Page 228 - Curvature and Homology

2 10 VI. CURVATURE AND HOMOLOGY: KAEHLER MANIFOLDS

From 1.10.4 it is seen that the Ricci curvature K in the direction of
the tangent vector X is given by

(QX, X)
= (X, X)

Therefore, in analogy with 5 1.10 a Kaehler manifold for which the
Ricci directions are indeterminate is called a Kaehler-Einstein manifold
and the Ricci curvature is given by

or, in terms of the fundamental form $2 and. the 2-form y5 determining
the 1" Chern class, y5 is proportional to $2, that is

Since y5 is closed, d~ A $2 = 0. Thus, if n > 1, K is a constant.

6.5. Holomorphic tensor fields

We have seen that there exist no (non-trivial) holomorphic p-forms
on a compact Kaehler manifold with positive Ricci curvature. In this
section this result is generalized to tensor fields of type (,P :) as follows:
Denote by A,, and Amin the algebraically largest and smallest eigen-
values of the Ricci operator & (cf. 5 3.2), respectively. Then, for a
holomorphic tensor field t of type (t g), if

everywhere and is strictly positive at least at one point, t must vanish,
that is no such tensor fields exist.
The idea of the proof is based on a part of the 'Bochner lemma'
(cf. V1.F) which for our purposes is easily established. (The non-
orientable case is more difficult to prove. The applications of this lemma
made by Bochner and others have led to many important results on the
homology properties of Riemannian manifolds). We shall state it as

Proposition 6.5.1. Let f be a function of class 2 on a compact and orientable
Riemannian manifold M. Then, if A f 0 (5 0) on M, Af vanishes
identically.