Page 134 - Curvature and Homology

116 III. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY
Eliminating Aa from (3.9.3) and (3.9.4) we obtain

(do,
- ' do) gw
2
Transvecting (3.9.2) with g*ir and substituting (3.9.5) in the resulting
equation we obtain C*hl = Cjk, where

Evidently, the qkl are the components of a tensor called the Weyl con-
fotmal mature tensor. Moreover, this tensor remains invariant under
a conformal change of metric. The case n = 3 is interesting. Indeed,
by choosing an orthogonal coordinate system (gij = 0, i # j) at a point
(cf. § 1.1 l), it is readily shown that the Weyl conformal curvature
tensor vanishes.
Consider a Riemannian manifold M with metric g and let g* be a
conformally related locally flat metric. Under the circumstances M
is said to be (locally) conformally pat. Clearly then, the Weyl conformal
curvature tensor of M vanishes. Conversely, if the tensor CjFl is a zero
tensor on M, there exists a function a such that g* = e2"g is a locally
flat metric on M. For, from (3.9.6)

Applying (1.10.2 1) and (1.10.22) we deduce

3
Di Pjk, = (n - L l
where we have put

Hence, for n > 3, Cijk = 0.