Page 133 - Curvature and Homology

3.9. CONFORMALLY FLAT MANIFOLDS 115
Corollary. On a compact and orientable Riemannian manifold, a necessary
and suflcient condition that the infinitesimal transformation X generate a
I-parameter group of motions is given by the equations
A5=2Q5 and S[=O.

3.9. Conformally flat manifolds

Let M be a Riemannian manifold with metric tensor g. Consider the
Riemannian manifold M* constructed from M as follows: (i) M* = M
as a differentiable manifold, that is, as differentiable manifolds M and
M* have equivalent differentiable structures which we identify; (ii) the
metric tensor g* of M* is conformally related to g, that is, g* = p2g
(p > 0). Since the quadratic form ds2 for n = 2 is reducible to the
form h[(d~l)~ + (d~~)~] (in infinitely many ways) the metric tensors of
any two 2-dimensional Riemannian manifolds are conformally related.
In the sequel, we shall therefore assume n > 2.
For convenience we write p = eo. It follows that the components gi,
and g*i, of the tensors g and g* are related by the equations

The components of the Levi-Civita connections associated with the
metric tensors g and g* are then related as follows:

A computation gives

Transvecting (3.9.2) with g4' we see that the components of the cor-
responding Ricci tensors are related by

Again, transvecting (3.9.3) with gjk we obtain the following relation
between the scalar curvatures R and R*: