Page 103 - Curvature and Homology

constant curvature K is Euclidean space (K = 0), hyperbolic space
(K < 0), or spherical space (K > 0).
Suppose M and M' are not isometric but rather that the map f defines
a homeomorphism which reproduces the metric except for a scalar
factor. We then say that M and M' are conformally homeomorphic.
A Riemannian manifold of constant curvature is called a space form.
The problem of determining the space forms becomes by virtue of the
above remarks a problem in the determination of (discontinuous) groups
of motions. A space form may then be regarded as a homogeneous space
G/H where G is the group of motions and H the isotropy subgroup
leaving a point fixed. It is therefore not surprising that the curvature
properties of a compact Riemannian manifold determine to some extent the
structure of its group of motions. In fact, it is shown that the existence or
rather non-existence of l-parameter groups of motions as well as
1-parameter groups of conformal transformations is dependent upon
the Ricci curvature of the manifold [4]. On the other hand, the existence
of a globally defined l-parameter group of non-isometric conformal
transformations of a compact homogeneous Riemannian manifold is a
sufficient condition for it to be a homology sphere. Indeed, it is then
isometric with a sphere [79].

3.2. Curvature and betti numbers

At this point, it is convenient to employ the symbol denoting a form
in the coefficients of the form as well.
Let a be a harmonic l-form of class 2 defined on a compact, orientable
Riemannian manifold M and consider the integral
(Aa, a) = da A *a (3.2.1)
over M. Since a is a harmonic form, Aa vanishes, and so
I
I
/M~a~*a=O. (3.2.2)
I
1 The expression of the integrand in local coordinates is given by
Aa A *a = (- gjk Dk Dj ai + ~!a,) E,~~...~~-, I\ du*l A ... A d~jn-1
a8
dui
= (- aigik Dk Dj ai + Rig ai a') *1 (3,2.3)
by virtue of the formulae (2.7.8), (2.7.9), and (2.12.4).
Lemma 3.2.1. For a regular I-form a on a compact and orientable
Riemannian manifold M