Let H be a complete simply connected Riemannian manifold without focal points and let I(H) be the group of isometries of H. In this work we extend to manifolds without focal points, results of which have already been proved for manifolds of non-positive curvature.
In Section 1 we introduce the notation as well as the results to be assumed.
In Section 2 a relationship between the flat factor in the de Rham decomposition of H and the Clifford translations is obtained (Theorem 2.1); this extends a result of Wolf [11] and it is the basic tool in this paper. It is first used to describe the homogeneous spaces without focal points as products of flat tori and simply connected manifolds without focal points.
In Section 3 we prove Theorem 3.2, that extends Theorem 2.4 of Chen and Eberlein [i], which relates certain properties of the subgroups D of I(H) that satisfy the duality condition. Its proof depends, in a crucial way, on the fact that, in H, the sum of two interior angles of a geodesic triangle (with possibly one vertex at infinity) is less than or equal to rc (Lemma 3.2.1). As a consequence we have a generalized version of the Center theorem that includes the manifolds of finite volume (Theorem 3.3).
The results in this paper are contained in the author's PhD Thesis [2] presented at the Instituto de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, Brazil. The author wishes to thank Professor Patrick Eberlein for very helpful discussions.
1. PRELIMINARIES
A complete Riemannian manifold M is said to have no focal points if for any ray ~ and any non-trivial Jacobi field J along 7 vanishing at t = 0, d (J(t), J(t) > > 0 for t > 0 where ( , ) denotes the inner product with respect to the Riemannian metric of M.
H will always denote a complete simply connected Riemannian manifold without focal points of dimension n, and all geodesics will have unit speed.
If M is not simply connected it can be represented as a quotient H/D, where