Page 229 - Curvature and Homology

Corollary. If A f 2 0 (2 0) on M, then f is a constant.
The proof is an easy application of lemma 3.2.1. For,

In order to establish the above result, we put f equal to the 'square
length' of the tensor field t. But first, a tensorjield of type (i z) is said to be
holomorphic if its components (with respect to a given system of local
complex coordinates) are holomorphic functions. This notion is evidently an
invariant of the complex structure. Since the rjk and qk are the only
non-vanishing coefficients of connection, the tensor field

of type (t 3 is holomorphic, if and only if, the covariant derivatives of t
with respect to 2 for all i = 1, a*-, n are zero.
Consider the tensor field t + f. If t is holomorphic,

Applying the interchange formula (1.7.21) it follows that

Transvecting (6.5.1) with gkl* we obtain

- t1-lp+lPj, . jq RGr.
p=l
Now, put