Page 135 - Curvature and Homology

3.9. CONFORMALLY FLAT MANIFOLDS 117

I
If g* = ee"g is a locally flat metric, both R*$, and R* vanish, and so
from (3.9.5)

The integrability conditions of the system (3.9.9) are evidently given by

It follows after substitution from (3.9.9) into (3.9.10) that Cgjk = 0.
Thus, the equations (3.9.9) are integrable.

Proposition 3.9.1. A necessary and su-t codtion that a Riemannian
manqold of dimension n > 3 be conformally jlat is that its Wcyl confotmal
mature tmor vanish. For n = 3, it is necessary and su.t that the
tensor Cijk = 0.
The conformal curvature tensor of a Riemannian manifold of constant
curvature is readily seen to vanish. Thus,

Corollary. A Riemannian manifold of constant curvature is conformally
$at poetidid n 2 - 3.
We now show that a compact and orientable conformally flat
Riemannian manifold M whose Ricci curvature is positive definite is a
homology sphere. This is certainly the case if M is a manifold of positive
constant curvature.
Indeed, since M is conformally flat, its Weyl conformal curvature
tensor vanishes. Hence, from formula (3.2.10), for a harmonic p-form a
n - 2p - '
F(a) = - aii:...is aft*, .is + p! R(a, a). (3.9.1 1)
R,
n-2 (n - 1) (n - 2)
Since the operator Q is positive definite let A,, denote the greatest lower
bound of the smallest eigenvalues of Q on M. Then, for any 1-form 8,
(Qp, 8) 2 A,, (8, /3) and the scalar curvature R = gij &, 2 n A,, > 0.
This latter statement follows from the fact that at the pole of a geodesic
coordinate system the scalar curvature R is the trace of the matrix (Rij),
(gij(p) = 4j).
Again, at a point P E M if a geodesic coordinate system is chosen
it follows from (3.9.1 1) that