Page 224 - Curvature and Homology

206 VI. CURVATURE AND HOMOLOGY: KAEHLER MANIFOLDS

We have proved
Theorem 6.2.1. On a compact Kaehler manifold of positive definite
Ricci curvature, a holomorphic form of degree p, 0 <p 5 n is necessarily
zero [4, 581.

6.3. Deviation from constant holomorphic curvature
In this section a class of compact spaces having the same homology
structure as Pn and of which Pn is itself a member is considered. They
have one common local property, namely, their Ricci curvatures are
positive. Aside from this their local structures can be quite different-
their classification being made complete, however, by means of a
condition on the projective curvature tensors associated with these
spaces. They need not have constant holomorphic curvature. If instead,
a measure W of their deviation from this property is given, and if the
function W associated with a space M satisfies a certain inequality
depending on the Ricci curvature of the space, M is a member of the
class.
Consider the Kaehler manifolds (M,g) and (M,gf) of complex
dimension n. If the matrices of connection forms w and o' canonically
defined by g and g', respectively are projectively related their coefficients
of connection are related by
pi
jk=Ilfk+pj8:+pkS: (6.3.1)
(cf. 5 3.1 1 ). Since

It follows easily that the tensor w with components

is an invariant of the connections w and of. For this reason we shall call
it the projective curvature tensor of (M, g) (or (M, g')). It is to be noted
that w vanishes for n = 1. Its vanishing for the dimensions n > 1 is