) + T(G) to make T(G) into a Lie group and T(H) into a Lie subgroup of T(G). Let T(a) be the differential of a. Then (T (G), T( H), T (a)) is a symmetric space and T(G)/T(H) = T(G/H) in a natural manner. For differential geometric properties of this construction and of similar constructions under more general circumstances, see Yano and Kobayashi
[IJ. Example 5.2. We shall exhibit a symmetric Lie algebra (9, with the following properties: (i) 9 is simple and is a vector space direct sum 91 + 90 with the relations
[90' 90J c 90'
[90' 91J c 91'
[90' 91J c 91'
[91, 91J C 90'
[91, 91J = 0,
[gl' 91J = 0;
(ii) The canonical decomposition 9
=
9+
9, a)
+ 91
m is given by
(iii) With respect to the KillingCartan form B of 9, the subspaces 91 and 91 are dual to each other and, moreover,
The following is an example of such a symmetric Lie algebra:
9 = sI(p
+ q; R),
90={(;1 ~J}, 91 {(~1 ~)}.
91={(~
;2)}.
=
where Xu, X 22 , X 12 , X 21 are matrices with p rows and p columns, q rows and q columns, p rows and q columns, q rows and p columns, respectively, and trace Xu + trace X 22 = o. There are twelve classes of symmetric Lie algebras of classical type and six symmetric Lie algebras of exceptional type possessing the three properties above. They can be also characterized by the
XI.
SYMMETRIC SPACES
243
properties that 9 is simple and that m contains a proper subspace invariant by ad (1») • For all these, see Berger [2J, KobayashiNagano [lJ, and Koh [lJ. We consider now symmetric Lie algebras (g, 1), a) such that 9 is simple and that ad (1») acts irreducibly on m. According to Berger's classification, there are 44 classes of such symmetric Lie algebras of classical type and 86 such symmetric Lie algebras of exceptional type (cf. Berger [2J). We give only one example. Example 5.3. Let 9 = sI(p + q; R) and
where Xu, X 12 , X 21 , X 22 are matrices of the sizes described in Example 5.2. We define an involutive automorphism a of 9 by setting
To obtain a symmetric Lie algebra (g, 1), a) with 9 solvable, it suffices to consider (g1 + g1' ~g1' a) with g1 solvable (cf. Example 2.1). The following construction gives another example. Example 5.4. Let 1) be a Lie algebra and 1)c its complexification, Le., l)c = 1) + i1). Choosing an arbitrary ideal 1)1 of 1), e.g., 1)1 = [1), 1)J, we set 9 = 1) + i1)l C 1)c. Let a: 9 + 9 be the complex conjugation, i.e., a(X + iY) = X  iY for X E 1) and Y E 1)1' Then (g, 1), a) is a symmetric Lie algebra. If 1) is solvable, so are 1)c and g.
6. Riemannian symmetric spaces A Riemannian manifold M is said to be Riemannian locally symmetric if it is affine locally syn1metric with respect to the Riemannian connection. Similarly, a Riemannian manifold M is said to be Riemannian (globally) symmetric if it is affine symmetric with respect to the Riemannian connection.
6.1. Let M be a Riemannian locally symmetric space. M, the symmet1)) Sx is isometric.
PROPOSITION
For each x
E
244
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Proof. Since SaJ is an affine transformation and its differential at x preserves the metric tensor at x, it suffices to prove the followIng.
Let M and M' be Riemannian manifolds andf: M + M' an affine mapping. If f*: TaJ(M) + Tf(aJ) (M') is isometric for some point x E M, thenf is isometric. Proof of Lemma. Let g and g' be the metric tensors for M and M', respectively. Let y EM and let T be a curve from y to x. Denote by the same letter T the parallel displacement along T. Set T' = f( T). For X, Y E Ty(M) we have LEMMA.
g;(y) (1* X,
1* Y)
= g~(aJ) (T' (f*X), T' (1* Y)) = g~(aJ) (j~ =
(TX),f* (T Y))
gaJ( TX, TY)
=
gy(X, Y).
QED. From Theorem 1.1 we obtain 6.2. A'Riemannian manifold is Riemannian locally symmetric if' and only if its 6urvature tensor field is parallel. THEOREM
From Theorem 1.2 we obtain 6.3. A complete, simply connected, Riemannian locally symmetric space is Riemannian symmetric. THEOREM
From Theorem 1.3 we obtain THEOREM
6.4.
A Riemannian symmetric space is complete.
6.5. Let M be a Riemannian symmetric space, G the largest connected group of isometries of M and H the isotropy subgroup of G at a point 0 of M. Let So be the symmetry of M at 0 and a the involutive automorphism of G defined by THEOREM
a(g) =
So
0
g
0
S;1
for g
E
G.
Let G(J be the closed subgroup of G consisting of elements fixed by a. Then (1) G is transitive on M so that M = GjH; (2) H is compact and lies between Gq and the identity component of Gq • Proof. (1) Although this may be obtained from Theorem 1.4, we shall prove it more directly. Let x, y E M and let T = xt , 0 < t < 4a, be a geodesic from x to y. (Since M is complete by Theorem 6.4,
XI.
SYMMETRIC SPACES
245
x andy can be joined by a geodesic by Theorem 4.1 of Chapter IV.) If we set then it is a Iparameter family of isometries by Proposition 6.1. Since io is the identity transformation, this Iparameter family of isometries it is contained in G. Clearly, fa maps x into y. (2) Let ~(M) be the group of isometries of M and ~o(M) the isotropy subgroup of :1(M) at o. By Corollary 4.8 of Chapter I or by Theorem "3.4 of Chapter VI, :1o(M) is compact. Being the identity component of I(M), G is closed in :1(M). Hence H = G tl ~o(M) is compact. The fact that H lies between Gfj and the identity component of Gfj can be proved exactly in the same way as Theorem 1.5. QED. For the study of Riemannian symmetric spaces, it is therefore natural to c~nsider symmetric spaces (G, H, 0') satisfying the condition that· adg (H) is compact. This means that the image of H under the adjoint representation of G is a compact subgroup of the group of linear transformations of g. The condition is satisfied if H is compact. The converse holds if the adjoint representation of G restricted to H is faithful, that is, if H meets with the center of G only at the identity element. In particular, if (G, H, 0') is effective, then adg (H) is compact if and only if H is compact. Assume that (G, H, 0') is a symmetric space with compact adg (H). Let 9 = 1) + m be the canonical decomposition. Since 1) and m are invariant by adg (H) (cf. Proposition 2.2) and since adg (H) is compact, 9 admits an adg (H)invariant inner product with respect to which 1) and m are perpendicular to each other. This inner product restricted to m induces a Ginvariant Riemannian metric on G/H. By Theorem 3.3, any Ginvariant Riemannian metric on GIH defines the canonical connection of GjH. Hence GjH is a Riemannian symmetric space. Remark. Proposition 6.1, Theorems 6.2, 6.3, and 6.4 are valid for the indefinite Riemannian case. Theorem 6.5, except the compactness of H, holds also in this case. But the proof of (1) has to be modified slightly; instead of a single geodesic from x toy, we
246
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
need a broken geodesic from x to y. I t is also possible to obtain (1) from Theorem 1.4. THEOREM 6.6. Let M be a simply connected Riemannian symmetric space and M = M o X M 1 X •.. X M k its de Rham decomposition, where M o is a Euclidean space and M 1 , ••• , M k are all irreducible. Then each factor M i is a Riemannian symmetric space. Proof. We first note that M is complete by Theorem 6.4 and the de Rham decomposition theorem (cf. Theorem 6.2 of Chapter IV) can be applied to M. It suffices to prove the following lemma. LEMMA. Let M 1 and M 2 be Riemannian manifolds. If their Riemannian direct product M] X M 2 is Riemannian symmetric, then both M 1 and M 2 are Riemannian symmetric. Proof of Lemma. Take arbitrary points 0 1 E M 1 and 02 E M 2 • Let s be the symmetry of M] X M 2 at (0 1 , O2 ). Let Xl E T o1 (M1 ) and set X = (Xb 0) E T(ol,o2)(M1 X M 2 ). Then the symmetry s maps the geodesic exp tX = (exp tX1 , 02) upon the geodesic exp (tX) = (exp (tX1 ), 02). It follows easily thats sends M 1 X {02} onto itself and induces a symmetry of M 1 at 0]. Hence, M 1 is Riemannian symmetric. Similarly for M 2 • QED.
7. Structure of orthogonal symlnetric Lie algebras Let (g, 1), a) be a symmetric Lie algebra. Consider the Lie a1gebra ad g (1») of linear endomorphisms of 9 consisting of ad X where X E 1). If the connected Lie group of linear transformations of 9 generated by adg (1») is compact, then (g, 1), a) is called an orthogonal symmetric Lie algebra. If (G, H, a) is a symmetric space such that H has a finite number of connected components and if (g, 1), a) is its symmetric Lie algebra, then adg (H) is compact if and only if (g, 1), a) is an orthogonal symmetric Lie algebra. Let 9 = 1) + m be the canonical decomposition of an orthogonal symmetric Lie algebra. Then 9 admits an adg (1»)invariant inner product with respect to which 1) and m are perpendicular. By an ad g (1»)invariant inner product, we mean an inner product ( , ) such that
([X, Y]), Z)
+ (Y,
[X, Z])
=
0
for X
E
1)
and
Thus, ad X is skewsymmetric with respect to (
Y, Z
,
E
g.
). The
XI.
SYMMETRIC SPACES
247
existence of such an inner product is obvious, since an inner product is adg (1))invariant if and only if it is invariant by the connected compact Lie group of linear transformations of 9 generated by adg (1)).
7.1. Let (g, 1), a) be an orthogonal symmetric Lie algebra and B the KillingCartan form of g. Let c be the center of g. If 1) () c = 0, then B is negativedefinite on 1). Proof. Let ( , ) be an adg (1))invariant inner product on 9 and fix an orthonormal basis of 9 with respect to ( , ). Then, for each X E 1), ad X is expressed by a skewsymmetric matrix (au (X) ) . We have PROPOSITION
i,j
i,i
and the equality holds if and only if X is in the center c.
QED.
U sing Proposition 7.1 we shall prove Let (g, 1), a) be an orthogonal symmetric Lie algebra such that 1) and the center of 9 have trivial intersection. Then (g, 1), a) is a direct sum of orthogonal symmetric Lie algebras (go, 1)0' a0) and (gl' 1)1' a1 ) such that (1) If go = 1)0 + mo is the canonical decomposition, then [m o, moJ = 0; (2) gl is semisimple. Proo£ We make a strong use of the following THEOREM
7.2.
The radical t ofa Lie algebra 9 is the orthogonal complement oj' [g, gJ with respect to the KillingCartan form B of g. LEMMA.
For the proof, see Jacobson [1; p. 73J or Bourbaki [1; p. 69J. Let t be the radical of 9 and let 9 = 1) + m be the canonical decomposition. Set m o = m () t. Let m1 be an adg (1))invariant subspace of m such that (vector space direct sum). (VYe have only to take the orthogonal complement of mo in m
248
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
with respect to an ad g (1»)invariant inner product.) Set 1)1
[m1, m1],
=
1)0 = {X E 1); B(X, 1)1) = O},
go
+ mo, 1)1 + mI·
1)0
=
gl =
Since B is negativedefinite on 1) by Proposition 7.1, 1)0 is a complement of 1)1 in 1). Hence 9 is a vector space direct sum of go and gl. From the definition of m o and m 1 we obtain (1) [1), mo] c mo,
(2) [1), mJ em]. Using 1)1 = [m 1, m1], the Jacobi identity and (2), we have
(3) [1), 1)1] c 1)1. From the invariance of Band (3) we obtain
B([1), 1)0]' 1)])
B(1)o, [1)], 1)])
c
B(1)o, 1)1)
B([m o, m1], 1») = B(m o, [m 1 , 1)])
c
B(t, [g, g]) = 0
c
=
O.
Hence,
(4) [1), 1)0] c 1)0. From (cf. Lemma)
and from Proposition 7.1, we obtain
(5) [m o, m1] = O. From 1)1 = [m 1, m1], the Jacobi identity and (5) we obtain (6) [m o, 1)1] = We shall prove
o.
(7) [1)0' m1] = O. Since [1)0' mt] c m1 by (2) and m1 (\ t = ml (\ t (\ m = m1 (\ mo = 0, it suffices to show [1)0' mt] c t. By the lemma, this will follow from B([1)o, m 1], g) = O. Since 1) and m are perpendicular to each other with respect to B (cf. Lemma in the proofofTheorem 3.4), we have only to prove B([1)o, mt], m) = O.
XI.
249
SYMMETRIC SPACES
This in turn follows from and
B([90' m1], m1)
C
B(90' em}, m1]) = B(90' 91) =
o.
From (1), (2), ... , (7) we see immediately that 9 is a (Lie algebra) direct sum of go and gl. From (cf. Lemma)
B([m o, mo], 9)
C
B([m o, 9], mo) c B([g, g], r)
=
0
and from Proposition 7.1, we obtain
(8) [mo, mo] = o. It remains to prove that gl is semisimple, i.e., go contains the radical r of g. Since r = (r (\ 9) + (r (\ m) = (r (\ 9) + mo, it suffices to prove r (\ 9 c 9o. But this follows from B(r (\ 9, 91) c B(r, 91) = B(r, [m 1, m1]) c B(r, [g, g]) = O. Hence (g, 9, a) is a direct sum of (go, 9o, 0'0) and (gl' 91" 0'1). To complete the proof, we remark that the connected Lie group of linear transformations of 9 generated by adg (9) is isomorphic to the direct product of the two Lie groups generated by adgo (9o) and by adg (91) and hence both (go, 9o, 0'0) and (gl' 91' 0'1) are orthogonal ~ymmetric Lie algebras. QED.
.
Remark. It is clear that in Theorem 7.2 if (g, 9, a) is effective, so are (go, 9o, 0'0) and (g1" 91' 0'1)· Consider an orthogonal symmetric Lie algebra (g, 9, a) with 9 semisimple. From Theorem 5.2 we see that (g, 9, a) is a direct sum of orthogonal symmetric Lie algebras of the following two kinds: (1) (g' + g', ~g', a), where g' is simple, ~g' = {(X, X) ; X E g'} and O'(X, Y) = (Y, X) for X, Y E g'; (2) (g, 9, a), where 9 is simple. Let (g' + g', ~g', a) be an orthogonal symmetric Lie algebra with g' simple. Let g' + g' = ~g' + m be the canonical decomposition. Then (1) ad (~g') is irreducible on m; (2) The KillingCartan form B of g' + g' is negativedefinite. PROPOSITION
7.3.
250
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Let m' be a subspace of m invariant by ad (8.g'). Then {X; (X, X) Em'} is an ideal of g'. Since g' is simple, it follows that either m' = 0 or m' = m, thus proving (1). Let B' be the KillingCartan form of g'. Then Proof.
B((X, Y), (X, Y)) = B'(X, X)
+ B'(Y, Y) for (X, Y)
E
g'
+ g'.
On the other hand, by Proposition 7.1 we have
o>
B((X, X), (X, X))
=
2B'(X, X) for (X, X)
E ~g',
I t follows that B' and B are negativedefinite.
X =F O.
QED.
7.4. Let (g, 1), 0') be an orthogonal symmetric Lie algebra with 9 simple. Let 9 = 1) + m be the canonical decomposition. Then (1) ad 1) is irreducible on m; (2) The KillingCartanform B of 9 is (negative or positive) definite on PROPOSITION
m. Proof. Choose an inner product ( , ) on m which is ad 1)invariant. Let f3 be the linear transformation of m defined by
(f3X, Y) = B(X, Y)
for X, Y
E
m.
Since B is symnletric and nondegenerate on m (cf. Lemma in the proof of Theorem 3.4), the eigenvalues of f3 are all real and nonzero. Let m = m 1 + ... + mk be the eigenspace decomposition. Then ml' ... , mk are mutually orthogonal with respect to ( ). Hence, if i =F j, then
o=
(m i , mj) = (f3m i , mj) = B( mi' mj),
which implies that m 1 , ••• , mk are mutually orthogonal with respect to B also. On the other hand, since Band ( , ) are invariant by ad 1), it follows that f3 commutes with ad 1) and that [1), m i ] c m i for every i. Hence, if i =F j, then
B([mi' mj], [mi' mj]) c B([m i , mj], 1)) = B(m i , [m j, 1)])
C
B(m i , mj ) = O.
Since B is negativedefinite on 1), it follows that [m i , mj ] = 0 for i =F j. Using this we verify easily that each [m i , mi ] + mi is an
XI.
251
SYMMETRIC SPACES
ideal of g. Since 9 is simple, we must have m = m 1, i.e., that fJ has only one eigenvalue. This proves that B is proportional to ( , ). Hence B is definite on m, thus proving (2). To prove (1), let m' be a subspace of m invariant by ad 1) and m" the orthogonal complement of m' in m with respect to ( , ) (and hence with respect also to B, since B is proportional to ( , )). Then m" is also invariant by ad 1). Hence B([m', m"], Em', m"]) c B([m', m"], 1))
= B(m', [m", 1)])
c B(m',
m")
=
O.
Since B is negativedefinite on 1), it follows that [m', m"] = O. Using this we verify easily that Em', m'] + m' is an ideal of g. Since 9 is simple, it follows that either m' = 0 or m' = m.
QED. As we can see from Example 5.2, it is essential that in Proposition 7.4 (g, 1), (1) is an orthogonal symmetric Lie algebra. The following proposition which holds for any symmetric Lie algebra may be considered as a converse to (1) of Propositions 7.3 and 7.4. 7.5. Let (g, 1), (1) be an ejfective symmetric Lie algebra and 9 = 1) + m the canonical decomposition. If ad 1) is irreducible on m, then one of the following three holds: (1) 9 = g' + g' with g' simple, 1) = ~g' and (1(X, Y) = (Y, X) for X, Y E g',(2) 9 is simple; (3) [m, m] = O. Proof. If 9 is semisimple, then we have either (1) or (2) by Theorem 5.2. It suffices therefore to prove that either 9 is semisimple or [m, m] = O. Assuming that 9 is not semisimple, let r be the radical of 9 and consider the repeated commutators of r: PROPOSITION
for i = 1, 2, .••• There exists a nonnegative integer k such that r k =F 0 but r k +1 = O. Since r is invariant by (1, so is r k • Hence rk = rk
(\
1)
+r
k
(\
m.
Since r is an ideal of g, so is r k and [1), r k (\ m] c r k ( \ m. Since ad 1) is irreducible on m, either r k (\ m = 0 or r k (\ m = m. If
r k n. m = 0, then r k is contained in 1), contradicting the assumption that (g, 1), 0') is effective. Hence, r k contains m. Since [rk, rkJ = 0, we have Em, mJ = O. QED.
Remark. Since [m, mJ + m is an ideal of g, it follows that 1) = Em, mJ in (1) and (2) of Proposition 7.5. Hence we may conclude that, for an effective symmetric Lie algebra (9, 1), 0'), ad ([m, mJ) is irreducible on m if and only if(l) or (2) of Proposition 7.5 holds. For a symmetric space (G, H, 0'), the irreducibility of ad ([ m, mJ) acting on m is precisely the irreducibility of the restricted linear holonomy group of the canonical connection on GIH (cf. Theorem 3.2). We shall therefore say that an effective symmetric Lie algebra (g, 1),0') is irreducible if ad ([m, mJ) is irreducible on m, i.e., if(l) or (2) of Proposition 7.5 holds. In general, an orthogonal symmetric Lie algebra (g, 1), 0') with 9 semisimple is said to be of compact type or noncompact type according as the KillingCartan form B of 9 is negativedefinite or positivedefinite on m. By Propositions 7.3, 7.4, and 7.5, every irreducible orthogonal symmetric Lie algebra is either of compact type or of noncompact type. From Theorem 5.2 we see also that every orthogonal symmetric Lie algebra (g, 1), 0') with 9 semisimple is a direct sum of an orthogonal symmetric Lie algebra of compact type and an orthogonal symmetric Lie algebra of noncompact type. (Of course, one of the factors might be trivial.) Since B is negativedefinite on 1) (cf. Proposition 7.1), (g, 1), 0') is of compact type if and only if B is negativedefinite on g. A Lie algebra 9 is said to be oj'compact type if its KillingCartan form B is negativedefinite on g. Hence, an orthogonal symmetric Lie algebra (g, 1), 0') is of compact type if and only if 9 is of compact type. It is known (cf. Hochschild [1; pp. 142144J) that a connected, semisimple Lie group G is compact if and only if its Lie algebra 9 is of compact type (see also Example 3.2 of Chapter X). This justifies the term "compact type." Example 7.1. We shall show that each point x of a Riemannian locally symmetric space M has a neighborhood which is isometric to a neighborhood of the origin of a certain Riemannian symmetric space GIH. Let m = Tx(M) and let 1) be the set of all linear endomorphisms U of m which, when extended to a derivation, map gx and R x upon 0 (the condition U . gx = 0 means that
XI.
SYMMETRIC SPACES
V is skewsymmetric). Then 9
253
+ 1)
can be made into a symmetric Lie algebra with a in the same manner as in Example 2.2. Let G be a simply connected Lie group with Lie algebra 9 and let H be the connected Lie subgroup corresponding to 1) so that (G, H, a) is a symmetric space for (g, 1), a). We show that K = ad (H) on m is compact so that (G, H, a) is a Riemannian symmetric space and (g, 1), a) is an orthogonal symmetric Lie algebra. By construction of g, ad (1») on m is nothing but the action of 1) on m. Thus it is sufficient to show that the group K of orthogonal transformations generated by 1) is compact. Consider the closure K of K in the orthogonal group on m and let 1) be its Lie algebra. If V E fj, then Ut = exp tV E K is the limit of a sequence of elements Un E K. Since Un maps Roo into itself, so does Ute This being the case for every t, we conclude that V· Roo = 0, i.e. V E 1). Thus K = K, proving our assertion that K = ad (H) on m is compact. Now in the argument at the end of §3, the linear isomorphism F is isometric and hence the affine isomorphism f is an isometry by the lemma for Proposition 6.1. =
m
8. Duality We shall first consider an arbitrary symmetric Lie algebra (g, 1), (f) which is not necessarily orthogonal. Let 9 = 1) + m be the canonical decomposition. If we denote by gC and 1)c the complexifications of 9 and 1), respectively, and by aCthe involutive automorphism of gC induced by a, then (gC, 1)c, aC) is a symmetric Lie algebra and (g, 1), 1 or ag ~ 1. Also, if A E O( 1, n), then from tASA = S we get det A = ±1. Let SO(I, n) = 0(1, n) n SL(n + 1; R). It is known that 0(1, n) has four connected components; a matrix A E 0(1, n) belongs to the identity component if and only if det A = 1 and ag > 1. In what follows we shall denote by G the identity component of 0(1, n), which is also the identity component of SO(I, n). The Lie algebra 9 of G is given by
0(1, n)
=
{X
E
gl(n
+ 1; R); tXS + SX =
Now the hypersurface Min Rn+l defined by F(x, x) disjoint union of two connected components:
M' = {xEM;xO > I}
and
O}. =
1 is the
Mil = {xEM;xO < I}.
