Page 227 - Curvature and Homology

smallest (largest) to the largest (smallest) value of K is 2 provided k > 0
(< 0). To see this, let K = K(X, Y) denote the sectional curvature
determined by the vector fields X = tAa/ axA and Y = rlAi3/azA.
Then,

where <X,Y) = gij&j* denotes the (local) scalar product of the
vector fields e'ia/azi and #* a/aZi in that order.
If we put

then

Hence, since 0 5 r 5 1,

from which we conclude that

if k is positive, and if k is negative

Theorem 6.4.1. The general- sectional curvature K in a manifold of
constant holomorphic curvature k satisjies the inequalqties (6.4.1 ) for k > 0
and (6.4.2) for k < 0 where the upper limit in (64.1) and the lower limit
in (6.4.2) are attained when the section is holomorphic [5].
Thus, for k > 0 the manifold is *-pinched. This result should be
compared with theorem 3.2.7.