Page 139 - Curvature and Homology

3.1 1. PROJECTIVE TRANSFORMATIONS
Hence, if M is compact and orientable

0 = (dV, 5) = (85, 85)
from which 85 = 0. We conclude (by theorem 3.8.2, cor.)
Theorem 3.10.1. In a compact and orientable Riemannian manifold an
infinitesimal aflne collineation is a motion [73].
Corollary. There exist no (non-trivial) I-parameter groups of aflne
collineations on a compact and orientable Riemannian manifold of negative
definite Ricci curvature.
This follows from theorem 3.8.1.
More generally, it can be shown that an infinitesimal affine collineation
defined by a vector field of bounded length on a complete but not
compact Riemannian manifold is an infinitesimal motion. We remark
that compactness implies completeness (cf. 5 7.7).

3.1 1. Projective transformations

We have defined an affine collineation of a Riemannian manifold M
as a differentiable homeomorphism f of M onto M preserving the
geodesics and the affine character of the parameter s denoting arc length
along a geodesic. If, more generally, f leaves the geodesics invariant,
the affine character of the parameter s not necessarily being preserved,
f is called a projective transformation.
A transformation f of M is aflne, if and only 'if

where w is the matrix of forms defining the affine connection of M, or,
equivalently in terms of a system of local coordinates

where the rfk are given by f *wj = r*:, duk, f * denoting the induced
dual map on forms. A transformation f of M is projective, if and only if
there.exists a covector pV) depending on f such that

where the p,(n are the components of pV) with respect to the given
local coordinates. Under the circumstances, o and f*w are called