XI.
SYMMETRIC SPACES
269
By Witt's theorem (cf. §3 of Chapter V) or by elementary linear algebra we know that 0(1, n) acts transitively on M and that G acts transitively on M'. We shall confine our attention to the action of G on M'. Let x EM', i.e. F(x, x) = 1 and XO > 1. The tangent space TrA M') is given, through the identification by parallel displacement in Rn+\ by the subspace of all vectors a E Rn+l such that F(x, a) = O. The restriction of F to Ta/M') is positivedefinite. Thus the form F restricted to the tangent space at each point of M' gives rise to a Riemannian metric on M' which is obviously invariant by G. For eo EM', the isotropy group is the subgroup H of G consisting of all matrices of the form where B
E
SO(n).
We have thus a diffeomorphism f of GjH onto M' which is compatible with the actions of G on GjH and on M'. If we define an involutive automorphism (J of G by (J(A) = SAS\ then the set offixed points coincides with H. Thus GjH is a symmetric space and the canonical decomposition of the Lie algebra is
0(1, n)
=
1)
+ m*,
where 1) is o(n) considered as a subalgebra of 0(1, n) and m* is the subspace of all matrices of the form
(~ ~),
where; is a (column) vector in Rn.
Identifying m* with Rn we see that ad (H) on m* is nothing but the action of SO(n) on Rn, that is,
Similarly, ad (1)) on m * is expressed by ad
(~ ~). (~ ~) = (;~ '(~~)).
270
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
The symmetric Lie morphic to the dual (cf. 10.1; in fact, we have o(n) im of o(n + 1)
(~ ~) i
algebra 0 (1, n) = 0 (n) + m* is iso§8) of o(n 1) = o(n) + m in Example the following isomorphism of the dual onto o(n) m*:
E
(~ ~~)
o(n)
E
zm
(~ ~)
+
+
(~ t~)
E
E
o(n),
m*.
The KillingCartan form gJ* of 0 ( 1, n) satisfies
where ~,
1]
ERn are identified with
the left hand side and with
(
0
~
0 t~) (0 ~ ( 0 ' 1]
_t~)
0
'
(0 1]
t1]) 0
E m* on
t1] )
OEm on the
right hand side. (This is a special case of what we saw in the proof of Theorem 8.4.) We may also verify directly that
gJ*(X, Y) = 2(n
1) (X, Y)
for X, Y E m*.
With the invariant metric arising from (X, Y) on m, G/H is a Riemannian symmetric space. The differentialf* off: G/H + M' at the origin maps the element of m* corresponding to ~ ERn upon
G)
E
T..(M') and is thus isometric. Since f is compatible
with the actions of G on G/H and on M' and since the metrics are invariant by G, we see thatfis an isometry of GIH onto M'. The curvature tensor R of G/H at the origin is expressed, under the identification of m* with Rn, by R(~, 1]) = ~ A 1]; thus GIH has constant sectional curvature 1. The geodesics of G/H starting from the origin are of the form
XI.
7T(exp t;), where;
exp
t;
E
m*
271
SYMMETRIC SPACES
=
Rn. For example, if
~
cosh t
sinh t o · . . 0
sinh t
cosh t o · . . 0
o
0
o
0
=
eI , then
=
so thatf(exp t;) = (cosh t, sinh t, 0, ... , 0) is a geodesic in M' starting from eo. All other geodesics starting from eo are obtained from this geodesic by the rotations in SO (n). Totally geodesic submanifolds in G/H can be found in the same manner as in Example 10.1. We can also proceed in the following manner (which is applicable to Example 10.1 as well). If m' is an arbitrary mdimensional subspace of m*, then [[ m', m'], m'] c m'. Indeed, identifying m * with R n, let CX I , • • • , CX m , CX m +l' • • • , CX n be a basis of m* such that {cx I , • • • , cx m } is a basis of m', then [CXi' cx i ] = cx i A cx i E o(n) and [[cx i , cx i ], cx k ] = (cx i A cx i ) cxk Em' for 1 < i, j, k < m. Thus [[m', m'], m'] c m'. Hence m' gives rise to an mdimensional complete totally geodesic submanifold of G/H through the origin. In M', any complete totally geodesic submanifold of dimension m through eo can be transformed by a rotation in SO(n) to the submanifold
{x
E
M'; xm +I
= ... =
xn
=
O}.
Recall that M' can be realized as the unit open ball Dn in Rn+l (see Theorem 3.2 of Chapter V). In this representation of M', the submanifold above is
Dm = {y = (yl, ... ,yn)
E
Dn;ym+I = ... =yn = O}.
Example 10.3. Oriented Grassmann manifold SO(p + q)/SO(P) X SO(q). Let Gp,q(R) be the Grassmann manifold of pplanes in RP+q whose differentiable structure is defined in the manner of Example 2.4 of Chapter IX only by replacing C by R throughout. .... Let Gp,q(R) denote the set of all oriented pplanes in Rp+q. In order to define its differentiable structure we modify the preceding
272
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
argument for Gp,q(R) as follows: Va is the set of all oriented pplanes S such that xa1 IS, ... , xap I S form an ordered basis of S compatible with the given orientation of S, where xl, ... , xp +q (instead of zl, ... , zp+q in Example 2.4 of Chapter IX) is the natural coordinate system in Rp+q. We call Gp,q(R) the oriented Grassmann manifold. It is the twofold covering of Gp,q(R) . ..., The rotation group G = SO(P + q) acts transitively on Gp,q(R). Let So be the pplane x p +q = ... = x p +q = 0 with the orientation given by the ordered coordinate system xl, ... , x'P. The isotropy group H at So consists of all matrices of the form ~
where V
E
SO(p)
and
V
E
SO(q).
Denoting this subgroup by SO(p) X SO(q), we have G[H = SO(P + q){SO(P) X SO(q) = Gp,q(R). We' see that G[H is a symmetric space if we define an involutive automorphism a of G by
a(A)
=
SASl,
where S
=
1 0) (o p
fq
The canonical decomposition of the Lie algebra: 9 given by
+ q), o(P) + o(q) = {(~ ;); U E o(P),
9
= o(p
~
=
m
={(:
and
~X);
•
V
E
=
1)
+ m is
O(q+
XEM(q,P;R)).
where M (q, p; R) denotes the vector space of all real matrices with q rows and p columns. The adjoint representation of H on m is given by
U ad ( 0
0) (0 tX ) (0 UtoXVl ). V X 0 = VXUl
The restriction to m of the inner product (A, B)
=
! trace AB
XI.
273
SYMMETRIC SPACES
on 0 (p + q) is, of course, invariant by ad (H) and induces an invariant metric on GIH. Discussions on the curvature tensor, geodesics, etc. are similar to Example 10.1 (See also Leichtweiss [lJ, Wong [4J.) We shall discuss another special case p = 2 and q = n in Example 10.6. Example 10.4. SOO(p, q)fSO(p) X SO(q). Here SOO(p, q) denotes the identity component of the orthogonal group O(p, q) for the bilinear form on Rp+q: p p+q
F(x,y)
=

L XjJi + L
j=1
j=p+1
x~j.
The involutive autolnorphism (J is defined in the same way as in Example 10.3. Discussions of this symmetric space are similar to the special case p = 1, q = n, which was treated in Example 10.2. Example 10.5 Complex projective space Pn(C). Before we discuss Pn(C) from the point of view of Hermitian symmetric spaces, we shall reconsider the complex structure and the metric in P n( C) in Examples 2.4 and 6.3 of Chapter IX from a more geometric viewpoint. In Cn+l with the natural basis eo, e1 , ••• ,en we consider
h(z, w) =
n
L
ZkWk
and
g'(z, w) = Re h(z, w),
k=O
which are related by h(z, w) = g'(z, w) + ig'(z, iw). In fact, g'(z, w) is nothing but the usual inner product when we identify Cn+l with R2n+2. For the unit sphere S2n+1 = {z E cn+l; h(z, z) = I} the tangent space T z (S2n+l) at each point z can be identified (through parallel displacement in Cn+ 1 ) with {w E Cn+ 1 ; g' (Z, W) = O}. Let T~ be the orthogonal complement of the vector iz E T z (S2n+l) , that is , T~
= {w
E cn+l; g'(z, w)
=
a
and
g'(iz, w) =
a}.
When we consider S2n+l as a principal fibre bundle over Pn(C) with structure group Sl as in Example 2.4 of Chapter IX, there is a connection such that T~ is the horizontal subspace at z. The natural projection 1T of S2 n+l onto Pn(C) induces a linear isomorphism of T~ onto Tp(Pn(C)), where p = 1T(Z). The Riemannian metric g' on S2n+l (that is, the metric on S2n+l induced by g') is invariant by the structure group. Thus we may define a
274
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Riemannian metric gp(X, Y)
=
~c g~(X',
g on P n (C)
by
Y'),
c>
°
fixed,
where z is any point of S2n+l with 7T(Z) = P and X', Y' are the vectors in T~ such that 7T* (X') = X and 7T* (Y') = Y. On the other hand, the complex structures J': w ~ iw in the subspaces T~, Z E S2n+l, are compatible with the action of S1, that is, J'(e i8w) = !i8(J'W). It follows that we may define a complex structure J in each tangent space Tp(P n(C)) by '"
J(X) = 7T(J'X'),
X
E
Tp(Pn(C)),
where 7T* (X') = X. Indeed, the almost complex structure so defined on P n(C) coincides with the usual complex structure of Pn(C). Since each of them is invariant by the group l!.(n + 1) acting transitively on Pn(C), it is sufficient to show that J at Po = 7T( eo) coincides with the usual complex structure at Po induced by a complexcbartp: (ZO, z1, ... , zn) E Uo ~ (t1, ... , tn) E Cn,where (ZO, ••• ,zn) are homogeneous coordinates, Uo = {z; ZO =F O}, and (tl, ... , tn ) are inhomogeneous coordinates, i.e., tk = Zk j ZO f~r 1 < k < n. We know that T: o is spanned by e1, ••• , em ie 1 , ••• ,ien as a real vector snace. Now 7T *(ek ) is the tangent vector at Po of the curve
17"((e o
+ sek)jVl + S2)
=
(1,0, ... , 0, s, 0, ... ,0),
where s is the (k + l)st homogeneous coordinate; we see that p* (7T* (e k )) is the kth element of the natural basis ofCn. Similarly, 17" * (ie k ) is the tangent vector at Po of the curve (1, 0, .: .. , 0, is, 0, ... ,0), and (P*(7T*_(ie k)) = ip*(7T*(ek)). Since J7T*(e k) = 17"* (iek) by definition", of J at Po for 1 < k < n, we conclude that p*J = ip*, that is, J is nothing but the complex structure induced by the chart p from the usual complex structure of cn. The metric is Hermitian with respect to 1, since J' is Hermitian with respect to g' on T~ for each z E S2n+l. Observe also that both g and j are invariant by the action of U( n + 1) on P n (C). At Po the expression of the metric in Theorem 7.8 (1) of Chapter IX reduces to (4jc) ~:=1 dta. dla. and coincides with our metric By
g
i.
XI.
275
SYMMETRIC SPACES
invariance of both metrics by U(n + 1), we conclude that our metric i coincides with that in Theorem 7.8 (1) of Chapter IX. For the action of U(n + 1) on Pn(C), the isotropy group at Po = 7T(eo) consists of all matrices of the form where B Denoting this subgroup by U(I)
E
U(n).
U(n) we have a diffeo1)/U(l) x U(n) onto Pn(C). The action of X
morphismfof U(n + U( n + 1) is not effective; if we take the almost effective action of SU(n + 1) on Pn(C), the isotropy group at Po is the subgroup S( U( 1) X U(n)) consisting of all matrices in U( 1) X U(n) with determinant 1. We shall, however, proceed with the representation G/H with G = U(n + 1) and H = U(l) X U(n). We define an involutive automorphism a of U(n + 1) by a(A)
SASl,
=
where S =
1 0) (o . In
The subgroup of all fixed elements is U(I) X U(n). Thus G/H is a symmetric space. The canonical decomposition of the Lie algebra is given by 9 = 1) + m, where 9 = u(n l)
= u(l)
and m =
{C
+ 1): all skewHermitian matrices of degree n + 1,
+ u(n)
= {(
:~ ); ~
E
~ ~), A + X =
0, B
E
u(n)},
e+
In the following discussion we shall identify m with Cn in this obvious manner. The adjoint action of H on m is expressed by ei8
ad
(
o
; E
m.
276
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
The inner product (A, B) = is ad (U (n

l trace AB = l trace AtB,
A, B
+ 1)) invariant; its restriction to
E
u(n
+ 1)
m is given by
where h(;, 'YJ) is defined in Cn just as in the beginning. We shall take
as an inner product on m invariant by ad H and get an invariant Riemannian metric g on G jH. We also consider an ad (H) invariant complex structure on m given by ; * i;, that is,
t~) (0 0 ( *
;
0
i;
i
t~ ) 0
Since GjH is symmetric, we know that the induced Ginvariant almost complex structure Jon G{H is integrable (see Proposition 9.1). We also see that g is Hermitian with respect to J, since it is so on m. We now wish to show that the invariant metric g and the invariant complex structure J on G{H correspond by f to g and J ..., on Pn(C). Since g and J are invariant by U(n + 1), it is sufficient to check our assertion for the differentialf* off at the origin of G{H. We see thatf* maps; E m = Cn upon

where (
~ ) belongs to T~o(S2"+1). Thus f*
and the complex structures.
preserves the metrics
XI.
277
SYMMETRIC SPACES
The curvature tensor R at the origin of G/ H is computed as follows:
R(;, 'YJ)'
;tig  'YJt~, + ,tij;  ,t~'YJ = h(', 'YJ);  h(', ;)'YJ + h(;, 'YJ)'  h( 'YJ, ;), =
=
c
4 {; A 'YJ
+ J; AJ'YJ + 2g(;, J'YJ)J}"
where g = (4/c) Re hand ; A 'YJ is the endomorphism such that (; A 'YJ) . , = g(', 'YJ);  g(', ;) 'YJ. It follows that the holomorphic sectional curvature of G/H is equal to the constant c (see §7 of Chapter IX). To find geodesics in Pn(C), let; E m be the element corresponding to e1 E Cn. Then cos t sin t
o
o so that the curve (cos t, sin t, 0, ... , 0) in homogeneous coordinates is a geodesic starting from Po = (1, 0, ... , 0). We note that it is contained in the complex projective line PI defined by zn = ... = zn = O. All other geodesics starting from Po are obtained by transforming this geodesic by U ( 1) X U (n). In order to find totally geodesic complex submanifolds of G/ H through the origin, we have to find a Jinvariant subspace m' of m such that [Em', m'], m'] em'. But, actually, any Jinvariant subspace m' of m satisfies this bracket condition. Indeed, if m' corresponds to a complex subspace of cn, let CX I , • • • , CXm, CX m+l' ••• , CXn be a basis of mover C such that CXI' ••• , CXm is a basis of m' over C. Then CX I , ••• , CXm, J CX I , ~ •• , J CXm form a basis of m' over R. We see that [[CXi'
cx;], CXk] = R(CXi'
CX;) • CXk
C
=

4 {CXi
A cx;
+ JCXi
A
Jcx;
+ 2g( CX i' JCXj)J}
. CXk
278
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
belongs to m' for I < i,}, k < m. Similarly, the triple brackets among aI' ... , am, Ja l , . • • , Jam belong to m', proving our assertion. If m' is the complex subspace spanned by el , . . . , em of m = Cn, then g' = m' + [m', m'] consists of all matrices of the form A' where A' E u(m + 1). (
0) 00'
The subgroup G' generated by g' is the subgroup of U(n consisting of all rnatrices of the form
0)
A' (
o
I n
where A'
E
U(m
+ 1)
+ 1).
'
m
Hence the corresponding totally geodesic complex submanifold in Pn(C) is Pm(C) = {(ZO, ... , zn) E Pn(C); zm+l = ... = zn = O}. It follows that any complete totally geodesic complex submanifold is an mdimensional complex projective subspace, where m is the complex dimension. Example 10.6 Complex quadric Qnl(C). In Pn(C) with homogeneous coordinates zO, z1, ... , zn the complex quadric Qnl(C) is a complex hypersurface defined by the equation
(ZO)2
+ (Zl)2 + ... + (zn)2
=
o.
Let g be the FubiniStudy metric with holomorphic sectional curvature c for Pn(C). Its restriction g to Qnl(C) is a Kaehler metric (Example 6.7 and §9 of Chapter IX). We shall represent Qnl(C) as a Hermitian symmetric space. First we fix the following notations. In Cn+l with the natural basis eo, el , . . . , en we denote by H (z, w) the complex bilinear form defined by n
H(z, w) =
.L ZkW \
Z
=
(Zk), W
= (w 'C ).
k=O
Then Qnl(C) = {7T(Z); Z E Cn+l  {O}, H(z, z) = O}, where 7T is the natural projection of Cn+l  {O} onto Pn(C). The unit vector
flo
=
(eo
+ iel )/V2 E S2n+l
satisfies H(flo, flo) = O. We let qo = 7T(flo) E Qnl(C). The tangent space T po (S2n+l) has a basis consisting of ie o  el , ieo + el ,
XI.
279
SYMMETRIC SPACES
+ ie e2, .• en' 'ie2, ... , ien, where the first vector is V2i{3oo The subspace T po defined in Example 10.5 is spanned ie o + e eo + ie e2, ... , en, ie2' ... , ie n. Let T;o be the subspace spanned eo
l,
0
,
l,
l ,
by e2' . en' ie2' .• linear isomorphism 0
•
,
0
,
ie n. The natural projection 1T induces a
1T*: T po ~ Tqo(Pn(C)) and a linear isomorphism
1T*: T;o ~ T qo ( QnI(C))
0
To prove the second assertion, it suffices to show that 1T*( T;o) is contained in T qo ( QnI(C)), since both spaces have dimension 2n  2. For each k, 2 < k < n, the curves in Pn(C)
at = (cos t, i, 0, ... , 0, sin t, 0,
0
••
,
0)
and
ht = (1, i cos t, 0, ... , 0, i sin t, 0, ... , 0), where sin t and i sin t appear as the (k + l)st homogeneous coordinates, lie in QnI(C). Thus their tangent vectors at t =
°
do
= 1T *(e k )
and
bo = 1T * (ie k )
are contained in T qo ( QnI(C)), proving our assertion. For an arbitrary point q E QnI(C), we take a point z E S2n+1 such that H(z, z) = and 1T(Z) = q. We may write z uniquely in the form z = x + iy, where x and yare orthogonal real vectors in Cn+l of length I/V2. In the space T~ defined in Example 10.5 let T; be the orthogonal complement (with respect to g') of the subspace spanned by ix + y and x + iy. We may then prove that 7r induces a linear isomorphism of T; onto T q ( QnI(C)); this can be done either directly as in the case where z = (eo + ie l ) /V2 or by using the transitive action of SO(n + 1) on QnI(C), which we discuss in the following paragraph. The group SO(n + 1), considered as a subgroup of U(n + 1), acts on Pn(C) and leaves QnI(C) invariant, since H(Az, Aw) = H(z, w) for alIA E SO(n + 1) andz,w E Cn+I.Actually,SO(n + 1) is transitive on QnI(C). In fact, for any z E S2n+1 such that H(z, z) = 0, we want to find A E SO(n + 1) such that 1T(Af3o) = 7r( z). In the unique representation z = x + iy we may normalize x andy and obtain two orthonormal real vectors (xo, (Xl such that
°
280
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
+ i(XI)
z). By extending (Xo, (Xl to an ordered orthonormal basis (Xo, (Xl' ••• , (Xn in Rn+l we get A E SO(n + 1) with these column vectors. We have then 7T(A({3o)) = 7T(Z). For the action of G = SO(n + 1) on QnI(C), the isotropy group H at qo turns out to be the subgroup SO(2) X SO(n  1) of SO(n + 1) consisting of all matrices of the form 7T ((Xo
=
7T (
where B
E
SO(n  1).
We have thus a diffeomorphismj: GjH + QnI(C). From Example 10.3 we know already that G{H is a symmetric space. The canonical decomposition of the Lie algebra is g
=
1)
+ m,
o
A
0
A
0
0
o
0
B
where g 1)
=
=
m =
+ 1)
o(n
+ o(n
0(2)
 1)
o
0
_t;
0
0
_t'YJ
,B
E
o(n  1)
;;, 'YJ are column vectors in
Rnl •
Identifying (;, 'YJ) E Rnl + Rnl with the matrix above in m, we see that ad (H) on m is expressed by ad (R(O) x B) . (;, 'YJ) = (B;, B'YJ)R(O), where the right hand side is a matrix product. Similarly, ad (1)) on
XI.
281
SYMMETRIC SPACES
m is expressed by
ad
0
It
0
It
0
0
0
0
B
• (~, 'YJ)
(B~, B'YJ) + (~, 'YJ) (
=
0 It
The differentialf* at the origin of G/H maps (;, "rJ)
(~L(7T(expt(~, 'YJ) • flo))
=
:A). E
m upon
7T*((~, 'YJ) • flo) o
= 17*
where the vector
o o ; + i'YJ
0
Tqo(Qnl(C)),
E
; + i'YJ belongs to T;o.
We define an inner product g on m
X
m by
where (~, ;') is the standard inner product in Rnl. We also define a complex structure J on m by
Both g and J are invariant by ad (H) and give rise to a structure of a Hermitian symmetric space on G/H. It is now an easy matter to verify thatfis an isomorphism ofG/H onto Qnl(C) as Kaehler manifolds. The curvature tensor R at the origin is given as follows:
o
It
0
o o o
0
It
where It = (;', 'YJ)  (;, 'YJ') and B = ~ {~
,
B
A ;'
+ 'YJ
A
'YJ'}, the
282
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
endomorphism ; A ;' being defined by using the standard inner product in Rnl. If we compute the Ricci tensor S by using the formula in Proposition 4.5 (2) of Chapter IX, we obtain _ (n  l)c S 2· g.
Thus Qnl(C) is an Einstein space. For n > 3, it can be shown that ad (H) on m is irreducible and therefore Qnl(C) is irreducible. For n = 3 we see that 0
0
a
b
0
0
b
a
a
b
0
0
b
a
0
0
m1 =
and
m2=
0
0
c
d
0
0
d
c
c
d
0
0
d
c
0
0
are invariant by ad (H). Indeed, Q2(C) can be shown to be isomorphic to P1(C) X P1(C) as Kaehler manifolds. Example 10.7. Complex hyperbolic space. In Cn+l with the standard basis eo, e1 , ••• , en we consider a Hermitian form F defined by n
F(z, w)
zOwO
=
+L
ZkWk.
k=l
Let U(I, n) = {A
E
GL(n
= {A
E
GL(n
+ 1; C); F(Az, Aw) = F(z, w), z, W E cn+l} + 1; C); tASA = S},
where
S=
1 (
o
XI.
283
SYMMETRIC SPACES
It is known that U(I, n) is connected. Its Lie algebra is
u(l, n) = {X
=
E
{C;
gI(n
+ 1; C); tXS + SX =
~); A
E
R,
~ en, B E
E
O}
u(n)}
Let M be a real hypersurface in Cn+1 defined by F(z, z) = l. We note that the group U(I, n) acts transitively on M. On the other hand, the group 8 1 = {e i6 } acts freely on M by z + ei8 z; let M' be the base manifold of the principal fibre bundle M with group 8 1 • For z EM, the tangent space Tz(M) is represented by
{w In particular, iz defined by T~
E
E
Cn+1; Re F(z, w)
Tz(M). Let
= {w = {w
E
E
T~
= O}.
be the subspace of Tz(M)
T~(M);
Re F(iz, w) = O} Cn+1; F(z, w) = O}.
We observe that the restriction of F to T~ is positivedefinite. We have a connection in M such that T~, z E M, are the horizontal subspaces. The natural projection n of M onto M' induces a linear isomorphism of T~ onto T 1T (z) (M'). The complex structures w + iw on T~, z EM, are compatible with the action of Sl and induce an almost complex structure J' on M' such that n *i = J'n *. (We shall later see that J' is integrable so that M' is a complex manifold.) Fixing any negative constant c, we define in each tangent space Tp(M'), P EM', an inner product g' by g'(X, Y)
=  ~ Re (F(X', Y')), c
where X', Y' E T~ and n(z) = P, n*(X') = X, n*(Y') = Y. The metric g' is Hermitian with respect to J'. Now the group G = U(I, n) acts transitively on M and hence on M'. Both g' and J' are invariant by G. The isotropy group at Po = n(eo) is the subgroup H = U(I) X U(n) of U(I, n) consisting of all matrices of the form
(
ei6
0)
o
B '
B
E
U(n).
284
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Thus we have a diffeomorphism f of G/H onto M'. The action of U(I, n) is not effective; if we take SU(1, n) = {A E U(1, n); det A = I}, which acts almost effectively on M', then the isotropy group at Po is
S(U(1)
X
U(n)) = {A
E
U(1)
If we take the conjugation by S
=
X
U(n); detA = I}.
1 0) (o In
as an involutive
automorphism of G, then H is precisely the subgroup of all fixed elements. Thus G/H is a symmetric space. The canonical decomposition of the Lie algebra is 9
=
1)
+ m,
where 9 = u( 1, n)
and
Identifying; expressed by
E
C n with the above matrix in m, ad (H) on m is
The complex structure; + i; on m = Cn is invariant by ad (H) and induces an invariant complex structure on the symmetric space G{H. The inner product on m given by
4 g(;, 'fJ) =   Re h(;, 'fJ), c where h(;, 'fJ) is the standard Hermitian inner product on m = Cn, is invariant by ad (H) and induces an invariant Hermitian metric on G/H. (The inner product g on m is, of course, related to the restriction of the KillingCartan form of u (1, n).) The differential f* at the origin of G{H maps ; E m upon
XI.
7T*( ~ ) ETpo(M'),
285
SYMMETRIC SPACES
where (
~ )E T;o' Thus g and J
on GIH cor
respond to g' and J' on M' by the diffeomorphism fIt follows that J' is actually integrable and! is an isomorphism of GIH onto M' as Kaehler manifolds. The curvature tensor R at the origin is expressed on m by
which shows that G/H has constant holomorphic curvature equal to c. If we take ; = el in Cn, then cosh t sinh t
o
o is a geodesic in M'. All geodesics of M' are obtained from this geodesic by the transformations in U(1, n). Any Jinvariant subspace m' of real dimension 2m satisfies the condition
[Em', m'], m'] c m' and gives rise to a totally geodesic complex submanifold of complex dimension m in M'. (The situation is analogous to that of Pn(C) = SU(n + 1)/S(U(I) x U(n)), which is the dual of our space SU(l, n)/S(U(I) x U(n)).) We may identify M' with the open unit ball in Cn:
Dn(C) =
{W E cn;
J.l w'w' < I}
by the mapping 7T(ZO,
z\ ...
1
Zn) E
M'
~
(wI, ... ,
n W ) E
Dn(C),
where F(z, z) = 1 and w k = zkfzo (note that F(z, z) implies zOzo = 1 + ~:=1 ZkZk > 0).
1
286
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Example 10.8. Complex Grassmann manifold Gp,q(C). We take up Example 6.4 of Chapter IX from the point of view of symmetric spaces, generalizing Example 10.5. Let M' = M' (p + q, p; C) be the space of complex matrices Z with p + q rows and p columns such that tZZ = I p (or, equivalently, the p column vectors are orthonormal with respect to the standard inner product in Cp+q). The group U (p) acts freely on M' on the right: Z + ZB, where B E U(p). We may consider Gp,q(C) = M as the base space of the principal fibre bundle M' with group U(P). On the other hand, the group U (p + q) acts on M' to the left: Z + AZ, where A E U (p + q). This action is transitive and induces a transitive action of U (p + q) on M. Let e1 , ••• , ep +q be the natural basis in Cp+q and let Zo be the matrix with e1 , ••• , ep as the column vectors. Then Zo EM'. The isotropy group at Po = 7T' (Zo) E M for the action of U (p + q) is then
{A
E
U(p
+ q);
AZo = ZoB
E
U(P)}
U(P)
and
for some B where B
E
C E U(q)).
which we shall denote by U(p) X U(q). Thus we have M = U(p + q)fU(p) X U(q). For each point Z EM', the tangent space Tz(M') is the set of all matrices with p + q rows and p columns such that t WZ + tZW = O. In Tz(M') we have an inner product g(W1 , W 2 ) = Re trace tW2 W 1 • Let T; = {ZA; A E u(P)} c Tz(M') and let T~ be the orthogonal complement of T; in Tz(M') with respect to g. The subspace T~ admits a complex structure W + iW. We see that 7T': M' + M induces a linear isomorphism of T~ onto T 1f 
M,
then the isotropy group at Po for the action of U (p, q) on M turns out to be U(P) x U(q)
{( :
~); B
E
U (p), C E U (q) }.
The involutive automorphism 0': O'(A) = SASl of U(p, q) defines U(P, q)/U(p) X U(q) = M as a symmetric space. The natural Hermitian structure can be defined in the manner analogous to Example 10.8. We shall indicate how we can identify our space M with the
space Dv,o defined in Example 6.5. If Z'
tZoZo = Lp
(
~: )
EM', then
+ tZ1Z1.
Hence t ZoZo is positivedefinite. This implies that if Z' EM', then Zo is nonsingular. Let Z = ZlZOl. We show that the p X P Hermitian matrix 1'11  tZZ is positivedefinite so that Z E D'11,q: 1p

tZZ = 1p  tZol tZ]ZlZOl = tZOl(tZoZo  tZ1Z1)ZOl = tZolZOl > O.
XI.
If ( Zo )B ZI1
SYMMETRIC SPACES
289
o
= ( W ) with BE U(p) in M', it follows that Z1Z 01 = WI
W 1 W 0 • Thus the mapping Z' =
(Z ) Z:
+
Zl Z 01 induces a map
ping ofMinto Dp,q. This is onetoone. To show that it is surjective, let Z E D p ,q. Then there is a p x p positivedefinite Hermitian matrix, say, P such that 1p  tZZ = P2. We let Zo = pl and
ZI = ZPl. Then Z' = ( Zo ) E M' as can be easily verified, and Z = ZIZ 01 • ZI The action of U(p + q) on M, when transferred to Dp,q by this identification, is what we defined in Example 6.5.
11. An outline of the classification theory By Theorem 6.6 the classification of the simply connected Riemannian symmetric spaces is reduced to that of the irreducible ones. Similarly, by Theorems 5.2 and 7.2 the classification of the effective orthogonal symmetric Lie algebras is reduced to that of the irreducible ones. We shall say that two Riemannian manifolds M and M' with metric tensors g and g', respectively, are homothetic to each other if there exists a diffeomorphism p of M onto M' such that p*g' = c2g, where c is a positive constant. Then we have
11.1. The homotheticequivalence classes of simply connected irreducible Riemannian symmetric spaces M are in onetoone correspondence with the isomorphism classes of effective irreducible orthogonal symmetric Lie algebras (g, ~, 0'), where the correspondence is given by Theorem 6.5. Proof. The proofis straightforward. Only the following remark will suffice. Given (g,~, 0') we construct as in §2 an almost effective symmetric space (G, H, 0'), where G is simply connected and H is connected so that M = G{H is simply connected. Since (g, ~, a) is irreducible, an invariant Riemannian metric on G/H is unique up to a constant factor. Let N be the discrete normal subgroup of G consisting of those elements which act trivially on M = G/H. Denoting G/ Nand H{N anew by G and H, respectively, we obtain an effective symmetric space (G, H, a) where M = G/H PROPOSITION
290
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
is simply connected. We claim that G is the largest connected group ofisometries of M. Let G1 = .3°(M) be the largest connected group of isometries of M. The symmetry of M at the origin induces an involutive automorphism 0'1 of G1 which extends the automorphism 0' of G (cf. Theorem 1.5). In· the canonical decompositions 9 = 9 + m and 91 = 91 + m1, we have m = mI· Hence (cf. the remark following Proposition 7.5) we get 9 = em, m] = [m 1, m1]
This implies 9
=
91 and G
=
G1 •
=
91.
QED.
By virtue of the duality theorem (cf. Theorem 8.5) we may further restrict our attention to the effective symmetric spaces (G, H, 0') with irreducible orthogonal symmetric Lie algebras (9, 9, 0') of noncompact type. If (9, 9, 0') is such a symmetric Lie algebra, then it belongs to one of the following two classes in Theorem 8.5: (III) (9, 9, 0') where 9 is a simple Lie algebra of noncompact type which does not admit a compatible complex structure; (IV) (9, 9, 0') where 9 is a simple Lie algebra of compact type, 9 = 9c is the complexification of 9 and 0' is the complex conjugation in 9 with respect to 9. In either of these cases, 9 is simple and is of noncompact type (cf. Appendix 9). If (G, H, 0') is an effective symmetric space with irreducible orthogonal symmetric Lie algebra (9, 9, a) of noncompact type, then G has trivial center (cf. Theorem 8.4 of Chapter VIII and Theorem 8.6) and H is a maximal compact subgroup of G (cf. Corollary 9.3 of Chapter VIII and Theorem 8.6). If (G1, HI' 0'1) is another effective symmetric space with irreducible orthogonal symmetric Lie algebra (91' 91' 0'1) of noncompact type and if 91 = 9, then G1 = G since both G and G1 have trivial center, and HI is conjugate to H in G by Corollary 9.3 of Chapter VIII. In the canonical decompositions 9 = 9 + m and 91 = 91 + m1, m and m1 are the orthogonal complements of 9 and 91' respectively, in 9 = 91 with respect to the KillingCartan form of 9 = 91. We can now conclude that, for every simple Lie algebra 9 of noncompact type, there is (up to an isomorphism) at most one irreducible orthogonal symmetric Lie algebra (g, 9, 0') of noncompact type. Assuming the result ofWeyl on the existence
XI.
SYMMETRIC SPACES'
291
of compact real form, we shall show that every simple Lie algebra 9 of noncompact type gives rise to an irreducible orthogonal symmetric Lie algebra (g, 1), 0') of noncompact type. A theorem of Weyl states that if 9 is a complex simple Lie algebra, there is a real simple Lie algebra u ofcompact type such that 9 is isomorphic to the complexijication u Cof u. Such a u is called a compact real form ofg (cf. Hochschild [1; p.167J). First we consider the case where 9 is a simple Lie algebra which admits a compatible complex structure. Let' 1) be a compact real form of g. Then (g, 1), 0'), where 0' is the complex conjugation with respect to 1), is an irreducible orthogonal symmetric Lie algebra of noncompact type. \ We consider now the case where 9 is a simple Lie algebra of noncompact type which does not admit a compatible complex structure. Let gC be the complexification of g; it is a simple Lie algebra of noncompact type. Let u be a compact real form of gC. Then (gC, u, 0'1), where 0'1 is the complex ~onjugation with respect to u, is an irreducible orthogonal symmetric Lie algebra of noncompact type. We denote by (G l , U, al) an effective symmetric space with symmetric Lie algebra (gC, u, a1 ). Denote by (f... the complex conjugation of gC with respect to g. Then (f... is an involutive automorphism of gC (as a real Lie algebra) and induces an involutive automorphism (f... of the simply connected Lie group G1 .... with Lie algebra gC. Since Gl is the quotient group of G1 by its center, (f... induces an involutive automorphism (f... ofGl • By Theorem 9.4 of Chapter VIII and Theorem 8.6, there exists a maximal compact subgroup K of G1 which is invariant by (f.... By Corollary 9.3 of Chapter VIII and Theorem 8.6, K is conjugate to U in G1 • It follows that f is also a compact real form of gC. Ifwe denote by 0' the complex conjugation of gC with respect to f, then (gC, f, 0') is an irreducible orthogonal symmetric Lie algebra of noncompact type. Since K is invariant by (f..., in the canonical decomposition gC = f + if associated with (gC, f, 0') both f and if are invariant by (f.... Since 9 consists of those elements of gC which are left fixed by (f..., we obtain 9 = (g ( l f) + (g () if).

If we set 1) = 9 ( l f and denote also by 0' the restriction of 0' to g, then we obtain an irreducible orthogonal symmetric Lie algebra (g, 1), 0') of noncompact type.
292
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
We have thus established the fact that the irreducible orthogonal symmetric Lie algebras (g, 9, 0') of noncompact type are in onetoone correspondence with the real simple Lie algebras 9 of noncompact type. The classification of real simple Lie algebras was first achieved by E. Cartan [17] in 1914. Using the theory of symmetric spaces, he gave a more systematic classification in 1929 (cf. E. Cartan [16]). A more algebraic and systematic proof was obtained by Gantmacher [1], [2]. A further simplification was achieved by Murakami [6] and Wallach [1], [2] independently along the same line. Using results of Satake [2], Araki [1] gave another systematic method of classification. The main difference between the method of Araki and that of MurakamiWallach lies in different choices of Cartan subalgebras. The results in Chapter XI on Riemannian symmetric spaces are largely due to E. Cartan [7], [15]. Generalizations to affine symmetric spaces are due to Nomizu [2]. For more details and references on Riemannian symmetric spaces, see He1gason [2]. For affine symmetric spaces, see Berger [2] and Koh [1].
CHAPTER XII
Characteristic Classes 1. Weil homomorphism Let G be a Lie group with Lie algebra g. Let Jk(G) be the set of symmetric multilinear mappings J: 9 X • • • X 9 + R such that J((ad a)t1, ... , (ad a)t k ) = J(t1' •.• ,tk ) for a E G and t1 , ••• , t k E g. A multilinear mapping J satisfying the condition above is said to be invariant (by G). Obviously, Jk(G) is a vector space over R. We set 00
J (G) = 2, Jk (G) • k=O
For J E Jk(G) and g
Jg( t 1,
•• • ,
E
Jl,(G) , we defineJg
E
J1c+l,(G) by
t k +l,)
(k
1
+ l)! ~J(ta(l" •.. , taCk»)g(taCk+1h • • • , taCk+Z»,
where the summation is taken over all permutations (J of (1, ... , k + l). Extending this multiplication to J(G) in a natural manner, we make J (G) into a commutative algebra over R. Let P be a principal fibre bundle over a manifold M with group G and projection p. Our immediate objective is to define a certain homomorphism of the algebra J (G) into the cohomology algebra H *(M; R). We choose a connection in the bundle P. Let OJ be its connection form and 0 its curvature form. For eachJ E Jk( G), letJ(O) be the 2kform on P defined by
J(O) (Xl' . • · , X 2k ) 1 (2k) 1 ~ 8 aJ(0(XaC l), X a(2 »), ... , 0(XaC2k1h X aC2k »)) for Xl' ..• , X 2k 293
E
Tu(P)
294
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
where the summation is taken over all permutations (J of (1, 2, ... , 2k) and Ca denotes the sign of the permutation (J. The purpose of this section is to prove the following theorem. 1.1. Let P be a principal fibre bundle over M with group G and projection 7T. Choosing a connection in P, let 0 be its curvature form on P. (1) For eachfE [k(G), the 2kformf(0)' on P projects to a (unique) closed 2kform, say J(O), on M, i.e.,f(O) = 7T*(f(O)); (2) If we denote by w(f) the element of the de Rham cohomology group H2k(M; R) defined by the closed 2kformf(0) , then w(f) is independent of the choice of a connection and w: [( G) ~ H *(M; R) is an algebra homomorphism. Proof. We first prove THEOREM
LEMMA
1.
l
A qform ep on P projects to 'a (unique) qform, say p,
on Mif ) (a) ep( Xl' , X q ) = 0 whenever at least one of the X/: is vertical; (b) ep(RaXl , , RaXq ) = ep( Xl' ... , X q ) for the rzght translation R a defined by each element a E G. Proof of Lemma 1. Let VI' ... , Vq be tangent vectors of M at a point of M. Define a qform p on M by p(Vl , ... , Vq ) = ep(Xl , ... , X q ), where Xl' ... , X q are tangent vectors of P at a point u E P such that 7T(Xi) = Vi for i = 1, ... ,q. We have to show that p( VI' , Vq ) is well defined (independent of the choice of Xl' , X q ). Let Yl , ••• , Y q E Tv(P) with 7T( Y i ) = Vi for i = 1, , q. To prove that ep(Xl , ... ,Xq ) = ep(Yl , ... , Y q ) we may assume by (b) that u = v. Since Xi  Yi is vertical for each i, it follows from (a) that ep(Xl , ... , X q ) = ep( Yl , X 2 , ••• , X q ) = ep(Yl , Y 2 , X 3 , ••• , X q ) = ep( Yl , . . . , Y q ).
=
...
This completes the proof of Lemma 1. We recall (p. 77 of Volume I) the definition of exterior covariant differentiation D: For any qform ep on P, we set
(Dep) (Xl' ... , X q +l ) = (dep) (hXl , ... , hXq +l ), where hXi is the horizontal component of Xi.
295
XII. CHARACTERISTIC CLASSES
2. If a qform cp on P projects to a qform ip on M, i.e., cp = 7T*(ip), then dcp = Dcp. Proof of Lemma 2. Let Xl' ... , X q + l be tangent vectors of P LEMMA
at a point of P. Then
(dcp) (Xl' ... , X q + l ) = (d7T*ip) (Xl' ... , X q + l ) = (7T* dip) (Xl' ... , X q + l ) = (dip) (7TXH ••• , 7TXq + l ) = (dip) (7ThXI , ••• , 7ThXq + l )
= (7T* dip)(hXI , • • • , hXq +l ) = (d7T*ip) (hXI , • • • , hXq + l ) = (dcp) (hXI , • • • , hXq +l) = (Dcp) (Xl' ... , X q +l ). This completes the proof of Lemma 2. Applying Lemmas 1, and 2 to j (0) we shall prove ( 1) of Theorem 1.1. Since the curvature form 0 satisfies (a) of Lemma 1 by the very definition of 0 (cf. p. 77 of Volume I), so doesj(O). Since 0 is a tensorial form of type ad G so that for a
E
G
and sincejis invariant by G,j(O) satisfies (b) of Lemma 1. Hence j(O) projects to a unique 2kform, say 1(0), on M. To prove that ](0) is closed, it suffices to prove thatj(O) is closed. But, by Lemma 2, we have d(j(O)) = D(j(O)). SinceD acts as a skewderivation on the algebra of tensorial forms of P and since DO = 0 (by Bianchi's identity, cf. Theorem 5.4 of Chapter II), it follows that D(j(O)) = O. (We recall that a form 0 on P is called a tensorial form if it is horizontal in the sense that O(XI , • • • , X q ) = 0 whenever at least one of the X/s is vertical, cf. p. 75 of Volume 1.) It is a routine matter to verify that w: I(G) ~ H*(M; R) is an algebra homomorphism. It remains to show that w is independent of the choice of a connection. Consider two connection forms W o and WI on P and define
for 0 < t < 1. The following lemma is immediate.
296
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
is a tensorial Iform of type ad G, i.e., (1) r:t..(X) = 0 if X is vertical; (2) R: (r:t..) = ad (aI) r:t.. for a E G. {Wt; 0 < t < I} is a Iparameter family of connection forms on P. LEMMA
3.
r:t..
Let D t be the exterior covariant differentiation and 0t the curvature form defined by the connection form Wt. Then we have LEMMA
4.
d
dt 0t = Dtr:t..·
Proof of Lemma 4. Volume I) we have
0t = dW t
By the structure equation (cf. p. 78 of
+ ![w t, Wt]
= dw o + t dr:t..
+ ![w t, Wt].
· d b· SInce dt W t = r:t.., we 0 taln
:e 0, = doc + Hoc, ro,] + Hro"
oc]'
By Proposition 5.5 of Chapter II, the right hand side in the equality above is equal to D t r:t... This completes the proof of Lemma 4. We are now in a position to show that w is independent of the choice of a connection. Given f E Jk(G) and gvalued forms PI' ... , Pk on P of degree, respectively, ql' •.. , qk' we define a formf( PI' ... , Pk) of degree qI + ... + qk on P by
f( PI' ... , Pk) (Xl' ... , X q1 +..·+ q J
= A
I
eqf( Pl(Xq(I),
••• ,
X q(ql»)' • . • ,
Pk(Xq (Ql+"'+Qk_l+l>, ••• , X(Ql+"'+Qk»))'
where A = I/(ql + + qk)!' the summation is taken over all permutations a of (1, , qI + ... + qk) and eq is the sign of the permutation a. The 2kform f( 0) defined at the beginning of this section may be considered as an abbreviated notation for f(O, ... ,0). We may express f( PI' ... ,Pk) as follows. Let E l' . . . , E r be a basis for 9 and write r
Pi =
I
j=1
p:Ej
for i = 1, . . . , k.
