Page 137 - Curvature and Homology

Theorem 3.9.2. Let M be a compact and orientable Riemannian mani$old
of positive Ricci curvature. If

then, bJM) vanishes [6, 74.

Corollary. M is a homology sphere if (3.9.14) hol& for all p, 0 < p < n.
This generalizes theorem 3.9.1.

3.10. Affine collineations

Let M be a Riemannian manifold with metric tensor g and C = C(t)
a geodesic on M defined by the parametric equations ui = ui(t),
i = 1, ..*, n. Denoting the arc length by s, that is ds2 =gadurdu*, the
equations of C are given by

where A(t) = (d2s/dt2)/(&/dt) and the rf, are the coefficients of the
Levi Civita connection (associated with the metric). By an afine
collineation of M we mean a differentiable homeomorphism f of M onto
itself which maps geodesics into geodesics, the arc length receiving
an affine transformation:

for some constants a # 0 and b. Clearly, iff is a motion it is an afhe
collineation. The converse, however, is not true in general, but, if we
assume that M is compact and orientable, an affine collineation is neces-
sarily a motion (theorem 3.10.1).
It can be shown that the affine collineations of M form a Lie group.
Let G denote a connected Lie group of affine collineations of M and L
its Lie algebra. To each element A of L we associate the 1-parameter
subgroup a, of G generated by A. The corresponding 1-parameter
group of transformations R,, on M induces a (right invariant) vector
field A* on M. The vector field A* in turn defines an infinitesimal
transformation 8(A*) of M corresponding to the action on M of a,.
Since the elements of G map geodesics into geodesics the Lie derivative
of the left hand side of (3.10.1) with s as parameter must vanish.