Page 255 - Curvature and Homology

EXERCISES

where f is a real-valued function on M. Then,
sf = Sdf,
and so, from (6.14.3)

Assume G(+) 2 0. Then, since A 2 0, Gdf 5 0. Applying VI.F.3, we
see that Gdf vanishes identically. Thus G(+) = - A S 0. Consequently,
G(+) = 0 and A = 0. Finally, if P*Q(y, v) is positive definite at each
point of M, 4 must vanish. For, by substituting (6.14.1) into (6.14.4),
we derive
qP*Q(y,v) = G(v).
Remark: If the bundle B is the product of M and C, B = M x C,
the usual formulas are obtained.

EXERCISES

A. &pinched Kaehler manifolds [2]

1. Establish the following identities for the curvature tensor of a Kaehler
manifold M with metric g (cf. I. I):
(a) R(X, Y) = R(JX JY),
(b) K(X,Y) = K(JX,JY),
(c) K(X, JY) = K(JX, Y),
and when X, Y, JX, JY are orthonormal
(d) g(R(X, JX) Y, JY) = - K(X, Y) - K( JX, Y).
To prove (a), apply the interchange formula (1.7.21) to the tensor J defining
the complex structure of M (see proof of lemma 7.3.2); to prove (b), (c), and (d)
employ the symmetry properties (1.1. (a) - (d)) of the curvature tensor.
2.
2. If the real dimension of M is 2n(n > I), and M is &pinched, then S I
To see this, let {X,]X,Y, JY) be an orthonormal set of vectors in the tangent
space Tp at P E M. Then, from (3.2.23)

Applying (1 .(d)) we obtain
1
8 5 K(X,Y) $7 (2 - 58)
from which we conclude that 8 5 3.