297
XII. CHARACTERISTIC CLASSES
If we set then r
:2
A··· A ail"'ik mit 1'1
ik • m I' k
We shall now complete the proof of Theorem 1.1 by showing the following LEMMA
5.
Let j
E [k (G).
II>
=
k
ff
If we set (ex,
Qt, • • • , Qt)
dt,
then projects to a (unique) (2k  1) form on M and d = j(o.1' . • . , 0.1) j(o.o, • .. , 0.0). Proof of Lemma 5. By Lemma 1, j(oc, o. H • • • , o.t) projects I)form on M. Hence projects to a (2k  I)form to a (2k on M. By Lemmas 2 and 4 and by Dto.t = 0 (Bianchi's identity), we obtain kd(j(oc, o.t, ..• , o.t)) =kDt(j(oc, o.H ..• , o.t)) = kj(Dtoc, o.t, · · . , o.t) = =
kf (~
Qt, Qt, • • • , Q t)
d dt (j (o. t, o.t) . . · , o.t))·
Hence,
kf(d(f(ex,
dll>
Qt, • • • , Qt)))
dt
(1 d
=
Jo
=
j(o. 1,
dt (j(o. t, · .. , o.t)) dt • • • ,
0.1 ) j(o.o, · .. , 0. 0 ).
This completes the proof of Lemma 5 and hence that of Theorem 1.1. r ::> g..1 ::> n. If a is an ideal of g, the KillingCartan form of a is the restriction of the KillingCartan form B of 9 to a. 3. Anyone of the following properties of 9 is equivalent to semisimplicity of g: (I) Its radical r = 0 (by definition) ; (2) Its maximum nilpotent ideal n = 0; (3) g..1 = 0, i.e., its KillingCartan form B is nondegenerate; (4) Every abelian ideal of 9 is 0; (5) 9 is isomorphic to a direct sum of simple Lie algebras; (6) Every finitedimensional representation of 9 is semisimple, i.e., completely reducible (H. Weyl). A Lie algebra 9 is said to be reductive if its adjoint representation is semisimple. 9 is reductive if and only if it is isomorphic to a direct sum of a semisimple Lie algebra and an abelian Lie algebra. A subalgebra 1) of 9 is said to be reductive in 9 if the restriction of the adjoint representation of 9 to 1) is semisimple. If 1) is reductive in 9 and if p is a finitedimensional semisimple representation of g, then the restriction of p to 1) is semisimple. If 1) is a subalgebra of 9 reductive in 9 and if f is a subalgebra of 1) reductive in 1), then f is reductive in g. If a subalgebra 1) of 9 is
APPENDIX
327
9
semisimple, it is reductive in 9 by property (6) of a semisimple Lie algebra.
If H is a compact subgroup ofa Lie group G, then the Lie algebra 9 ofH is reductive in the Lie algebra 9 ofG. In fact, ifAda denotes the adjoint representation of G, then Ada (H) is compact, and there is a (positivedefinite) inner product in 9 which is invariant by Ada (H). If a is a subspace of 9 which is invariant by ad g (9), then its orthogonal complement with respect to this inner product is also invariant by ad g (9). In particular, the Lie algebra 9 of a compact Lie group is reductive. 4. A subalgebra $ of a Lie algebra 9 is called a Levi subalgebra if 9 = r + $ (vector space direct sum), where r is the radical of g. Since $ is isomorphic to g/r, it is semisimple. The theorem of Levi states that every Lie algebra 9 has a Levi subalgebra. We prove the following result which is needed in §5 of Chapter XI: If A is a compact group of automorphisms of a Lie algebra g, then 9 has a Levi subalgebra which is invariant by A. The proofis by induction on the dimension of the radical r of g. (i) Case [r, r] o. We set 9 = g/[r, r] and = r/[r, r]. Then is the radical of g. Since A leaves r invariant, it acts on 9 as an automorphism group. By inductive hypothesis, there is a Levi subalgebra $ of 9 which is invariant by A. If we denote by p the natural homomorphism 9 + 9 = g/[r, r], then [r, r] is the radical of pl(S). Since A leaves Pl(S) invariant, A may be considered as an automorphism group of pl(S). Again by inductive hypothesis, there is a Levi subalgebra $ of pl(S) which is invariant by A. Then $ is a Levi subalgebra of 9 which is invariant by A. (ii) Case [r, r] = o. We fix one Levi subalgebra $0 of g. Let $ be an arbitrary Levi subalgebra of g. For each X E $0' we write
"*
r
r
X=f(X) +Xs, Then f:
$0 +
where f(X)
E
rand
r is a linear mapping. If X, Y
[Xs, Ys]
=
[X f(X), Y f(Y)]
=
[X,
YJ 
E $0'
[X,f(Y)]  [f(X),
Xs
E $.
then we have
YJ.
Hence,
(*) f([X, Y])
=
[X,f(Y)]
+ [f(X),
Y]
for X, Y
E $0.
328
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Conversely, letf: So + r be a linear mapping satisfying (*). Then the mapping X E So + X f(X) Egis an isomorphism of So into g, and its image is a Levi subalgebra of g. Let V be the space of all linear mappingsf: So + r satisfying (*). It is now easy to see that the construction off given above gives a onetoone correspondence between the set of Levi subalgebras S and the vector space V of linear mappings f: So + r satisfying (*). (The reader who is familiar with the Lie algebra cohomology theory will recognize that V is the space 2 1 (so, r) of lcocycles of So with respect to the representation p of So defined by p(X) U = [X, UJ for X E So and U E r. But this fact will not be used here.) Every automorphism of 9 sends each Levi subalgebra of 9 into a Levi subalgebra of g. Hence, the group A acts on the set of Levi subalgebras of 9 and, consequently, on V. The problem now is to find a fixed point of A acting on V. We shall show that A is a group of affine transformations acting on V. Let ex be an element of A. Letfo,f~,f, apdfa be the elements of V corresponding to Levi subalgebras so, ex(so), sand ex(s), respectively. Let X E So and denote by ii(X) the socomponent of ex(X) with respect to the decomposition 9 = r + So so that ex(X) = fJ(X) + ii(X) where fJ(X) E r. If we rewrite this as ii(X) = fJ(X) + ex(X), we may consider fJ(X) as the rcomponent of ii(X) with respect to the decomposition 9 = r + ex(so). Since f~ corresponds to ex(so) by definition, fJ(X) is equal tof~(ii(X)). Hence, we have
ii(X)
=f~(ii(X))
+ ex(X).
The right hand side may be written as {f~(ii(X)) + ex(f(X))} + {ex(X)  ex(f(X))}. Since X f(X) belongs to s from tlle definition off, {ex(X)  ex(f(X))} belongs to ex(s) , and{f~( (sX)) + ex(f(X))} is the rcomponent of ii(X) with respect to the decomposition 9 = r + ex(s). Hence Since X is an arbitrary element of so, we have
fa
= f~
+ ex
0
f
0
iiI.
This shows that f E V + fa E V is an affine transformation of V with translation partf~ and linear partf + ex f iiI. In general, if A is a compact group of affine transformations of a vector space 0
0
APPENDIX
9
329
V, then there is a fixed point of A in V. Although this is a special case of Theorem 9.2 of Chapter VIII since V admits an inner product with respect to which A is a group of Euclidean motions, we can give a more explicit expression for a fixed point. Let fl be the Haar measure on A with total measure fl (A) = 1. Then the point of V given by
is left fixed by A. In our application in §5 of Chapter XI, A is a finite group oforder 2 generated by an involutive automorphism IX of g. In this case, the formula above for a fixed point reduces to
t(fo + loa). The result we obtained here has been proved by Taft [1], [2] for a wider class of algebras with finite automorphism groups. 5. We shall prove the following fact quoted in §ll of Chapter XI. If 9 is a real simple Lie algebra, then either 9 admits a compatible
complex structure (i.e., a complex structure J such that J([X, YJ) = [X, Jy] for X, Y E g) or its complexification gC is simple over R (and hence over C also). Assuming that gC is not simple, let a be any nontrivial ideal of gC. Then a (\ 9 is an ideal of 9 and hence is equal to either 9 or O. If a (\ 9 = g, then a :::) [a, gC] :::) [g, 9
+ ig]
= [g, g]
+ i[g, g]
=
9
+ ig
= gC,
which is a contradiction. Hence, a (\ 9 = O. It is an easy matter to verify that {Y E g; iY E a (\ ig} is an ideal of g. If this ideal is equal to g, then a:::) [a, gCJ :::) [ig, g + ig] = ig + 9 = gC, which is a contradiction. Hence, a (\ ig = O. We define two linear mappings p and q of a into 9 by
p(X
+ iY)
=
X
and
q(X
+ iY)
=
Y
for X
+ iY Ea.
The image of p is an ideal of 9 and the kernel of p is a (\ ig. Hence, p is bijective. Similarly, q is also bij ective. We define a linear isomorphism J: 9 ~ 9 by J = q 0 pl so that every element of a is of the form X + iJX, where X E g. It is straightforward to verify that J is a compatible complex structure in g. 6. A real semisimple Lie algebra 9 is said to be of compact type if its KillingCartan form B is negativedefinite. We shall show
330
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
that if 9 is the Lie algebra of a compact Lie group G, then the KillingCartan form B of 9 is negative semidefinite. Since Ad (G) is compact, there is a positivedefinite inner product (, ) in 9 which is invariant by Ad (G). For any X E g, ad (X) is then skewsymmetric. (ad (X) Y, ad (X) Y) < 0 for all Since (ad (X) ad (X) Y, Y) Y E g, it follows that (ad (X)ad (X)) is negative semidefinite. Its trace B(X, X) is therefore nonpositive, which proves our assertion. Since B is nondegenerate for a semisimple Lie algebra g, we can conclude that if G is a compact Lie group with semisimple Lie algebra g, then 9 is a compact type. Converse?J, if G is a connected Lie group with semisimple Lie algebra 9 of compact type, then G is compact (c£ Example 3.2 of Chapter X). Ifa Lie algebra 9 admits a compatible complex structure J, then ad (JX) ad (JY) = ad (X) ad (Y) and hence B(JX, JY) = B(X, Y), and B cannot be negativedefinite. 7. A real subalgebra I) of a complex Lie algebra 9 is called a real form of 9 if 9 = I) + il) and I) () il) = 0 so that 9 = I)c. Every complex semisimple Lie algebra 9 has a real form which is of compact type (Theorem of Weyl). The proof requires the knowledge of root system, but is otherwise simple (cf. Hochschild [1; p. 167]).
NOTES
Note 12. Connections and holonomy groups (Supplement to Note 1) 1. In Theorem 7.2 of Chapter IV we proved that if M is a connected, simply connected, and complete Riemannian manifold such that its (restricted) affine holonomy group . (P/2)
dv
=
L11::>.1 dv + Lg(df, df)
dv
=
O.
339
NOTES
Since t1f = 0 as we have shown, we obtain
tg(cif, df) dv
=
0,
which implies df = 0 everywhere, that is,jis a constant function. A function j is said to be harmonic if t1j O. The lemma shows that a compact Riemannian manifold has no harmonic function except for constant functions. The lemma of Hopf was used successfully in the work of Bochner, Lichnerowicz, Yano, and others (see Yano and Bochner [1] and Yano [2]). Lichnerowicz [3, p. 4] gave a generalization to the case of a tensor field: if K is a tensor field with components K:::: and if n
'" ~ giiKs'" t''';i;; 
0,
i,;=l
then K::::;i =
0
(that is, VK
=
0).
We shall give some examples of computation for Li. Let Mbe an ndimensional Riemannian manifold isometrically immersed in R n+p. Since the computation is done locally, we identify a point x of M with the position vector x of the corresponding point in R n+p. Let a be any constant vector in R n+p and consider the function j(x) = (x, a) on M. (If a is the ith vector of the standard basis e1 , ••• , en +p ofRn+p, thenjis the ith coordinate function of the immersion.) We compute Lif. For any vector field Y on M, we have
Yj = (Y, a), where Y on the right hand side is the R n+Pvalued vector function which represents the vector field Y on M. For any vector field X on M, we have
XYj =
(V~Y,
a)
(V xY, a)
+ (cx(X,
Y), a)
by using the formula of Gauss (§3, Chapter VII) V~Y = V xY
+ cx(X, Y),
where cx(X, Y) is the second fundamental form of M. Since
340
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
we obtain
(rt: (X, Y), a).
XYf  (V x Y)f
The left hand side is equal to (V2f) (; Y; X). If {Xl' ..• , X n } is an orthonormal basis in Tro(M), then n
/)"f
=
2. gij (V2f)(;
Xi; X,)
i,,=l
n
=
2: (V2f)(;
n
Xi; Xi) =
i=l
If ~l> ••• if we set
,
~p
(2: rt:(Xi, Xi), a) i=l
is an orthonormal basis in the normal space at x and p
rt:(X, Y)
=
2, hk(X,
Y)~k'
k=l
then n
2, h'C(Xi, Xi)
=
trace A k ,
i=l
where Ak is the operator corresponding to hk • Thus 1
n
 2, rt:(Xi, Xi) n i=l
is nothing but the mean curvature normal r; at x (cf. the remark following Example 5.3 of Chapter VII). Thus
Ji.f
n(r;, a)
for f(x) = (x, a).
Example 1. We may obtain another proof of the result that there is no compact minimal submanifold in a Euclidean space (cf. the remark cited above). Indeed, the mean curvature normal r; of M in R n+p is zero if and only if /)"f = 0 for every function f of the form (x, a). If M is compact, the harmonic function f must be a constant. By taking a ei , where {el , • • . , en + p } is an orthonormal basis of the Euclidean space, we see that the ith coordinate function of the immersion is a constant. Thus M reduces to a single point. Example 2. Let Sn(r) be the hypersphere of radius r > 0 in Rn+l: (X O)2 + (X l )2 + . .. (xn) 2 r 2• As in Example 4.2 of Chapter VII, we take the outward unit normal ~ = xlr and get
341
NOTES
A~ =
(ljr)1. Thus the mean curvature normal 1  (trace n
A~)~ =
'YJ
is equal to
xjr 2 •
We have for a functionf of the formf(x)
=
(x, a)
(1) Ilf = (njr 2 )f. In particular, each coordinate function Xi, as a function on sn(r), is an eigenfunction for the eigenvalue njr 2 of the Laplacian Il. We have also
(2) V2j + (flr 2 )g
=
0,
that is, (V2j) (; Y; X) + (flr 2 )g(X, Y) and Y. This follows from ( V 2f) (; Y; . 2 admits a nonzero solution f of the differential equation (2) ij' and only if M is isometric to sn(r). If M is a connected, compact, orientable Einstein space of scalar curvature 1, then M admits an eigenfunction f with Ilf =  nf if and only if M is isometric to Sn ( 1). For these results and the corresponding results for the complex projective space Pn(C), see Obata [1] and the references cited there (also see Note 11). We note that the Laplacian in Obata [1] is the negative of our Laplacian. Example 3. Let M be a kdimensional submanifold of sn(r) in Rn+l. In addition to the unit normal ~ = xjr to sn(r), choose n  k vector fields ~j which are normal to M and tangent to Sn(r) such that ~, ~1' ••• , ~nk are orthonormal at every point of M.
342
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
We denote by D, vo, and V the Riemannian connections of Rn+t, Sn (r), and M, respectively. For vector fields X and Y tangent to M we have and V~Y
nk
= VxY
+ 'Lhj(X,
Y);j
j=l
so that nk
DxY = VxY
+ h(X, Y); + 'Lhi(X,
Y);j,
j=l
where h is the second fundamental form for Sn(r) in Rn+1 and hI, ... , hn  k are the second fundamental forms of M as a submanifold of Sn (r) . For any constant vector a in Rn+\ consider f(x) = (x, a) as a function on M. By the same sort of computation as in Example 1 we get for the Laplacian ~ for M k
k
~f = ('L h(X~, Xi);
+ 'L
i=l
nk
'L hj(Xi , Xi) ;j, a),
i=l j=l
where Xl' ... , X k is an orthonormal basis of the tangent space of M at a point. Of course, we have k
'L h(Xi , Xi)
=
kfr 2 ,
i=l k
nk
and'L 'L hi(Xi , Xi);j is equal to k~, where
~
is the mean curvature
i=l j=l
vector of M in Sn(r). Thus we obtain ~f =
 (kfr 2 )f +
(k~,
a).
In particular, for each coordinate function function on M, we have
Xi
considered as a
0 3 isometrically immersed in Rn+l is actually imbedded as a hypersphere if it has nonzero constant sectional curvature. For the case n = 2, we have
1. Let M be a connected complete 2dimensional Riemannian manifold isometrically immersed in R3. If M has constant curvature k =I= 0, then (a) k > 0; (b) M is imbedded as a sphere. THEOREM
Part (a) is due to Hilbert [2]; we omit the proof. We shall prove here part (b) assuming that k > O. Under this assumption, it follows that M is compact (in fact, a sphere or a projective plane) by virtue of Theorem 1 of Note 4. By replacing M by its universal covering space if necessary, we may assume that M is actually a sphere with the usual metric; our assertion amounts to saying that an isometric immersion of a sphere into R3 is nothing but an ordinary imbedding (rigidity of a sphere). Thus it is a special case of the theorem of CohnVossen [1], [2] (see Chern [18]). We shall prove the rigidity of a sphere in the following form:
2.
Let M be a 2dimensional connected compact Riemannian manifold of constant curvature k. If M is isometrically immersed in R3, then M is imbedded as a sphere (and k > 0). THEOREM
344
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
We prepare a few lemmas, for which we assume M to be orientable and choose a field of unit normals so that the second fundamental form h and the corresponding symmetric operator A are defined globally on M.
1. For each x E M, let A(X) and p,(x) be the eigenvalues of A such that A(x) > P, (x). The functions A and p, are continuous on M and differentiable on W {x EM; A(X) > p,(x)}. Proof. If (xl, x 2) is a local coordinate system around a point x, LEMMA
then A can be expressed by a matrix
a(xt, X2) ( c(xl, x 2)
b(xt, X2)) d(xl, x 2) .
The eigenvalues A and p, are the roots of the quadratic equation
b)
t  a = t 2 rpt + 1p, c t  d where rp(xl, x 2) = a(xt, x 2) + d(xt, x 2) and 1p(xt, x 2) = a(xt, x 2) d(xt, x 2)  b(xt, x 2)c(xl, x 2) are differentiable functions together with a, b, c, and d. Our assertion easily follows from the formula for the roots of the quadratic equation.
f(t)
LEMMA
det (
2. Assume A(xo) > p, (xo). Then there exist differentiable
vector fields X and Y on a neighborhood U
of x 0 such that
g(X, X) = g( Y, Y) 1, g(X, Y) = 0, AX = AX, and AY = p,Y at each point of U. Proof. Let V be a neighborhood of Xo in which A > p,. Choose a tangent vector Zo at X o such that AZo i= AZo and AZo =1= p,Zo (this is possible because A p, at x o). Extend Zo to a differentiable vector field Z on V. Then X = AZ  pZ and Y = AZ  AZ are differentiable vector fields on V such that AX = AX and AY = p,Yat every point. By our choice of Zo, X and Yare not 0 at xo, and therefore they are not 0 in a neighborhood U of xo• Since X and Yare eigenvectors of A for the distinct eigenvalues, X and Yare orthogonal at each point. By normalizing X and Y we get the vector fields we want.
345
NOTES LEMMA
In the notation of Lemma 2 we have
3.
(1) VxX = aY,
VyX = bY,
VxY = aX,
VyY = bY,
where a and b are given by a = YA/(A  p,)
and
b = Xp,/(A  p,) j
(2) the curvature k is given by k = (a 2 + b2)  Xb + Ya. Proof. (1) From g{X, X) = 1, we obtain g(VxX, X) = 0 so that VxX = aY for a certain function a. Similarly, g(VyX, X) = 0 so that VyX = bY for a certain function b. From g(Y, y) = 1, we get VxY = cX and V yY = dX in the same fashion. From g(X, Y) = 0, we get g(VxX, Y) + g(X, VxY) = 0 so that a + c = o. Similarly, g(VyX, Y) + g(X, VyY) = 0 so that b + d = o. The functions a and b can be determined by using the equation of Codazzi (V xA) (Y) = (VyA) (X). (2) The results in (1) give R(X, Y)Y = (a 2 + b2)X  (Xb)X + (Ya)X and hence k = g(X, R(X, Y)Y) = a 2  b 2  Xb + Ya. 4. Assume A(X o) > p,(xo). If A has a relative maximum at Xo and p, has a relative minimum at x o, then the curvature k is < 0 at Xo• Proof. For the vector fields X and Y in Lemma 1 we have Xl = Xp, = YA = Yp, = 0 at Xo• By computation based on Lemma 3, we have at X o LEMMA
Ya
=
y 2 A/(A  p,),
and . Since A has a relative maximum at x o, we have y 2A < a at XO• Since p, has a relative minimum at XO• we have X2p, > a at Xo. Thus k(x o) < 0 from the formula above. Still assuming that M is orientable, we now prove Theorem 2. Since M is compact, there is a point where the second fundamental form is (positive or negative) definite (cf. Proposition 4.6 of Chapter VII). Thus k = Ap, is positive. Hence the constant
346
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
curvature k is positive. Since M is compact, the function A has a maximum at x o, and at Xo the function ft has a minimum. If A(X o) > ft(X o), then Lemma 4 gives a contradiction to k > O. Hence A(X o) = ft(x o), which implies that A(X) = ft(x) for every x EM. Every point being umbilical, we see that M is a sphere by Theorem 5.1 of Chapter VII. If M is not orientable, let £1 be the orientable double covering space of M. Applying the reasoning above to £1, we see that £1, immersed in Rn+l as a double covering of M, is umbilical at every point. Hence M is umbilical at every point and is a sphere by Theorem 5.1 of Chapter VII. vVe can make use of Lemma 4 to prove another classical theorem. A connected compact surface in R3 is called an ovaloid if the curvature is positive everywhere. An ovaloid whose mean curvature is a constant is a sphere. (Note that an ovaloid is necessarily orientable; the condition H = constant is independent of the choice of the field of unit normals, i.e., just a change of sign.) This result is due to Liebmann [1]. We shall here prove a more general result due to Aleksandrov [2] and Chern [9].
