Page 140 - Curvature and Homology

1 22 111. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY
projectively related affine connections. On the other hand, two affine
connections o and w* are said to be projectively related if there exists
a covariant . vector field p, such that in the given local coordinates

Let M be a Riemannian manifold with metric g. If there exists a
metric g* on M such that the connections o and w* canonically defined
by g and g* are projectively related, then, by means of a straightforward
computation, the tensor w whose components are

is an invariant of the projectively related affine connections, that is,
the tensor w* corresponding to the connection o* projectively related
to o coincides with w. This tensor is known as the Weyl projective
curvature tensor. Its vanishing is of particular interest. Indeed, if w = 0,
the curvature of M (relative to g or g*) has the representation

Hence,
1
R,kl = (Rj~l,l - Rjl girl

from which, by the symmetry properties of the Riemannian curvature

Transvecting with gil we deduce that

Substituting the expression (3.1 1.4) for the Ricci curvature in (3.1 1.3)
eives

Thus, M is a manifold of constant curvature.
Conversely, assume that M (with metric g or g*) has constant curva-
ture. Then, its curvature has the representation (3.11.5) and its Ricci
curvature -is given by (3.1 1.4). Substituting from (3.1 1.4) and (3.1 1.5)
into (3.1 1.2), we conclude that the tensor w vanishes.