Page 17 - Curvature and Homology

INTRODUCTION xvii

geometry, in particular, Kaehler geometry along the lines proposed by
S. Chern. Its influence on the homology structure of the manifold is
discussed in Chapters V and VI. Whereas the homology properties
described in Chapter I11 ar8 similar to those of the ordinary sphere
(insofar as betti numbers are concerned), the corresponding properties
in Chapter VI are possessed by P, itself. Families of hermitian manifolds
are described, each including P, and retaining its betti numbers. One
of the most important applications of the effect of curvature on homology
is to be found in the vanishing theorems due to K. Kodaira. They are
essential in the applications of sheaf theory to complex manifolds.
A conformal transformation of a compact Riemann surface is a holo-
morphic homeomorphism. For compact Kaehler manifolds of higher
dimension, an element of the connected component of the identity of
the group of conformal transformations is an isometry, and consequently
a holomorphic homeomorphism. More generally, an infinitesimal con-
formal map of a compact Riemannian manifold admitting a harmonic
form of constant length is an infinitesimal isometry. Thus, if a compact
homogeneous Riemannian manifold admits an infinitesimal non-iso-
metric conformal transformation, it is a homology sphere. Indeed, it is
then isometric with a sphere. The conformal transformation group is
studied in Chapter 111, and in Chapter VII groups of holomorphic as
well as conformal homeomorphisms of Kaehler manifolds are in-
vestigated.
In Appendix A, a proof of de Rham's theorems based on the concept
of a sheaf is given although this notion is not defined. Indeed, the proof
is but an adaptation from the general theory of sheaves and a knowledge
of the subject is not required.