3. If an ovaloid M is isometrically immersed in R 3 in such a way that ft = f (A), where A and ft are the functions giving the eigenvalues of A as in Lemma 1 and f is a nonincreasing function (i.e. t < S implies f(t) > f(s)), then M is imbedded as a sphere. Proof. Let X o be a point where A has a maximum. Then A(X) < A(X o) and hence ft(x) > ft(x o) for every x, that is, ft has a minimum at Xo• If A(X o) > ft(x o), then Lemma 4 gives k(x o) < 0, which is a contradiction to the assumption k > O. Thus A(X o) = ft(x o) and A(X) = ft(x) for every x as in the proof of Theorem 2. Theorem 3 includes the case of constant mean curvature H (A + ft = constant) as well as Theorem 2 (ft = kJA where k is a positive constant). Hopf [7] proved that a connected, compact orientable surface in R3 such that H = constant is a sphere provided that its genus is O. Aleksandrov [3] proved this result without the assumption on the genus. We now consider the case where the constant curvature k is 0 in Theorem 1. Pogorelov [3], [4] and HartmanNirenberg [1] proved that M is immersed as a plane or a cylinder. Massey [I] gave a more elementary proof. HartmannNirenberg actually THEOREM
347
NOTES
proved a general theorem concerning hypersurfaces in a Euclidean space whose Gaussian mapping has Jacobian offixed sign >0 (or < 0) and derived the following result: Iff is an isometric immersion of an ndimensional connected completeflat Riemannian manifold Minto R n+l, then f( M) is of the form C X R nl, where R nl is a Euclidean subspace of dimension n  1 and C is a curve lying on a plane R 2 perpendicular to R nl. A more direct proof generalizing the method of Massey can be found in Nomizu [12] in which some related problem was treated. Sacksteder [1] and O'Neill [5] studied a generalization of the above theorem to the case where M has nonnegative sectional curvature everywhere or Mis of larger codimension. Hartman [1] proved the following general result in this direction. Let f be an isometric immersion of an ndimensional complete Riemannian manifold M into R n+p. Assume that (1) the sectional curvature of M > 0 for all planes; (2) the index of relative nullity v(x) is a positive constant, say, v. Then there exist an (n  v) dimensional complete Riemannian manifold M', an isometric immersionf' of M' into Rn+jJv, and an isometry cp of M onto M' X RV such that the diagram
M
f
>
1(p M'xRv
Rn+p
~id j'xid )
Rn+pvx Rv
is commutative, where id denotes the identity mapping. (For the notion of index of relative nullity, see Note 16.) For rigidity of hypersurfaces, see also Chern [18], [22]; ChernHanoHsiung [1]; Hsiung [1][4]; Katsurada [1][4]; Aeppli [1].
Note 16. Index of nullity For a Riemannian manifold M the index dimension fl (x) of the subspace
of nullity
To(x) = {X E Ta;(M); R (X, Y) = 0 for all Y
E
at x is the
Ta;(M)}.
If M is isometrically immersed in a Euclidean space Rn+p, where
348
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
n = dim M, the index of relative nullity vex) at x is the dimension of the subspace
T;(M)
=
{X
E T~(M);
A1;(X)
= 0 for all ; E
T;(M)},
where T~(M)..l is the normal space to M at x. These notions were defined in ChernKuiper [1]. When the codimension p is 1, take a unit normal; at x. The subspace Til (x) is just the null space of A = A~ so that n  vex) is equal to the rank of A, namely, the type number at x. In Theorem 6.1 of Chapter VII, we showed that if the type number at x is > 2, then v(x) = fl(X). We prove the following result of Chern and Kuiper. PROPOSITION
1. For an arbitrary codimension p, we have
vex) < flex) < vex)
+ p.
Proof. As in the case of codimension 1 we have Til c To by the equation of Gauss. Hence vex) < fl(X). Let S be the orthogonal complement of Til in the space To so that To = Til + S. Ifwe set m = dim S, then m = fleX)  vex) and we wish to prove m < p. We observe that Til can also be defined as
{X
E T~(M);
ex(X, Y)
=
0 for all Y E
T~(M)};
this follows easily from Proposition 3.3, (2) of Chapter VII. Consider the mapping ex: S X S ~ T; (normal space). If p < m, then the proof of the lemma for Theorem 4.7 of Chapter VIr gives a pair of vectors X, YES, not both 0, such that
ex(X, X) = oc( Y, Y)
and
ex(X, Y)
=
o.
Since XES c To, we have R(X, Y) = O. From the formula
g(R(X, Y)Y, X)
=
g(ex(X, X), ex(Y, Y))  g(ex(X, Y), ex(X, Y))
349
NOTES
it follows that (ex(X, X), ex(Y, Y)) = 0, that is, ex(X, X) = ex(Y, Y) = O. Since R(X, Z) = 0 for an arbitrary Z in Tx(M), we have
o=
(ex(X, X), ex(Z, Z))  (ex(X, Z), ex(X, Z)) =  ( ex (X, Z), ex (X, Z)). Thus ex(X, Z) = 0 for every Z E Tx(M), showing X E Til. Thus X E T" n S and X = O. Similarly, Y = 0, contradicting the fact that X and Yare not both O. We have thus proved p > m. g(R(X, Z)Z, X)
=
The nullity spaces To on a Riemannian manifold were studied by Maltz [1], Rosenthal [1]. As in A. Gray [4] the index of nullity can be defined for any curvaturelike tensor field L of type (1, 3) on a Riemannian manifold M. A tensor field L of type (1, 3) is curvaturelike ifit has the same formal properties as the curvature tensor field R, namely, (1) L(X, Y) is a skewsymmetric endomorphism; (2) L(X, Y) = L( Y, X) ; (3) 6L(X, Y)Z = 0; (4) 6(V xL)(Y, Z) = O. Here 6 denotes the cyclic sum over X, Y, and Z. For L, define the nullity at x as the dimension of the nullity space
To(x) = {X
E
Tx(M); L(X, Y) = 0 for all Y
E
Tx(M)}.
We have then
2. For a curvaturelike tensor field L, (1) the orthogonal complement TI(x) of To(x) in Tx(M) is spanned by L(X, Y)Z, where X, Y, and Z are arbitrary vectors at x; (2) if dim To(x) = constant on M, then the distribution To is involutive and totally geodesic (that is, V xY E Toftr any vector fields X, Y belonging to To). PROPOSITION
Note 17. Type number and rigidity of imbedding Let V be a real vector space of n dimensions with inner product. We identify A2 V with the space E( V) of skewsymmetric endomorphisms of V: (x A Y)Z = (y, Z)x  (x, z)y. Given a system {AI' ... ' A k } of symmetric endomorphisms that are linearly independent, we define the type number of the
350
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
system to be the largest integer r for which there are r vectors Xl' ••• ,Xr in V such that the kr vectors Acx(x i ), 1 < (I.. < k, 1 :::;;: i < r, are linearly independent. When k = 1, the type number of the system {A} of one single endomorphism A is just equal to the rank of A. We have
1. Let {AI' ... ' A k} and {AI' .•. ' A k} be two systems of symmetric endomorphisms and assume that {AI' ••. , A k} is linearly independent with type number > 3. If THEOREM
that is, if k
2 Acx(x)
k
A
Acx(Y)
2 Acx(x)
=
cx=1
A
Acx(Y)
for all x,Y
E
V,
cx=1
then there exists an orthogonal matrix S = (spcx)
of degree k such that
k
1 < (I.. < k. 2 spcxAp, p=1 This theorem as well as the following proof are reformulations of Chern [5].
Acx
=
Suppose Y1' •.. 'Yk' Zl' .•. , Zk are linearly independent and let f = ~~=1 Ycx A zcx. Considering f as an element in E( V), we have (i) range off = subspace spanned by {Y1' .•. 'Yk' Zl' , Zk}. (ii) If f is equal to ~~=1 jcx A zcx' where Y1' ... , jk' Zl' , Zk are in V, then j1' .•. , jk' Zl' , Zk are linearly independent and LEMMA
1.
SP{j1' . • . , jk' Zl'
, Zk} = SP{Y1' ... 'Yk' Zl' ... , Zk},
where Sp{· ..} denotes the subspace spanned by the elements in {...}. Proof. Sincef(x) = ~~=I{(X, zcx)Ycx  (x, Ycx)zcx}, it is clear that range f c Sp{Yb •.. 'Yk' Zl' ... , Zk}. To prove the inclusion in the other direction, ta1{e any f1 and choose x E V such that (x,Yp) = 1 (x,Ycx) = 0
for all
(I..
for all
(1...
and
# f1
351
NOTES
This is possible since {Ycx' zcx}, 1 < (l < k, are linearly independent. Then we have f(x) = Yp, showing that Yp E range j. Similarly, zp E range f for each fJ. Thus
SP{Y1' ... 'Yk' Zl' ... , Zk} c rangef. To prove (ii), we first observe that range Zl' ... , Zk} so that by (i) we have
SP{Y1' ... 'Yk' Zl' ... , Zk} Since {Ycx' zcx}, 1
k
and
i=l
a1,3.. = 0
1 <j < n.
Then
n
= L
i,i=l
aiixi
A
Xi =
L (a ii
i
 aU)x i
A
Xi·
Since ~:=1 Xi A Yi = 0 and since Xi A Xi are linearly independent, we have au  au = 0 for all i < j. Thus for 1 < i < k
and j > k.
This shows thatYi' 1 < i < k, are linearly dependent on Xk •
Proof of the theorem.
Let
Xl) X 2 ,
Xl' ••• ,
xa E V be such that Acx(xi ) ,
352
I
4, we have k
L Aa(x
.
k j )
A
Aa(xi ) =
a=1
_
L Aa(x
_ j )
A
Aa(x i )
a=1
I.e., 1 < i < 3. We know that Aa (x 1 ), 1 < a. < k, are linearly independent. From the above_equation for i = 1, we conclude by Lemma 2 that Aa(xj )  Aa(xj ) is a linear combination of A p (x 1 ), 1 < fJ < k. Similarly, we see that Aa(xj )  Aa(xj ) is a linear combination of A p (x 2 ), 1 < fJ < k. Since A p (x 1 ), A p (x 2 ), 1 < fJ < k, are linearly independe~t (as are Ap (x 1 ), Ap (x 2 ), 1 < fJ < k), it follows that Aa(xj )  Aa(xj ) = 0, 1 < a. < k, j > 4, proving our assertion. This completes the proof of Theorem 1.
Letf: M ~ R n+p be an isometric imbedding of an ndimensional Riemannian manifold M into a Euclidean space R n+p • We shall deal only with a local theory and use the notations developed in §3 of Chapter VII. The mapping ~ ~ A g of Tx(M) 1.. into the vector space of all symmetric endomorphisms of Tx(M) was defined in Proposition 3.3 of Chapter VII. Let k(x) be the rank of this linear mapping. If we denote by No (x) the null space of the linear mapping ~ + A g, then the orthogonal complement of No(x) in Tx(M) 1.. is called the first normal space at x. Clearly, k(x) is equal to the dimension of the first normal space at x. If we choose a basis ~I' ••• , ~k' where k = k(x), in the first normal space at x, then the corresponding endomorphisms AI' ... , A k are linearly independent. Now the local rigidity theorem by Allendoerfer [1], which is an extension of the result on a hypersurface (Corollary 6.5 of Chapter VII), can be stated as follows. 
THEOREM
2. Let f and f be two isometric imbeddings of an n
dimensional Riemannian manifold M into a Euclidean space R n+p • Assume that, for a neighborhood U of a point x 0 E M, we have (1) the dimensions of the first normal spaces at x E U for both f and f are equal to a constant, say, k; (2) the type number off at each x E U is at least 3.
354
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Then there is an isometry hood of X O•
T
of R n+p such that f =
T
0
f on a neighbor
We note that Theorem 1 can be utilized for the proof of Theorem 2. See also Chern [15].
Note 18. Isometric imbeddings The theory of surfaces in 3dimensional Euclidean space differs considerably from the general theory of submanifolds in Ndimensional Euclidean space in its method and results (see Note 15) . The problem of isometric imbedding of a 2dimensional Riemannian manifold into the 3dimensional Euclidean space, in particular the socalled Weyl problem, will not be discussed here. The first result in the general isometric imbedding is the theorem of Janet [1] and E. Cartan [14] which states that a real analytic Riemannian manifold M of dimension n can be locally isometrically irnbedded into any real analytic Riemannian manifold V of dimension in(n + 1). (See Burstin [1] for comments on Janet's proof.) The generalization to the Coo case is open even when V is a Euclidean space. A global isometric imbedding theorem was obtained by Nash [2]:
1.
Every compact ndimensional Riemannian manifold M of class Ck (3 < k < 00) can be Ckisometrically imbedded in any small portion of a Euclidean space RN, where N = in(3n + 11). Every noncompact ndimensional Riemannian manifold M ofclass Ck (3 < k < 00) can be Ckisometrically imbedded in any small portion of a Euclidean space RN, where N = tn(n + 1)(3n + 11). THEOREM
For a Clisometric imbedding, the dimension of the receiving space can be very smalL The result of Nash [1] and Kuiper [5] states:
2. Let M be a compact ndimensional Riemannian manifold of class Cl with boundary (which can be empty). If M can be Climbedded in a Euclidean space RN, N > n + 1, then it can be Clisometrically imbedded in RS. THEOREM
In particular, M can be locally Clisometrically imbedded in Rn+l. A Clisometric imbedding can be quite pathological. An ndimensional torus can be Clisometrically imbedded in Rn+l but
NOTES
355
it cannot be C4isometrically imbedded in R2nl (Tompkins [1], cf. Corollary 5.3 of Chapter VII). Once we know that every Riemannian manifold can be isometrically imbedded in a Euclidean space of sufficiently large dimension, we naturally seek for a Euclidean space of smallest possible dimension in which a Riemannian manifold can be isometrically imbedded. Hilbert [2] proved that a complete surface of constant negative curvature cannot be C4isometrically immersed in R3, as is mentioned in Note 15 as well. Hilbert's result has been generalized by Efimov [1] to a complete surface of bounded negative curvature. The result of Chern and Kuiper [1] generalizing the theorem of Tompkins mentioned above says that a compact ndimensional Riemannian manifold with nonpositive sectional curvature can not be isometrically immersed in R2nl (cf. Corollary 5.2 of Chapter VII). A further generalization by O'Neill [2] states that if N is a complete simply connected Riemannian manifold of dimension 2n  1 with sectional curvature K N < 0, then a compact ndimensional Riemannian manifold M with sectional curvature K M < K N cannot be isometrically immersed in N. Otsuki [2] proved that an ndimensional Riemannian manifold M of constant negative curvature cannot be (even locally) isometrically immersed in R 2 n2. For more results in this direction, see Otsuki [2], [3]. The holonomy group gives also some restrictions on isometric immersions. If a compact ndimensional Riemannian manifold M can be isometrically immersed in Rn+\ then its restricted linear holonomy group must be SO(n) (see Kobayashi [8]). If a compact ndimensional Riemannian manifold M can be isometrically immersed in Rn+2, then its restricted linear holonomy group must be SO(k) X SO(n  k) except when n =·4 (see R. L. Bishop [1]). If a compact ndimensional Riemannian manifold M with restricted holonomy group SO(k) X SO(n  k) is isometrically imbedded in Rn+2, then the imbedding is a product of two imbeddings of hypersurfaces, with an exception occurring in the case k = 1 or n  k = 1 (see S. B. Alexander [1]). For a noncompact ndimensional Riemannian manifold M isometrically immersed in R n+l, there are certain restrictions on the holonomy group (see DolbeaultLemoine [1]). According to R. L. Bishop [1], when a noncompact ndimensional Rieman
356
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
nian manifold M is isometrically immersed Rn+2, its restricted linear holonomy group is either of the form SO(k) X SO(n  k) or U(m), where n = 2m (unless M is flat). The isometric imbedding problem is easier if the receiving space is allowed to be infinitedimensional. Bieberbach [4J proved that a simply connected complete surface H 2 of constant negative curvature can be isometrically imbedded in a Hilbert space. This was generalized by Blanusa [1J to an isometric imbedding of a simply connected complete ndimensional sRace H n of constant negative curvature into a Hilbert space. Later Blanusa [2J improved his own result by showing that H n can be isometrically imbedded into R6n5 and H 2 into R6. A holomorphic isometric imbedding of a Kaehler manifold into a complex Euclidean space is not always possible. A compact complex manifold of positive dimension can never be immersed in a complex Euclidean space. According to Propositions 8.3 and 8.4 of Chapter IX, a Kaehler manifold which can be holomorphically and isometrically immersed into a complex Euclidean space must have nonpositive holomorphic sectional curvature and negative semidefinite Ricci tensor. Bochner [2J proved that a certain class of Kaehler manifolds (including at least all classical bounded symmetric domains with Bergman metric) cannot be holomorphically and isometrically immersed into a finitedimensional complex Euclidean space. A systematic study of imbedding a Kaehler manifold into a complex Hilbert space and other infinitedimensional spaces was done by Calabi [1J. It is interesting to note (cf. Bochner [2J and Kobayashi [14J) that the Bergman metric of a complex manifold M is so defined that M is holomorphically and isometrically imbedded in a complex projective space (generally of infinite dimensions) in a natural manner. Consider now the following generalization of the isometric imbedding problem. Let M be a Riemannian manifold and G a group of isometries of M. The problem is to find not only an isometric imbedding of M into a Euclidean space RN but also a compatible imbedding of G into the group 3(RN) of Euclidean motions of RN. Such an imbedding (rather a pair of imbeddings) is called an equivariant isometric imbedding of (M, G). If G is transitive on M so that M = GjH and if the linear isotropy representation of H is irreducible, then a differentiable imbedding of Minto RN
357
NOTES
with a compatible imbedding of G into ~(RN) IS Isometric since there is, up to a constant factor, only one Riemannian metric on GjH invariant by G. It is known (Lichnerowicz [3]; pp. 158167) that a Hermitian symmetric space GjH of compact type can be equivariantly and isometrically imbedded into RN, where N = dim G. This has been extended to almost all symmetric spaces of compact type (Nagano [10] and Kobayashi [25]). Finally we mention a paper by Friedman [1] in which a generalization of the J anetCartan theorem to an indefinite Riemannian metric is obtained.
Note 19. Equivalence problems for Riemannian manifolds A transformation f between two Riemannian manifolds M and M' is said to be strongly curvaturepreserving iff maps VrnR upon V'rnR' for every m = 0, 1, 2, ... , where VrnR and V'rnR' denote the mth covariant differentials of the curvature tensor fields R of M and R' of M', respectively. Nomizu and Yano [3], [4] proved
1. A strongly curvaturepreserving diffeomorphism between irreducible and analytic Riemannian manifolds M and M' of dimension > 2 is a homothetic trans,formation. THEOREM
This combined with Lemma 2 on p. 242 of Volume I yields
A strongly curvaturepreserving diffeomorphism of a complete irreducible analytic Riemannian manifold M of dimension > 2 onto itself is an isometry. COROLLARY.
The corresponding result for infinitesimal transformations was proved in Nomizu and Yano l2]. See Tanno [1] for generalizations to the case of an indefinite metric. We say that a Riemannian manifold M is strongly curvaturehomogeneous if, for any points x and y of M, there is a linear isomorphism of Tx(M) onto Ty(M) which maps gx (the metric at x) and (vrnR)x, m = 0, 1, 2, ... upon gy and (vrnR)y, m = 0, 1, 2, .... Then THEOREM
2. If a connected Riemannian manifold M is strongly
curvaturehomogeneous, then it is locally homogeneous. complete and simply connected, then it is homogeneous.
If;
moreover, M is
358
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Theorem 2 was originally proved by Singer [2] when M is complete and simply connected. The local version as stated in Theorem 2 was later obtained by Nomizu [9]. Kulkarni [1] studied the following question related to Theorem 1 above and Schur's theorem (Theorem 2.2 of Chapter V) : When there exists a diffeomorphism which preserves sectional curvature between two Riemannian manifolds M and M', are M and M' isometric?
Note 20. GaussBonnet theorem 1. The GaussBonnet theorem for a compact orientable 2dimensional Riemannian manifold M states that
1 2
7T
1 ,v
K dA
=
x(M),
where K is the Gaussian curvature ofthe surface M, dA denotes the area element of M, and x( M) is the Euler number of M. This is usually derived from the GaussBonnet formula for a piece of surface. Let D be a simply connected region on M bounded by a piecewise differentiable curve C consisting of m differentiable curves. Then the GaussBonnet formula for D states
f/
g
ds
i~ (17 
mi )
+ InK dA
=
217,
where kg is the geodesic curvature of C and (Xl' ••• , (Xm denote the inner angles at the points where C is not differentiable. Triangulating M and applying the GaussBonnet formula to each triangle we obtain the GaussBonnet theorem for M. 2. The first step toward a generalization of the GaussBonnet theorem to higher dimensional manifolds was taken by H. Hopf [2]. The following lemma of Hopf is basic.
Let N be an ndimensional compact manifold in R n with boundary aN. Then the degree of the spherical map of Gauss aN ~ Snl is equal to the Euler number x(N) of N. For the proo:f, see, for instance, Milnor [5J. Now consider an ndimensional compact submanifold M (without boundary) in Rn+k. Let NI!. denote the closed eneighborhood of M. For e sufficiently small N e is a differentiable manifold LEMMA.
359
NOTES
with boundary. Since X(Ne) = X(M), the lemma above implies that the degree of the spherical map of Gauss aNe + Sn+kl is equal to the Euler number X(M) of M. Consider the special case where k = 1 and Mis orientable. Then aNe consists of two components. Let dbe the degree of the spherical map M + Sn. Then the degree of the spherical map aNe + Sn is given by d + (1) nd. Hence
We shall now express the degree d by integrals. Let yO, yl, ... , yn be the natural coordinate system in Rn+l. The volume element of the unit sphere Sn is given by n
L (1)jJi dy0
/'..
A ••• A
dyi
A ••• A
dyne
i=O
Let ~ be the spherical map M + Sn; it can be given by n + 1 functions ~o, ~1, ... ,~n on M with (~O) 2 + (~l) 2 + ... + (~n) 2 = 1. Then d =
i
n/'..
L~
i
d~ 0 A • • • A d~ i A • • • A d~ n •
Mi=O
The formula of Weingarten states (cf. §3 of Chapter VII) ~* (X)
= A(X)
for every vector field X on M,
where A = A g is the symmetric transformation of each tangent space Tx(M) defined by the second fundamental form. By a simple calculation we obtain
d
=
~
rK
wnJM
n
dv
(w n
=
volume of the unit nsphere),
where K n is the Gaussian curvature of M and dv denotes the volume element of M. For n even, we can replace d by ix( M) and express K n as a polynomial of the Riemannian curvature. The resulting formula may be considered a generalization of the GaussBonnet formula. We note that 2Kn dv is equal to the nform y defined in §5 of Chapter XII for the vector bundle T( M) and the Riemannian connection of M.
