Page 132 - Curvature and Homology

114 III. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY
Theorem 3.8.1. There are no non-trivial (global) 1-parameter groups
of conformal transformations on a compact and orientable Riernam'an
manifold M of dimension n 2 2 with negative definite Ricci curvature
[4, 731.
For, let X be the infinitesimal conformal transformation induced
by a given 1-parameter group of conformal transformations of M and 8
the 1-form defined by X by duality. Then t(5) vanishes, and so by
(3.8.2) and (3.8.3)
(Af + (1 - A)d8f - 2Qf, f) = 0.
n
A computation gives

and consequently, if (Q5, 5) 5 0 then, for n 2, we must have
(Qt.,f) = 0, Sf = 0, DX = 0.

Moreover, if the Ricci curvature is negative definite we conclude that
5 = 0, that is X vanishes.
We have proved in addition that if the Ricci quadratic form is negative
semi-definite, then a vector field X on M which generates a I-parameter
group of conformal transformations of M is necessarily a parallelcfield.

Corollary. There are no (globat) 1-parameter groups of motions on a
compact and orientable Riemannian manifold of negative definite Ricci
mature.
We have seen that an infinitesimal conformal transformation on a
Riemannian manifold M must satisfy the differential equation

Conversely, if M is compact and orientable, and 8 is a 1-formon M
which is a solution of equation (3.8.4), then by (3.8.2) and (3.8.3)
(t(f), t(5)) = 0 from which t(5) = 0, that is 8(f)g + (2/n)(65)g = 0. It
follows that the vector field X dual to 5 is an infinitesimal conformal
transformation. We have proved [73]

Theorem 3.8.2. On a compact and orientable Riemannian manifold a
necessary and suflcient condition that the vector field X be an infinitesimal
conformal transformation is given by
n
At. + (1 - 2)dSf = 2Qf.