360
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
3. A further generalization was obtained by Allendoerfer [1] and Fenchel [1] independently. Let M be an ndimensional compact orientable submanifold ofRn+k and N e the closed eneighborhood of M for a sufficiently small e. Let d be the degree of the spherical map oNe ~ Sn+kl and K n+k  1 the Gaussian curvature of the hypersurface oNe. Then by the result above, we have
!X(oNe ) = d =
r
1
Wn+kl JaNe
K n+1 k dv,
where dv denotes the volume element of oNe. Assume that nand k  1 are both even. Since 0N e is a sphere bundle over M, we have
On the other hand, integrating K n +k  1 along the fibres we obtain (by a nontrivial calculation)
r
1
W n+k 
1
JaNe
K n +k  1 dv
=f y, M
where y is the nform defined in §5 of Chapter XII. Hence,
X(M) =
r y.
JlJ;I
The assumption that k  1 is even is not restrictive. If k  1 is odd, it suffices to imbed Minto Rn+k+l in a natural manner. 4. As we have mentioned in Note 18, every Riemannian manifold can be isometrically imbedded in a Euclidean space of sufficiently high dimension. But this imbedding theorem was not established at the time when Allendoerfer and Fenchel obtained the GaussBonnet theorem for submanifolds of Euclidean spaces. But the local imbedding theorem of Janet and Cartan (c£ Note 18) was then available. In 1943, Allendoerfer and Wei! [1] obtained the GaussBonnet theorem for arbitrary Riemannian manifolds by proving a generalized GaussBonnet formula for a piece of a Riemannian manifold isometrically imbedded in a Euclidean space. 5. The socalled intrinsic proof was obtained by Chern [4] in 1944. Let M be a 2mdimensional orientable compact Riemannian manifold and S( M) the tangent sphere bundle over M, i.e., the
361
NOTES
bundle of unit tangent vectors of M. Let p: S(M) + M be the projection. Chern constructs a (2m  I)form 7T on S(M) such that d7T = p*(y) and that the integral of 7T along each fibre of S(M) is 1. Let X be a unit vector field on M with isolated singularities at Xl' ... , Xk • Let aI, ••• , ak be the index of X at Xl' ... , X n • By a theorem of Hopf, al + . . . + ak is equal to the Euler number X(M) (see for instance, Milnor [5]). The cross section X of S( M) may be considered as a submanifold of S( M) with boundary, and its boundary ax is given by alSl + ... + akSk where Sl' ... , Sk are the fibres of S( M) at Xl' ... , X k • Hence
r y =Jx p*(y) =Ix d7T =J =i ail 7T =iai
JM
oX
=
7T
i=l
Si
i=l
X(M).
We see in Chern's proof the birth of the concept of transgression. Chern [23] extended the GaussBonnet theorem to the case of indefinite Riemannian metric. Our axiomatic proof may be easily generalized to include also the case of indefinite Riemannian metric. See also Eells [1], BishopGoldberg [2], Avez [1] on the GaussBonnet theorem. On some estimates of the Euler number with the aid of the GaussBonnet theorem, see Chern [14], Berger [14], BergerBott [1]. 6. CohnVossen [3] investigated global properties of noncompact Riemannian manifolds of dimension 2. Among other things he proved that for a complete orientable Riemannian manifold M of dimension 2 the inequality
Lr
< X(M)
holds provided that the integral exists as an improper integral. No generalization to higher dimensions is known. 7. We also mention a result in Milnor [6] that a compact orientable surface of genus g > 2 does not admit any affine connection with zero curvature. See also Benzecri [2].
Note 21. Total curvature As in Note 20, let M be an ndimensional manifold immersed in a Euclidean space RN and B the bundle of unit normal spheres
362
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
over M. The degree of the canonical map v: B defined by
~ SNl
was
is the volume of the unit (N  I)sphere SNl and G(~) is the determinant of the symmetric transformation A~ defined by the second fundamental form of M. The total curvature of M immersed in RN by an immersion mapf: M ~ RN is defined by where
OJNl
T(M,j, RN)
=
I
llGI dvB •
OJNl B
We shall relate the total curvature of M with the topology of M. Given a function
L fJk(M)
> b(M),
k=O
where the infimum is taken Jor all immersions J and Jor variable N. Chern and Lashof [1], [2] proved the inequality T(M,h RN) > fJ(M). The equality infj T(M,h RN) = fJ(M) is due to Kuiper [7]. The inequality fJ(M) > ~fJk(M) is evident and the inequality ~ fJk(M) > b(M) is a Morse inequality (see, for instance, Milnor [3]). An immersion j~ is said to be minimal if T(M, h R N) = fJ(M). This is not to be confused with the term used on p. 34. We list other known facts on total curvature. THEOREM
2.
Let J: M
*
R Nand J': M'
*
R N' be immersions.
Then T(M X M,J X J', RN x RN') = T(M,h RN)T(M',J', RN'). In particular, letting M' be a point, we obtain
Let J: M * RN be an immersion and i: RN R N+k an imbedding oj R N as a linear subspace oj R N+k. Then COROLLARY.
T(M, i h RN+k) 0
=
*
T(M,h RN).
The corollary above is due to ChernLashof [2] and its generalization (Theorem 2) is due to Kuiper [7]. The following result is also due to ChernLashof [2]. 3. Let M be an ndimensional compact manifold and J: M * RN an immersion. Then T(M,h RN) = 2 if and only if the immersionJ is an imbedding andJ(M) is a convex hypersurface in a linear su1Jspace R n+1 of' R N. THEOREM
As pointed out by Kuiper [7], Theorem 3 implies that if a manifold M homeomorphic to a sphere can be minimally immersed into RN, then M is diffeomorphic to an ordinary sphere. (For a manifold M homeomorphic to a sphere admits a function with only two critical points, that is, M satisfies fJ(M) = 2.) We know of no general method to characterize those compact manifolds which can be minimally immersed into a Euclidean space. Here are some manifolds which admit minimal im. merSlons.
364
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
THEOREM 4. (Kuiper [9]). Every orientable closed surface and also every nonorientable closed surface with Euler number < 2 can be minimally immersed in R3. The real projective plane and the Klein bottle cannot be minimally immersed in R3. In an earlier paper, Kuiper [7] exhibited a minimal immersion of the real projective plane in R4. THEOREM 5. (Kobayashi [23]). Every compact homogeneous Kaehler manifold can be minimally imbedded into a Euclidean space of sufficiently high dimension. For further properties of the minimal imbedding in Theorem 5 such as the dimension of the receiving Euclidean space and the equivariance, see Kobayashi [23]. Tai [1] constructed minimal imbeddings of all projective spaces and the Cayley planes. Takeuchi and Kobayashi [1] obtained minimal irnbeddings of all Rspaces, generalizing Theorem 5 and the result of Tai. For results in the theory of knots in connection with total curvature, see Ferus [1]. An article by Kuiper [11] lists a number of unsolved problems on total curvature. See also Kuiper [10], Otsuki [7], Wilson [1].
Note 22. Topology of Riemannian manifolds with positive curvature Let b be a positive number with 0 < b < 1. An ndimensional Riemannian manifold M is said to be bpinched if its sectional curvature K satisfies Ab < K < A for some positive number A. The constant A in the inequalities above is not essential. In general, the sectional curvature K of a Riemannian metric g and the sectional curvature K of the Riemannian metric g = cg (where c is a positive constant) are related by K = Klc. We can therefore "normalize" the metric so that if M is bpinched then b < K < 1. In Volume I, we proved the following theorem of Hopf [1] (cf. Theorem 3.1 of Chapter V and Theorem 7.10 of Chapter VI). THEOREM 1. A complete, simply connected Ipinched Riemannian manifold M is isometric to an ordinary sphere.
NOTES
365
Recently, ]. Wolf [7] obtained a complete classification of the complete Ipinched Riemannian manifolds. We know very little about bpinched Riemannian manifolds in general. The following list seems to exhaust all that we know at present.
2. A complete bpinched Riemannian manifold M with b > 0 is compact and has a finite fundamental group. THEOREM
Theorem 2 is due to Myers [1] and is proved in Theorem 5.8 of Chapter VIII. Actually, the theorem of Myers is stronger than the one stated above since it assumes only that the eigenvalues of the Ricci tensor are bounded below by a positive constant. The following result is due to Synge [2]: 3. A complete bpinched Riemannian manifold M (with b. > 0) of even dimension is either (1) simply connected or (2) nonorientable with 7Tl (M) = 2 2 • THEOREM
Theorem 1 suggests that if a Riemannian manifold M is complete and bpinched with b sufficiently close to 1, then M is similar to a sphere in one sense or another. The first result in this direction was obtained by Rauch [1] who proved that a complete simply connected Riemannian manifold which is 0.75pinched is homeomorphic to a sphere. For an evendimensional manifold, this pinching number was improved to b = 0.54 ... by Klingenberg [7] who made the first systematic study of cut loci since Myers and ]. H. C. Whitehead. Improving Klingenberg's method, Berger [6, 7] finally obtained 4. Let M be a complete simply connected Riemannian manifold of even dimensions which is bpinched. If b > 1, then M is homeomorphic to a sphere. If b = 1, then M is either homeomorphic to a sphere or isometric to a compact symmetric space of rank 1. THEOREM
The first part of Theorem 4 was obtained independently also by Toponogov [3]. Refining his results on cut loci and using Berger's proof of Theorem 4, Klingenberg [5, 6] obtained
5. A complete simply connected Riemannian manifold oj odd dimensions which is ipinched is homeomorphic to a sphere. THEOREM
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FOUNDATIONS OF DIFFERENTIAL GEOMETRY
The first parts of Theorem 4 and Theorem 5 are known as "Sphere Theorem." Since all complex and quaternionic projective spaces and the Cayley plane are known to be ipinched, Theorem 4 is the best possible result. On the other hand, in the odddimensional case it is not known whether Theorem 5 can be improved. The proof of the Sphere Theorem has been partly simplified since then by Tsukamoto [3J. Klingenberg [8J gave a new approach to the Sphere Theorem. Let M be a complete simply connected Riemannian manifold. He proved that if the conjugate locus of one particular point of M is similar to that of a compact simply connected symmetric space M', then H*(M; Z) is isomorphic to H*(M'; Z). In particular, he showed that if the sectional curvature along the geodesics emanating from one particular point of M is bpinched with b > i, then M has the homotopy type of a sphere and hence is homeomorphic to a sphere when dim M > 5 (by Smale's solution of the generalized Poincare conjecture, Smale [lJ) .. I n the Sphere Theorem, it is not known if M is diffeomorphic with an ordinary sphere. But Gromoll [1 J proved the following THEOREM 6. There exists a sequence of numbers i = b1 < b2 < b3 < ... , lim bA = 1, such that if M is a complete simply connected nA.. 00
dimensional Riemannian manifold which is bn _ 2 pinched, then M is diffeomorphic to an ordinary sphere. In the hope of the eventual classification of bpinched Riemannian manifolds, it is important to obtain as many properties as possible of a bpinched Riemannian manifold. An interesting conjecture by Chern is that a complete bpinched Riemannian manifold M of even dimensions has positive Euler number X(M). For dim M = 2, the conjecture is trivially true. For dim M = 4, we have X(M) = 2 + dim H2(M; R) > o. In fact, if M is compact with positive Ricci tensor, then Hl(M; R) = 0 by Theorem 5.8 of Chapter VIII and H3(M; R) = 0 by the Poincare duality. For dim M = 4, X(M) > 0 may be also verified by the GaussBonnet formula (cf. Chern [14J). Also using the GaussBonnet formula Berger [14J proved that if M is a complete bpinched Riemannian manifold of dimension 2m, then IX(M) I < 2m (2m) ! bm • Using harmonic forms, Berger [6J proved that, for
NOTES
367
a complete Riemannian manifold of odd dimension 2m + 1 which is 2(m  1)/(8m  I)pinched, its second Betti nurnber vanishes. There are very few examples of Riemannian manifolds with positive sectional curvature. The only compact simply connected manifolds which are known to carry Riemannian metrics of positive sectional curvature are the ordinary spheres, the complex projective spaces, the quaternionic projective spaces, the Cayley plane and the two homogeneous spaces (of dimension 7 and 13) discovered by Berger [8] (cf. also Eliasson [1]). In particular, one does not know if there is any compact product manifold M = M' X M" which can carry a Riemannian metric of positive sectional curvature. This question is open even for the product of two 2spheres S2 X S2. Closely connected with the concept of manifold of positive sectional curvature is that of manifold of positive curvature operator. At each point x of M the Riemannian curvature tensor R defines a linear transformation of A 2 Tx(M) into A 2 T:(M) which sends X A Y into R(·, ., X, Y). By the duality between A2 Tx(M) and A2 T:(M) defined by the Riemannian metric, this linear transformation can be identified with a linear endomorphism of the space of 2forms A2 T: (M), called the curvature operator at x. We shall denote by Px the curvature operator at x. Since R(X, Y, U, V) = R( U, V, X, Y), Px is symmetric (i.e., selfadjoint) with respect to the inner product defined on A2 (M) by the Riemannian metric. Hence the eigenvalues of Px are all real. Let A and A be two real numbers. By the notation A < Px < A we shall mean that the eigenvalues of Px are contained in the closed interval [A, A]. Similarly, by A < p(M) < A we shall mean that the eigenvalues of Px for all x E M are contained in [A, A]. If {Xi} is an orthonormal basis for Tx(M) and {~*} is the dual basis for T: (M), then g(Px(xt A Xi), xt A Xi) R(X1 , X 2, Xl' X 2). Hence, if 0 < A < p(M) < A, then M is (AI A) pinched (cf. Berger [11]). The following result is due to Berger [11].
r:
7. Let M be a compact Riemannian manifold with positive curature operator, i.e., 0 < A < p(M) < A. Then H2(M; R) = o. If the curvature operator is nonnegative, i.e., 0 < p(M) < A, then the harmonic 2forms of M are parallel tensor fields. THEOREM
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FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Using the GaussBonnet formula, K. Johnson proved the following
8. If M is a compact Riemannian manifold of even dimension with positive curvature operator, then its Euler number X(M) is positive. THEOREM
The following result is due to Bochner and Yano (BochnerYano [1] and YanoBochner [1; p. 83]). THEOREM
with
9.
If M is a compact ndimensional Riemannian manifold
o
5 with positive curvature is diffeomorphic with R n • THEOREM
of
Note 23. Topology of Kaehler manifolds with . . pos"tt"tve curvature Let M be a Kaehler manifold of complex dimension n. In addition to the sectional curvature and the holomorphic sectional curvature we introduced in §7 of Chapter IX, we define the Kaehlerian sectional curvature of M as follows. Let p be a plane in Tx(M), i.e., a real 2dimensional subspace of Tx(M). Let X and Y be an orthonormal basis. As in §7 of Chapter IX we define the angle rx(p) between p and J(p) by cos rx(p) = Ig(X, JY) I. We know (cf. Proposition 7.4 of Chapter IX) that the sectional curvature of a space of constant holomorphic sectional curvature 1 is given by i (1 + 3 cos 2 rx (p) ). Denote by K (p) the sectional curvature of M. Then it is quite natural to set
K*(p)
=
4K(P)j(1
+ 3 cos
2
rx(p)).
369
NOTES
We shall call K* (p) the Kaehlerian sectional curvature of the plane section p. For a Kaehler manifold M we consider three kinds ofpinchings. We say that M is ~pinched if it is so as a Riemannian manifold (c£ Note 22). We say that M is ~Kaehler pinched ifthere is a constant A such that ~A < K* (P) < A for all planes p. We say that M is such that ~A
~holomorphically
< K* (p) < A
pinched if there is a constant A
for all planes p invariant by J.
Now let X and Y be arbitrary tangent vectors of M at x. If we define Q(X) by Q(X) = R(X, JX, X, JX), then by polarization we obtain (cf. Bishop and Goldberg [2])
R(X, Y, X, Y)
=
a.l2[3Q(X  Q(X
+ JY) + 3Q(X 
+
JY)
Y)  Q(X  Y)
 4Q(X)  4Q(Y)]. From this identity we can deduce certain relations between the three pinchings introduced above. In particular (cf. Berger [5], Bishop and Goldberg [2]), if M is ~holomorphically pinched, then !(3~  2)A < K(p) :::;: A for all planes p. We shall now list implications of various pinching assumptions. 1. A complete Kaehler manifold M of complex dimension n is holomorphically isometric with the complex projective space P n (C) with a canonical metric if any one of the following three conditions is satisfied: THEOREM
(I ) M is ipinched; (2) M is Iholomorphically pinched; (3) M is IKaehler pinched.
As stated at the end of §7 of Chapter IX, case (2) follows from Theorems 7.8 and 7.9 (cf. Igusa [1]). Condition (3) clearly implies condition (2). By a simple algebraic manipulation, we obtain condition (2) from condition (1) (cf. Berger [5]).
370
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
2.
A complete bholomorphically pinched Kaehler manifold with b > 0 is compact and simply connected. THEOREM
The proof of Theorem 2 is essentially the same as that for the theorem of Myers (Theorem 5.8 of Chapter VIII) as pointed out by Tsukamoto [1]. (In the proof of Theorem 3.3 of Chapter VIII, we set Y = J( 7) and then argue in the same way as in the proof of Theorem 5.8 of Chapter VIII.) The theorem of Myers can be sharpened in the Kaehlerian case as follows (cf. Kobayashi [17]):
3. A compact Kaehler manifold M with positivedefinite Ricci tensor is simply connected. THEOREM
The proof makes use of Myers' theorem, the RiemannRoch Theorem of Hirzebruch [3], and the following result of Bochner [1] (cf. Lichnerowicz [3]): 4. A compact Kaehler manifold M with positivedefinite Ricci tensor admits no nonzero holomorphic pforms for p > 1. In other words, HP.O(M; C) = Oftr p > 1. THEOREM
An analogous theorem for a compact Kaehler manifold with positive holomorphic sectional curvature is not known. The following result is due to Bishop and Goldberg [3]. 5. Let 1\;[ be a complete Kaehler manifold which is either bpinched with b > a or bholomorphically pinched with b > i. Then THEOREM
dimH2(M; R) = 1. Since the direct product of two copies of complex projective space is iholomorphically pinched, i in Theorem 5 cannot be lowered. It would be of some interest to note that a iholomorphically pinched Kaehler manifold has positivedefinite Ricci tensor (Berger [5]). Theorem 5 implies that if M is a product of two compact complex manifolds, then M cannot admit a Kaehler metric of positive sectional curvature or of holomorphic pinching
>1·
A Kaehlerian analogue of the Sphere Theorem would be that if M is a compact Kaehler manifold of dimension n with positive sectional curvature, then Mis holomorphically homeomorphic to Pn(C); it is, however, yet to be proved. Rauch [3] obtained some
NOTES
371
preparatory theorems in this direction. Do Carmo [1] obtained a Kaehlerian analogue of Rauch's comparison theorenl and used it to prove that a compact ndimensional Kaehler manifold M with Kaehlerian pinching t5 > 0.8 has the same Z2 cohomology ring as the complex projective space Pn(C). This Kaehlerian pinching number has been improved by different methods. Klingenberg [8] proved that if M has Kaehlerian pinching t5 > 0.64, then M has the same homotopy type as Pn(C). Kobayashi [20] gave the Kaehlerian pinching number ~ = 0.571. . ... To date, the following result of Klingenberg [8] gives the best estimate for Kaehlerian pinching.
6. A compact ndimensional Kaehler manifold M with Kaehlerian pinching >1f!6 = 0.562 ... has the same homotopy type as the complex projective space Pn (C) . While Klingenberg makes use of Morse theory, Kobayashi reduces the problem to the Sphere Theorem by establishing the following theorem. THEOREM
7. If M is a compact Kaehler manifold with Kaehlerian pinching > t5, then there exists a principal circle bundle P over M with a Riemannian metric whose (Riemannian) pinching is greater than t5J (4 3t5). THEOREM
Thanks to Theorem 5, the proof of Theorem 7 (Kobayashi [20]) can be simplified considerably. The proof of Theorem 7 gives also the following result. 8. If M is a compact Kaehler manifold with holomorphic pinching >t5, then there exists a principal circle bundle P over M with a Riemannian metric whose (Riemannian) pinching is greater than (3t5 2)/(4  3t5). THEOREM
Theorem 8 is an improvement by Bishop and Goldberg [1] of a similar result obtained by Kobayashi [20]. From Theorem 8 we obtain 9. A compact ndimensional Kaehler manifold M with holomorphic pinching >t has the same homotopy type as the complex projective space P n (C) . THEOREM
Berger [19] proved that an ndimensional compact KaehlerEinstein manifold M with positive sectional curvature is isometric
372
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
to Pn(C) with a canonical metric. For a compact Kaehler manifold with constant scalar curvature, the socalled Ricci form is harmonic. From Theorem 5, it follows (Bishop and Goldberg [4]) that a compact Kaehler manifold with constant scalar curvature and with positive sectional curvature is an Einstein manifold. Hence,
10.. An ndimensional compact Kaehler manifold with constant scalar curvature and with positive sectional curvature is holomorphically isometric to Pn(C) with a canonical metric. THEOREM
Analyzing the proof of Berger [19], Bishop and Goldberg [4] obtained also the following result.
11. An ndimensional compact Kaehler manifold with constant scalar curvature and with holomorphic pinching > t is holomorphically isometric to Pn (C) with a canonical metric. THEOREM
For a Kaehler manifold of dimension 2 we have the best possible result due to Andreotti and Frankel (see Frankel [3]) :
12. A 2dimensional compact Kaehler manifold with positive sectional curvature is holomorphically homeomorphic to the complex projective space P 2 (C). THEOREM
The proof relies on the known classification of algebraic surfaces. Goldberg and Kobayashi [3] introduced the notion of holomorphic bisectional curvature. Given two Jinvariant planes/p and P' in T,,/ M), the holomorphic bisectional curvature H(p, P') is defined by H(p,p') = R(X, JX, Y, JY), where X (resp. Y) is a unit vector in p (resp. p'). It is a simple matter to verify that R(X, JX, Y, JY) depends only on p andp'. It is clear that H(p, p) is the holomorphic sectional curvature determined by p. By Bianchi's identity (Theorem 5.3 of Chapter III) we have
H( p, p')
=
R(X, Y, X, Y)
+ R(X, JY, X, JY).
The right hand side is a sum of two sectional curvatures (up to constant factors). We may therefore consider the concept of
NOTES
373
holomorphic bisectional curvature as an .intermediate concept between those of sectional curvature and holomorphic sectional curvature. In Theorems 5, 10, and 12 above, the assumption of positive sectional curvature may be replaced by that of positive holomorphic bisectional curvature (cf. GoldbergKobayashi [3]).
Note 24. Structure theorems on homogeneous complex manifolds The first systematic study of compact homogeneous complex manifolds was done by H. C. Wang [4] who classified completely the Cspaces, i.e., the simply connected compact homogeneous complex manifolds. One of his main results is stated in the following theorem.
1. Let K be a connected compact semisimple Lie group, T a toral subgroup of K, and C( T) the centralizer of Tin K. Let U be a closed connected subgroup of K such that (C( T))s c U e C( T), where (C( T))s denotes the semisimple part ofC( T). Then the coset space KIU, if evendimensional, has an invariant complex structure. Conversely, every Cspace can be thus obtained. i THEOREM
A Cspace KIU is a holomorphic fibre bundle over a homogeneous Kaehler manifold KIC( T) with complex toral fibre U/(C( T))s' According to Goto [2], the base space K/C( T) is projective algebraic and rational. We may say that a Cspace K/ U lies between K/(C(T))s (called an Mspace in Wang [4]) and a homogeneous Kaehler manifold K/C( T). This result was later generalized by GrauertRemmert [1]:
2. Every compact homogeneous complex manifold M is a holomorphic fibre bundle over a homogeneous projective algebraic manifold V with a complex parallelizable fibre F. THEOREM
A complex manifold is said to be complex parallelizable if its tangent bundle is complex analytically trivial. According to H. C. Wang [5], a compact complex parallelizable manifold is of the form GID, where G is a complex Lie group and D is a discrete subgroup of G (see Example 2.3 of Chapter IX). Wang has shown also that the following three conditions on a Cspace KIU are equivalent: (1) U = C( T); (2) the second Betti
374
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
number of KjU is nonzero; (3) the Euler number of KjU is nonzero. In particular, a Kaehlerian Cspace is necessarily of the form KjC( T). The following result is due to Matsushima [4J: 3. Every compact homogeneous Kaehler manifold is a Kaehlerian direct product of a Kaehlerian Cspace and afiat complex torus. THEOREM
By a homogeneous Kaehler manifold we mean of course a Kaehler manifold on which the group of holomorphic isometric transformations is transitive. BorelRemmert [lJ generalized the result of Matsushima to a compact Kaehler manifold on which the group of holomorphic transformations is transitive. (See also Aeppli [2J.) In the noncompact case, results are not so complete. Let M = GjH be a homogeneous complex manifold with an invariant volume element v. In terms of a local coordina.te system z\ ... , zn of M, v may be written in the form v = V dz 1 A ... A d:z 1 A ... A d:z n • Then we define a Hermitian form h = 2 ~ hij dz i d:z i by hij
=
0 2 log Vjozi ot}.
The Hermitian form h is welldefined, independently of the local coordinate system chosen. (Note that the same construction was used in obtaining the Bergman metric from the Bergman kernel form in §6 of Chapter IX). Koszul [2J considered the case where the canonical Hermitian form h is nondegenerate. If H is compact, then a homogeneous complex manifold M = GjH admits always an invariant volume element v and hence the canonical Hermitian form h. Among other things Koszul proved: 4. Let M = GjH be a homogeneous complex manifold with G connected and semisimple and H compact. If the canonical Hermitian form h is nondegenerate, then (1) H is connected,(2) the center of G is finite ,(3) the number of the negative squares in the canonical Hermitian form h is equal to the difference between the dimension of a maximal compact subgroup of G and the dimension of H. THEOREM
If h is positivedefinite in Theorem 4, then H is a maximal compact subgroup of G. Hence
NOTES
375
Let M = GjH be a homogeneous complex manifold with G connected and semisimple and H compact. If the canonical Hermitianform h is positivedefinite, then GjH is a Hermitian symmetric space of noncompact type. This corollary was also obtained by Borel [1] and was later generalized by Hano [2] to the case where G is unimodular. If M is a homogeneous bounded domain in Cn, then M is of the form GjH with H compact and its canonical Hermitian form h is nothing but the Bergman metric of M and hence is positivedefinite. PyatetzkiShapiro [1, 2] discovered homogeneous bounded domains in Cn which are not symmetric. HanoKobayashi [1] considered the case where h is degenerate and obtained the following COROLLARY.
5. Let M = GjH be a homogeneous complex manifold with an invariant volume element v. Then there is a closed subgroup L of G containing H such that (1) GjL is a homogeneous symplectic manifold, i.e., admits a closed 2form of maximum rank invariant by G; (2) LjH is a connected complex submanifold of GjH and is complex parallelizable. THEOREM
In the fibration of G/H over GjL with fibre LjH, the fibres are the maximal integral submanifolds of GjH defined by the distribution {X E T(GjH); h(X, .) = O}. It is not known whether the base space GjL is homogeneous complex and the fibration is holomorphic. In the special case where G is semisimple and GjH is compact, G j L is a homogeneous Kaehler manifold and the fibration is holomorphic (see Matsushima [7]). In Theorems 4 and 5, the group G is real and usually far from being complex as in the case where M = Gj His a bounded domain. Matsushima [6] considered homogeneous complex manifolds GjH where G is a complex Lie group and H is a closed complex subgroup, and he characterized those which are Stein manifolds.
Note 25. Invariant connections on homogeneous spaces 1. In §2 of Chapter X, we discussed the existence of invariant affine connections on reductive homogeneous spaces. Conversely,
376
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
given a differentiable manifold M with an affine connection we may ask under what conditions M admits a transitive group of affine transformations. Theorem 2.8 in Chapter X provides one answer. Generalizing this theorem, Kostant [5] gave the following result. Let V and V' be two affine connections on a differentiable manifold M, and let S be the tensor field of type (1, 2) which is the difference of V' and V, that is, S(X, Y) = VxY  VxY, as in Proposition 7.10 of Chapter III. We say that V is rigid with respect to V' if S is parallel with respect to V' (i.e., V'S = 0). A result of Kostant is: Let V be an affine connection on a simply connected manifold M. Then M is a reductive homogeneous space of a connected Lie group with V as an invariant affine connection if and only if there is an aiJine connection V on M such that (1) V'is invariant under parallelism (see p. 262 of Volume I) JO (2) V is rigid with respect to V' JO (3) V'is complete. A similar result was obtained by Molino [1]. We say that an affine connection V is locally invariant with respect to another affine connection V' if each of the tensor fields T, R, vm T, vmR, 1 < m < 00 (namely, the torsion and curvature tensor fields and their successive covariant differentials) is parallel with respect to V'. A result of Molino states: If an analytic affine connection V on a real analytic manifold M is locally invariant with respect to another affine connection, then M with V is locally isomorphic to a homogeneous space with a certain invariant affine connection. 2. NguyenVan Hai [2], [3], [4] studied conditions for the existence of an invariant affine connection on a (not necessarily reductive) homogeneous space. His result in [5] generalizes some results in Nomizu [8], [9] and is related to the problem of characterizing an affine connection which admits a transitive group of affine transformations. The problem of characterizing a Riemannian manifold which admits a transitive group of isometries was studied by Ambrose and Singer [2], Singer [2], Nomizu [8], [9] (see Note 19). 3. In Corollary 5.4 of Chapter X, we proved that a simply connected naturally reductive homogeneous space GjH with an invariant Riemannian metric is irreducible (as a Riemannian manifold) if G is simple. Let GjH be a Riemannian homogeneous space such that G is compact and the Euler number X(Gj H) is non zero (and I
NOTES
377
hence positive). If G is simple, GJH is an irreducible Riemannian manifold (Hano and Matsushima [1]). Conversely, if GJH is an irreducible Riemannian manifold, G is simple provided that G is effective on GJH (Kostant [4]). Kostant gave an example which shows that the condition on X(GJH) cannot be dropped. For detailed analysis of the holonomy groups and their reducibility for naturally reductive homogeneous spaces, and for arbitrary Riemannian homogeneous spaces, see Kostant [2] and [4], respectively. 4. As part of the general problem of determining all complete Riemannian manifolds with strictly positive curvature (see Note 22), Berger [8] studied a compact homogeneous space GIH with an invariant Riemannian metric which is naturally reductive in the manner of Corollary 3.6 of Chapter X. In the notation there, the condition that the sectional curvature is positive implies that [X, Y] i= 0 for any linearly independent vectors in the subspace m. Studying this algebraic condition for the Lie algebra, he 0 btained all such spaces and showed, among other things, that with two exceptions such spaces GIH are homeomorphic, but not necessarily isometric, to compact Riemannian symmetric spaces of rank 1. For example, S2n+1 admits a Riemannian metric of the type considered which is not of constant curvature. For the exceptional cases, he showed that they are not homeomorphic to compact Riemannian symmetric spaces of rank 1. 5. It is known that a Kaehlerian homogeneous space GIH of a reductive Lie group G is the direct product of Kaehlerian homogeneous spaces W o, WI' ... , W m' where W o is the center of G with an invariant Kaehlerian structure (hence a complex torus) and WI' ... , W m are simply connected Kaehlerian homogeneous spaces of simple Lie groups (Borel [1], Lichnerowicz [1], Matsushima [4]). Hano and Matsushima [1] showed that this decomposition is precisely the de Rham decomposition given in Theorem 8.1 of Chapter IX; they showed that if G is simple, then GIH is irreduciblea result to be contrasted with the Riemannian case described in 3 of this note. They also proved that if GJH is a Kaehlerian homogeneous space with semisimple G, then the decomposition ofthe Lie algebra 9 = m + 1) such that ad(H)m = m is unique and that if the corresponding canonical connection coincides with the Kaehlerian connection, then GIH is Hermitian symmetric.
378
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
6. Let G(H be a Riemannian homogeneous space where G is connected '" and effective on G/ H. If the connected linear isotropy group HO is irreducible, then G/H is simply connected and Riemannian symmetric provided that G(H is noncompact (a result of Matsushima; for the proof, see the appendix in Nagano [8]).
Note 26. Complex submanifolds 1. The results of ThomasE. Cartan and Fialkow on Einstein hypersurfaces M of Rn+\ n > 3, in Theorem 5.3 of Chapter VII were extended by Fialkow [1] to the case of hypersurfaces in a real space form (i.e. a Riemannian manifold of constant curvature c). The analogous problem for complex hypersurfaces in a complex space form (i.e. a Kaehler manifold of constant holomorphic sectional curvature) was studied by B. Smyth [1]. The standard models of complex space forms are the complex Euclidean space Cn with flat metric, the complex projective space Pn(C) with FubiniStudy metric (see Example 6.3 of Chapter IX and Example 10.5 of Chapter XI), and the unit ball Dn(C) with Bergman metric (see Example 6.5 of Chapter IX and Example 10.7 of Chapter XI). Smyth showed that an Einstein complex hypersurface in a complex space form is locally symmetric, and he proved the following classification theorem:
The only (simply connected) complete Einstein complex hypersurfaces M in Cn+l [resp. D n+1 (C)], n > 2, are Cn [resp. Dn(C)]. The only complete Einstein complex hypersurfaces Min Pn+1 (C) , n > 2, are Pn(C) or complex quadrics (see Example 10.6 ofChapter XI). THEOREM.
The corresponding local theorem was proved by Chern [27]. Takahashi [3] showed that the condition that M is Einstein can be relaxed to the condition that the Ricci tensor S of M is parallel. Kobayashi [28] obtained the following partial generalization: If M is a complete complex hypersurface with constant scalar curvature in Pn+1(C), n > 2, then M is either a projective hyperplane Pn(C) or a complex quadric. 2. Continuing Smyth [1], Nomizu and Smyth [1] extended the above results covering the case n = 1 and also removing the assumption of simplyconnectedness for M in the classification theorem. They also proved that the linear holonomy group of any
NOTES
379
complex hypersurface in D n+1 (C) Crespo Pn+l(C)] is isomorphic to U(n) Crespo U(n) or SO(n) X S\ the second case arising only when M is locally a complex quadric]. The linear holonomy group of a complex hypersurface in Cn+l was also studied by Kerbrat [1]. Nomizu and Smyth [1] also studied local rigidity and other related problems for complex hypersurfaces. 3. Let f be a Kaehlerian (i. e. holomorphic and isometric) immersion of a Kaehler manifold M into a Kaehler manifold M. As we saw in Proposition 9.2 of Chapter IX, the difference f!,.(P) of the holomorphic sectional curvatures of M and M for a holomorphic plane spanned by unit vectors X and J X of M is equal to 11\l(X, X) 1 2, where \l is the second fundamental form. Thus f!,.(p) < O. O'Neill [4] studied Kaehlerian immersions for which f!,. (p) is a constant for all holomorphic planes p and proved that such an immersion is totally geodesic (i.e., \l = 0) if m < n(n + 3) /2, where nand m are the complex dimensions of M and AIJ. He also showed that this result no longer holds if m = n(n + 3)/2. 4. For more remarks and results on Kaehlerian immersions, see Note 18. For other results and problems on complex submanifolds of a complex projective space, see Chern [19] and Pohl [2].
Note 27. Minimal submanifolds An isometric immersion f of a Riemannian manifold Minto another Riemannian manifold M is said to be minimal if the mean curvature normal (or its length, which is the mean curvature) defined in §5 of Chapter VII is O. We also say that M is a minimal submanifold immersed in M. It is locally obtained as an extremal for ndimensional volume for deformations with a fixed boundary, where n = dim M. (See Eisenhart [1].) A totally geodesic submanifold is, ofcourse, minimal. We showed that there is no compact minimal submanifold in a Euclidean space (§5 of Chapter VII). See also Example 1 of Note 14 for another proof based on the Laplacian. The same result holds when the Euclidean space is replaced by a simply connected Riemannian manifold with nonpositive sectional curvature (see O'Neill [2]). See also Myers [5].
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FOUNDATIONS OF DIFFERENTIAL GEOMETRY
Frankel [4] proved the following. Let M be a compact minimal submanifold immersed in a complete Riemannian manifold M. Then (1) If M has nonpositive sectional curvature, the fundamental group 'TTl (M) is infinite. (2) If M has positive definite Ricci tensor, the natural homomorphism 'TTl(M) + 'TTl(M) is surjective. In Note 14 we showed that a kdimensional submanifold M of Sn(r) c Rn+l is minimal if and only if A' L1X~
k. = x\ 2
r
o
c2, Studies in Math. Analysis and Related Topics, pp. 377387. Stanford Univ. Press, 1962. (MR 26 #2991.) STUDY, E. [1] Ktirzeste Wege im komp1exen Gebiete, Math. Ann. 60 (1905), 321377. Suss, W. [1] Zur relativen Differentia1geometrie V. Tohoku Math. J. 30 (1929), 202209.
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Summary of Basic Notations We summarize only those basic notations that are used most frequently throughout Volumes I and II. 1. l:~, l:i,1' ... , etc. stand for the summation taken over i or i, j, ... , where the range of indices is generally clear from the context. 6 denotes the cyclic sum, e.g. 6R(X, Y) Z = R(X, Y)Z + R(Y, Z)X + R(Z, X)Y.
2. Rand C denote the real and complex number fields, respectively. , xn ) Rn: vector space of ntuples of real nurnbers (xl, Cn: vector space ofntuples of complex numbers (zl, , zn) (x,y): standard inner product ~i xji in Rn (~i xji in cn) GL (n; R) : general linear group acting on R n gl(n; R): Lie algebra of GL(n; R) GL(n; C): general linear group acting on Cn gl(n; C): Lie algebra of GL(n; C) O(p, q): orthogonal group for ~r=l (X i )2 + ~f~;+l (X i )2 on Rp+q
O(n)
0(0, n): orthogonal group for the standard inner product in Rn o(p, q): Lie algebra of O(p, q) o(n): Lie algebra of O(n) U(p , q).• unitary group for  ~~ xiii + ~~+q xiii on Cp+q U(n) = U(O, n): unitary group for the standard inner product in Cn u(p, q): Lie algebra of U(p, q) u(n): Lie algebra of U(n) T~( V): tensor space of type (r, s) over a vector space V T (V) : tensor algebra over V Vc: complexification of a real vector space V An: space Rn regarded as an affine space A(n; R) : group of affine transformations of An a(n; R): Lie algebra of A(n; R) =
~=1
455
~=p+l
456
SUMMARY OF BASIC NOTi\TIONS
G(n, p) : Grassmann manifold of nplanes in Rn+ p G(n, p): Grassmann manifold of oriented nplanes in Rn+p V( n, p): Stiefel manifold of n frames in R n+p G11 q( C) : complex Grassmann manifold ofpplanes in Cp+q q: space of complex matrices ,  Z with q rows and p columns such that 1p  tZZ is positivedefinite Pn(C) = Gn,l(C) : complex projective space D n = Dn,l: unit disk in Cn
D:
3. M denotes an ndimensional differentiable manifold. Tao,(M) : tangent space of M at x ~(M): algebra of differentiable functions on M ~(M): Lie algebra of vector fields on M :1:( M) : algebra of tensor fields on M 1)(M): algebra of differential forms on M T(M): tangent bundle of M L (M) : bundle of linear frames of M O(M) : bundle of orthonormal frames of M (with respect to a given Riemannian metric g) o = (Oi): canonical Iform on L(M) or O(M) A (M) : bundle of affine frames of M T;( M) : tensor bundle of type (r, s) of M
.f*:
differential of a differentiable mapping f f*w: the transform of a differential form w by f xt : tangent vector of a curve X t at the point X t Lx: Lie differentiation with respect to a vector field X
4. For a Lie group G, GO denotes the identity component and 9 the' Lie algebra of G. La: left translation by a E G R a : right translation by a E G ad a: inner automorphism by a E G; also adjoint representation in 9 P(M, G): principal fibre bundle over M with structure group G
A *: fundamental vector field corresponding to A
E
9
w: connection form
.Q: curvature form
E(M, F, G, P) : bundle associated to P(M, G) with fibre F
457
SUMMARY OF BASIC NOTATIONS
5. For an affine (linear) connection on M, 8 = (8'L) : torsion form r}k: Christoffel symbols 'Y(x) : linear holonomy group at x E M (x): affine holonomy group at x E M Vx: covariant differentiation with respect to a vector (field) X
R: curvature tensor field (with components RJkl)' giving rise to curvature transformations R( X, Y). Also, the Riemannian curvature tensor:
R(X, Y, Z, W)
=
g(R(Z, W) Y, X)
T: torsion tensor field (with components T;k)
S: Ricci tensor field (with components R i j = ~k R:ki ) K(P): sectional curvature for a 2plane p in Tx(M), where M is a Riemannian manifold ~{(M): group of all affine transformations of M a(M): Lie algebra of all infinitesimal affine transformations ofM ~ (M) : group of all isometries of a Riemannian manifold M i(M) : Lie algebra of all infinitesimal isometries 6. For a manifold M immersed in a Riemannian manifold N, Tx(M)..l: normal space to M at x T( M) ..1: normal bundle of M O(N, M): bundle of adapted frames x( M) ..1: set of vector fields normal to M cx(X, Y) : second fundamental form defined as a mapping
A~:
endomorphism such that g(A~X, Y) = g(a(X, Y), ~), where ~ E Tx(M)..l hi(X, Y): symmetric bilinear forms on Tx(M) defined by
cx(X, Y) =
L hi(X,
Y) ~i'
where ~1' ••• , ~'P is an orthonormal basis of Tx(M)..l 'YJ:
1 mean curvature normal = n ~.'L (trace A l:) Sl
~i ' n =
dim M
458
SUMMARY OF BASIC NOTATIONS
K n : Gaussian curvature of a hypersurface M (n t(x) : type number
=
dim M)
7. For a manifold M with an almost complex structure J, T~(M) : complex tangent space at x ~,o: space of tangent vectors of type (1, 0) T~,l: space of tangent vectors of type (0, 1) (£( M) : set of complex rforms (£P,q(M) : set of complex forms of degree (p, q) C( M) : bundle of complex linear frames with structure group GL(n: C) U(M): bundle of unitary frames (with respect to a Hermitian metric) with structure group U(n) 8. For a homogeneous space GjH, ,.., H: linear isotropy group 9 = m + I): decomposition of Lie algebra such that
em,
I)]
c
m,
X m and Xl) denoting the mcomponent and the I)component of X E g, respectively B: KillingCartan form a: involutive automorphism of G which defines a symmetric space (G, H, a) (g, I), a) : symmetric Lie algebra (g*, I), a*) : dual of symmetric Lie algebra (g, I), a)
9. For a vector bundle E over M, ci(E) : ith Chern class c(E) : total Chern class ch (E): Chern character Pk(E) : kth Pontrjagin class X(E) : Euler class
Index for Volumes I and II (Note: For example, 1201 refers to p. 201 of Volume I, whereas 201 refers to p. 201 of Volume II.) Abelian Lie algebra, 325 variety, 131 Absolute parallelism, 1122 Adapted frame, 2, 54 Adjoint representation, 140 Affine connection, 1129 generalized, 1127 invariant, 375 invariant by parallelism, 194 rigid, 376 frame, 1126 holonomy group, 1130, 331 locally symmetric, 222 mapping, 1225 parallel displacement, 1130 parameter, 1138 symmetric, 223 space, 1125 tangent, 1125 transformation, 1125, 1226 infinitesimal, 1230 Allowable imbedding, 52 Almost cocomplex structure, 383 complex connection, 143 complex manifold, 121 natural orientation of, 121 complex mapping, 122, 127 complex structure, 121 conjugate, 123 integrability of, 124, 321 integrable, 124, 321 invariant, 216 on spheres, 138140 torsion of, 123
Almost effective action, 187 Hamiltonian manifold, 149 Hermitian manifold, 147 Kaehler manifold, 149 product structure, 384 symplectic manifold, 149 Alternation, 128 Analytic continuation, 1254 Arclength, 1157 Atlas, 12 complete, 12 Augmented index, 89 Automorphism of an almost complex structure, 127 a connection, 181 a Gstructure, 1307, 186, 333 a Lie algebra, 140 a Lie group, 140 Autoparallel submanifold, 53 Bergman metric, 163 Bianchi's identities, 178, 1121, 1135 Bonnet's theorem, 78 Bounded domain, 162, 375 symmetric, 263 Bundle associated, I55 holonomy, 185 homomorphism of, I53 induced, 160 normal, 3 of adapted frames, 2, 54 of affine frames, 1126 of complex linear frames, 141 of normal frames, 2 of orthonormal frames, 160 459
460
INDEX FOR VOLUMES I AND II
Bundle of unitary frames, 152 principal fibre, I50 reduced, I53 sub , I53 tangent, I56 tensor, I56 trivial, I50 vector, 1113 Hermitian, 178 orientable, oriented, 314 Riemannian, 315 Cspace, 138, 373 canonical complex structure, 115 connection, 1110, 1301, 7, 230 on a symmetric space, 230 decomposition (= de Rham decomposition), 1185, 1192, 171, 246, 263, 331 decomposition of a symmetric Lie algebra, 226 flat connection, 192 form on L(M), 1118 Hermitian form, 374 invariant connection, 1110, 1301, 192, 230 invariant Riemannian metric, 1155 linear connection, 1302 metric, 1155 Iform on a group, 141 parameter of a geodesic, 1162 Cayley numbers, 139 Center of gravity, 109 Characteristic class, 293 Chart, 12 Chern character, 311 Chern class, 305 Christoffel's symbols (rj,,), 1141 Clifford translation, 105 Codazzi, equation of, 2526 Compactopen topology, 146 Compact real form, 291 Compact type Lie algebra of, 204, 252, 329 symmetric Lie algebra of, 252
Compact type symmetric space of, 252, 256 Comparison theorem of Rauch, 76 Complete linear connection, 1134 Riemannian manifold, 1172 Riemannian metric, 1172 vector field, 113 Complex affine (locally) symmetric, 259 conjugation, 116 contact form, 385 differential form, 124, 125 holomorphic, 129 Grassmann manifold, 133, 160, 286 homogeneous space, 220, 373 hyperbolic space, 282 hypersurface, 378, 379 Lie algebra, 120, 329 Lie group, 130 linear frame, 141 manifold, 13, 121 parallelizable, 132, 373 projective space, 134, 159, 273 quadric, 278, 378 structure (on a vector space), 114 structure (on a manifold), 122 submanifold, 164, 175, 378 tangent space, 124 tangent vector, 124 of type (1, 0) or (0, 1), 125 torus, 131, 159 Complexification of a Lie algebra, 329 of a vector space, 116 Components of a linear connection, 1141 of a Iform, 16 of a tensor (field), 121, 126 of a vector (field), 15 Conformal transformation, 1309 infinitesimal, 1309 Conjugate almost complex structure, 123 complex manifold, 123 point, 67, 71 multiplicity of, 88
INDEX FOR VOLUMES I AND II
Connection, 163 affine, 1129 canonical, 1110, 1301, 7, 230 canonical flat, 192 canonical invariant, 1110, 1301, 192, 230 canonical linear connection, 1302 flat, 192 form, 164 generalized affine, 1127 Hermitian, 178 in normal bundle, 4, 15 induced, 182 invariant, 181, 1103, 376 by parallelism, 1262, 194 LeviCivita, 1158 linear, 1119 metric, 1117, 1158 ( ), 198 natural torsionfree, 197 normal invariant, 208 (+ ), 198 Riemannian, 1158 torsionfree, 332 universal, 1290, 332 (0), 199 Constant curvature, 1202 space of, 1202, 1204, 24, 71, 264, 268 surface of, 343 Constant holomorphic sectional curvature, 168 space of, 134, 159, 169, 282 Contact form, 381 complex, 385 Contact structure, 381 almost, 382 complex, 385 Contraction, 122 Contravariant tensor (space), 120 Convex hypersurface, 40 strictly, 40 neighborhood, 1149, 1166 Coordinate neighborhood, 13 Covariant derivative, 1114, 1115, 1122 differential, 1124
461
Covariant differentiation, 1115, 1116, 1123 tensor (space), 120 Covector, 16 Covering space, 161 Critical point, 362 index of, 362 nondegenerate, 362 Cross section, I57 adapted to a normal coordinate system, 1257 Cubic neighborhood, 13 Curvature, 1132 constant, 1202 form, 177 Gaussian (GaussKronecker), 33 holomorphic bisectional, 372 holomorphic sectional, 168 Kaehlerian sectional, 369 mean, 33 operator, 367 principal, 32 recurrent, 1305 scalar, 1294 sectional, 1202 tensor (field), 1132, 1145 Riemannian, 1201 total, 362 transformation, 1133 Cut locus, 100 Cut point, 96 Cylinder, 1223 Euclidean, 1210 twisted, 1223 Degree (P, q), 125 Derivation of ID(M), 133 of ;reM), 130 of tensor algebra, 125 Derived series of a Lie algebra, 325 Descending central series of a Lie algebra, 325 Development, 1131 Diffeomorphism, 19 Differential covariant, 1124
462
INDEX FOR VOLUMES I AND II
Differential form, 16, 17 complex, 124, 125 holomorhpic, 129 of a function, 18 of a mapping, 18 Direct product of symmetric spaces, 228 Direct sum of symmetric Lie algebras, 228 Discontinuous group, 144 properly, 143 Distance function, 1157 Distribution, 110 involutive, 110 Divergence, 1281, 337 Dual symmetric Lie algebra, 253 Effective action of a group, 142, 187 almost, 187 Effective symmetric Lie algebra, 226 Effective symmetric space, 225 almost, 225 Einstein hypersurface, 36, 378 manifold, 1294, 336, 341 Elliptic linear Lie algebra, 334 space form, 1209, 264 nframe, 6 Equation of Codazzi, 25, 26, 47 Gauss, 23, 26 Jacobi, 63 Equivalence problem, 1256, 357 Equivariant isometric imbedding, 356 Euclidean cylinder, 1210 locally, 1197, 1209, 1210 metric, 1154 motion, 1215 subspace, 1218 tangent space, 1193 torus, 1210 Euler class, 314 Exponential mapping, 139, 1140, 1147
Exterior covariant derivative, 177 covariant differentiation, 177 derivative, 17, 136 differentiation, 17, 136 Fibre, I55 bundle, principal, I50 metric, 1116 transitive, 1106 Finite type, linear Lie algebra of, 333 First normal space, 353 Flat affine connection, 1209 connection, 192 canonical, 192 linear connection, 1210 Riemannian manifold, 1209, 1210 Form curvature, 177 1form, 16 rform, 17 tensorial, 175 pseudo, 175 torsion, 1120 Frame adapted, 2, 54 affine, 1126 complex linear, 141 linear, I55 normal,2 orthonormal, 160 unitary, 152 Free action of a group, 142 Frobenius, theorem of, 110, 323 FubiniStudy metric, 160, 274 Fundamental theorem for hypersurfaces, 47 2form of a Hermitian manifold, 147 vector field, I51 Gauss equation of, 23, 26, 47 formula, 15, 18 spherical map of, 9, 18 theorema egregium of, 33
INDEX FOR VOLUMES I AND II
GaussBonnet theorem, 318, 358 Gaussian (GaussKronecker) curvature, 33 Geodesic, 1131, 1146 minimizing, 1166 totally, 1180, 54 Gradient, 337 Grassmann manifold, 6, 9, 271 complex, 133, 160, 286 of oriented pplanes, 9, 272 oriented, 272 Green's theorem, 1281 Gstructure, 1288, 332, 333 Hamiltonian manifold, 149 almost, 149 Harmonic function, 339 Hermitian connection, 178 inner product, 118 locally symmetric, 259 manifold, 147 metric, 146 symmetric, 259 vector bundle, 178 Holomorphic, 12 bisectional curvature, 372 form, 129 sectional curvature, 168 transformation, 336 vector field, 129 Holonomy bundle, 185 Holonomy group, 171, 172 affine, 1130 homogeneous, 1130 infinitesimal, 196, 1151 linear, 1130 local, 194, 1151 of a Kaehler manifold, 173 of a submanifold, 355 restricted, 171, 172 Holonomy theorem, 185 Homogeneous complex manifold, 220, 373 coordinate system, 134 holonomy group, 1130
463
Homogeneous Kaehler manifold, 374, 376 Riemannian manifold, 1155, 1176, 200, 208, 211, 376 space, 143 complex, 220 Kaehlerian, 374, 376 naturally reductive, 202 reductive, 190, 376 symmetric, 1301, 225 strongly curvature, 357 Homomorphism of fibre bundle, I53 symmetric Lie algebra, 227 symmetric space, 227 Homothetic, 289 transformation, 1242, 1309 Hopf manifold, 137 Hyperbolic space form, 1209, 268 complex, 282 H ypersurface, 5 Codazzi equation of, 26, 30 complex, 378 Einstein, 36 fundamental theorem for, 47 Gauss equation of, 23, 24, 30 Gaussian curvature of, 33 in a Euclidean space, 17, 29 mean curvature of, 33 principal curvature of, 32 Ricci tensor of, 35 rigidity of, 45 second fundamental form of, 13 spherical map of, 9 type number of, 42 Imbedding, 19, I53 isometric, 1161, 354 equivariant, 356 Immersion, 19 isometric, 1161, 354 Kaehlerian, 164 minimal (in mean curvature), 376 minimal (in total curvature), 363 Indefinite Riemannian metric, 1155 invariant, 200
464
INDEX FOR VOLUMES I AND II
Index, 89 augmented, 89 form, 81 of a critical point, 362 of nullity, 347 of relative nullity, 348 theorem (of Morse), 89 Induced bundle, 160 connection, 182 Riemannian metric, 1154 Infinite type, linear Lie algebra of, 333 Infinitesimal affine transformation, 1230 automorphism of an almost complex structure, 127 a Gstructure, 186 holonomy group, 196, 1151 isometry, 1237 variation of a geodesic, 63 Inhomogeneous coordinate system, 134 Inner product, 124 Integrability conditions of almost complex structure, 125, 145, 321, 324 Integrable almost complex structure, 124, 321, 324 Integral curve, 112 manifold, 110 Interior product, 135 Invariant affine connection, 375 almost complex structure, 216 by parallelism, 1262, 194 connection, 181, 1103, 375 indefinite Riemannian metric, 200 polynomial, 293, 298 Riemannian metric, 1154 Involutive distribution, 110 Lie algebra, 225 orthogonal, 246 Irreducible group of Euclidean motions, 1218 Riemannian manifold, 1179 symmetric Lie algebra, 252
Irreducible weakly, 331 Isometric, 1161 imbedding, 1161, 354, 355, 356, 379 equivariant, 356 immersion, 1161, 354, 355 Isometry, 146, 1161, 1236, 335 infinitesimal, 1237 Isotropy group, linear, 1154, 187 subgroup, 149 Jacobi equation, 63 field, 63, 68 Kaehler manifold, 149 almost, 149 pseudo, 149 nondegenerate, 175 metric, 149 Kaehlerian homogeneous, 374, 376 pinching, 369 sectional curvature, 369 KillingCartan form, 115·5, 252, 325 of o(n + 1), 266 of o(n, 1), 270 of a symmetric Lie algebra, 250 Killing vector field, 1237 Klein bottle, 1223 Laplacian, 338 Lasso, 173, 1184, 1284 Leibniz's formula, 111 Length function, 79 LeviCivita connection, 1158 Levi decomposition, 238, 327 subalgebra, 327 Lie algebra abelian, 325 complex, 120, 329 dual symmetric, 253 effective symmetric, 226 involutive, 225
INDEX FOR VOLUMES I AND II
Lie algebra irreducible symmetric, 252 nilpotent, 325 of (non) compact type, 204, 252, 329 orthogonal symmetric, 246 reductive, 326 semisimple, 325 simple, 325 solvable, 325 symmetric, 225, 238 Lie derivative, 129 Lie differentiation, 129 Lie group, 138 complex, 130 Lie subgroup, 139 Lie transformation group, 141 Lie triple system, 237 Lift, 164, 168, 188 horizontal, 164, 168, 188 natural, 1230 Linear connection, 111 9 frame, I55 complex, 141 holonomy group, 1130 isotropy group, 1154, 187 isotropy representation, 187 Local basis of a distrib~tion, 110 coordinate system, 13 Locally affine, 1210 Euclidean, 1197, 1209, 1210 symmetric, 1303, 222, 243, 259 Lorentz group, 268 Lorentz manifold (metrif), 1292, 1297 Manifold, 12, 13 complex (analytic), 13, 121 differentiable, 12, 13 Hermitian, 147 Kaehler, 149 oriented, orientable, 13 real analytic, 12 sub, 19 symplectic, 149 MaurerCartan, equations of, 141
465
Maximal nilpotent ideal, 325 Mean curvature, 33 constant, 346 normal, 34, 340, 341 Metric connection, 1117, 1158 Minimal immersion (in mean curvature), 379 (in total curvature), 363 Minimal sUbmanifold, 34, 340, 342, 379 Minimum point, 96 (  )connection, 199 Mobius band, 1223 M ultiplicity of a conjugate point, 88 of an index form, 89 Naturallift of a vector field, 1230 Natural torsionfree connection, 197 Naturally reductive homogeneous space, 202 Nilpotent Lie algebra, 325 Noncompact type Lie algebra of, 204, 252 symmetric Lie algebra of, 252 symmetric space of, 252, 256 Nondegenerate Kaehler manifold, 173 Nonpositive curvature, space of, 29, 70, 102, 109 Nonprolongeable, 1178 Normal bundle, 3 coordinate system, 1148, 1162 frame, 2 invariant connection, 208 space, 2 Nullity index of, 347 index of relative, 348 of a bilinear form, 89 Iparameter group of transformations, 112 subgroup, 139 Orbit, 112 Orientable, oriented, vector bundle, 314 Orientation, 13 natural, 121
466
INDEX FOR VOLUMES I AND II
Orthonormal frame, 160 Orthogonal symmetric Lie algebra, 246 of (non) compact type, 252 Ovaloid, 346 Paracompact, I58 Parallel cross section, 188 displacement, 170, 187, 188 affine, 1130 tensor field, 1124 Parallelism absolute, 1122 complex, 132, 373 invariant by, 1262, 194 Partition of unity, 1272 Pinched, pinching, 364, 369 Holomorphic, 369 Kaehlerian, 369 (+ )connection, 198 Point field, 1131 Polynomial function, 298 invariant, 293, 298 Pontrjagin class, 312 Positive curvature, space of, 74, 78, 88, 364 Positive Ricci tensor, space of, 74, 88 Principal curvature, 32 Principal direction, 32 Projectable vector field, 218 Projection, covering, I50 Projective space, I52, 134, 159 Prolongation (of linear Lie algebra), 333 Properly discontinuous, 143 Pseudoconformal transformation, 335 Pseudogroup of transformations, 11, 12 PseudoKaehler manifold, 149 Pseudotensorial form, 175 Quotient space, 143, 144 Radical of a Lie algebra, 238, 325 Radon measure, 108 Rank of a mapping, 18
Rauch, comparison theorem of, 76 Real form of a complex Lie algebra, 291,330 Real projective space, I52 Real representation of GL(n; C), 115 of SL(n; C), 151 Recurrent curvature, 1305 tensor, 1304 Reduced bundle, I53 Reducible connection, 181, 183 Riemannian manifold, 1179 structure group, I53 Reduction of connection, 181, 183 of structure group, I53 Reduction theorem, 183 Reductive homogeneous space, 190 naturally, 202 Lie algebra, 326 subalgebra, 326 Restriction of tensor field, 57, 58 de Rham decomposition, 1185, 1192, 171, 246, 263, 331 Ricci form, 153, 183 Ricci tensor, 1248, 1292, 35 of a Kaehler manifold, 149 Riemannain connection, 1158 curvature tensor, 1201 homogeneous space, 1155, 1176, 200, 208, 211, 376 locally symmetric, 232, 243 manifold, 160, 1154 metric, 127, 1154, 1155 canonical invariant, 1155 indefinite, 1155 induced, 1154 invariant, 1154 symmetric, 232, 243 vector bundle, 315 Rigid, 45 affine connection, 376
INDEX FOR VOLUMES I AND II
Rigidity, 349 theorem, 43, 46, 343, 353 Scalar curvature, 1294 Schur, theorem of, 1202 Kaehlerian analogue of, 168 Second fundamental form, 13, 20 of a complex hypersurface, 175 Sectional curvature, 1202 holomorphic, 168 Kaehlerian, 369 Segment, 1168 Semisimple Lie algebra, 325 Simple Lie algebra, 325 Simple covering, 1168 Skewderivation, 133 Solvable Lie algebra, 325 Space form, 1209 Sphere theorem, 366 Spherical map of Gauss, 9, 18, 358 Standard horizontal vector field, 1119 Stiefel manifold, 6 Strongly curvature preserving, 357 Strongly curvature homogeneous, 357 Structure constants, 141 equations, 177, 178, 1118, 1120, 1129 group, I50 Subbundle, I53 Submanifold, 19, 1 autoparallel, 53 complex, 164, 175, 378 totally geodesic, 1180, 54, 234 Symmetric affine (locally), 222, 223 complex affine (locally), 259 Hermitian (locally), 259 (homogeneous) space, 225 Lie algebra, 225, 238 dual, 253 effective, 226 irreducible, 252 orthogonal, 246 locally, 1303, 222, 232 Riemannian (locally), 1302, 232, 243
467
Symmetric space, 225 effective, 225 almost effective, 225 subspace, 227 Symmetrization, 128 Symmetry, 1301, 222, 225 Symplectic manifold, 149 almost, 149 homogeneous, Symplectic structure on T*(M), 165 Synge's formula, 87 Tangent affine space, 1125 bundle, I56 space, 15 complex, 124 vector, 14 Tensor algebra, 122, 124 bundle, I56 complex, 124 contravariant, 120 covariant, 120 field, 126 product, 117 space, 120, 121 Tensorial form, 175 pseudo, 175 Torsion form of an affine connection, 1120 of an almost complex structure, 123 of two tensor fields of type (1, 1), 138 tensor (field) of an affine connection, 1132, 1145 translation, 1132 Torsionfree connection, 332 natural, 197 Torus, 162 Complex, 131, 154 Euclidean, 1210 twisted, 1223 Total curvature, 362 Total differential, 16 of the length function, 79
468
INDEX FOR VOLUMES I AND II
Totally geodesic submanifold, 1180, 54 of a symmetric space, 234, 237 Transformation, 19 Transition functions, I51 Transvection, 236 Trivial fibre bundle, I51 Twisted cylinder, 1223 torus, 1223 Type ad G, 177 of tensor, 121 (1, 0), complex vector of, 125 (0, 1), co mplex vector of, 125 Type number, 42, 349 Umbilic (umbilical point), 30 Unitary frame, 152 Universal factorization property, 117 Variation of a geodesic, 63 infinitesimal, 63
Vector, 14 bundle, 1113 orientable, (oriented), 314 Riemannian, 315 field, 15 holomorphic, 129 Vertical component, 163 subspace, 163, 187 vector, 163 Volume element, 1281 Weakly irreducible, 331 Wei! homomorphism, 297 Weingarten's formula, 15 Weyl, theorem of, 204, 291, 330 Weyl group, 305 Whitney sum, 306 formula, 306, 315 (O)connection, 199
Errata for
Foundations of Differential Geometry, Volume I p. 9
Line 10 from bottom: "independent of p a neighborhood of M in M'"
E
M" should read "constant in
p. 16
Line 2 from bottom: "constantvector" should read "constant vector"
p.21
About the middle of the page:
"el, = I:j A{ el' should read "el, "t = I:j eJ" should read "el, 
BJ
"KJL: J:: J:" should read
"K;~.:: t: J:=~"
p. 22
Line 3 from bottom:
p. 46
Line 10 from bottom: "d«((Jn(b), ((In(b))'' should read "d«((Jn(b), ((IN(b))''
p. 53
Line 12 through line 14 should read "or injection if the induced mapping ]: M' ~ M is an imbedding and if f: G' ~ G is a monomorphism. By identifying P' with"
p. 64
Line 7: "w«Ra)*X)" should read" (R:w)(X)"
p. 84
Line 1: "a horizontal curve" should read "horizontal curves"
p. 89
Line 2 from bottom: "  2w(Xt, X7)" should read "  2U(Xt, X~)"
p. 106
In the proof of Proposition 11.4, the last 6 lines beginning with "On the other hand, ..." should read as follows: Now note that the mapping Xd ~ X EX (P) is induced by the action of K on P to the left and hence satisfies the condition [~] =  [X, Y] (in contrast to the situation in Proposition 4.1, p. 42, where the group acts on the right so that we have a Lie algebra homomorphism). Thus we have
wuo([X, Y])
=
wuo(  [x:Y])
=

A([X, Y])
so that 20u o(X, Y) = [A(X), A(Y)]  A([X, Y]).
"[X, Y]"
should read" [X, Y]"
p. 111
Line 5:
p. 118
Line 10: "ith column" should read ')'th column"
p. 118
Line 11: ')·th row" should read "ith row"
p. 131
Line 4 of Proposition 4.1: "C t" should read "6/' 469
470
ERRATA FOR VOLUME I
p. 136
Line 11 from bottom: "p. 611" should read "pp. 6162"
p. 149
Line 3 of Corollary 8.6 should read "dw = (  t)r A(Vw) for wE~r(M),"
p. 181
Line 5: ''y
p. 235
Equation in (2) of Proposition 2.6:
p. 244
Line 5 from bottom: "infinitesiaml" should be "infinitesimal"
p. 247
Line 10: "respect this" should read "respect to this"
p. 255
Line 2 from bottom: Insert "of the same dimensions" at the end
p. 256
Delete the six lines of Remark.
p. 280
Line 12: "A
p.283
Lines 7: "dv(XI ,
p. 283
Lines 9 and 10 from bottom: The equation should read as follows:
dv(alax l ,
E
M" should read ''y
E
E
M' "
"+ R(X, Y)" should be "  R(X, Y)"
g" should read "A E g"
•••
••• ,
,
X n ) = 1" should read "dv(X1,
alaxn) =
L: in ci
l
ilo ...•
=
~. e
ilo ... ,
'£ i CII •• ·cnn
••
X n ) = lin!"
••• ,
·c~n dv(Xil' k
•••
,
X in )
1
dv(XI, . . . , X n ) = det (Ct ) ;;! = VGln!,
1.n
•
p. 289
Line 3 from bottom: "there exists" should read "there exist"
p. 293
Line 1 of Proposition 2: "then i" should read "then it"
p. 310
Line 12: "3o(M)" should read "3°(M)"
p.315
Berger, M. [lJ: "(1953)" should read "(1955)"
p. 318
Line 1: "Momorimoto, A." should read "Morimoto, A."
p. 318 ,Line 3 from bottom: "Forchungen" should read "Forschungen" p.319
Line 17: "Lie algebra in" should read "Lie algebra of"
p. 321
Line 2 from bottom: "dreidmensionalen" should read "dreidimensionalen"
p. 322
Line 4 from bottom: "infinitesimalgeometrie" should read "Infinitesimalgeometrie"
p. 322
Line 3 from bottom: "Einornung" should read "Einordnung"
p. 322
Line 2 from bottom: "(1912)" should read" (192t)"