Group-like decompositions of Riemannian bundles

Abstract

Horocycles, at least in the hyperbolic plane, have been studied for many years. However, it was Hedlund who first used them in topological dynamics and ergodic theory ([9], [i0]), and it was he who had the courage actually to apply them in tile case of non-constant curvature. It is this assurance, subsequently exploited by E. Hopf, Anosov, and Sinai ([12, [13], [I], [2]), that has led me, through the back door, so to speak, to a viewpoint of Riemannian geometry about which I intend to speak today• The claim is that arbitrary Riemannian manifolds have much in common with Riemannian symmetric spaces, and that horocycles enable us to construct analogues of the Iwasawa and Bruhat decompositions of semisimple Lie groups when the curvature is strictly negative. In §i we establish the notation and express the Cartan structure equations in vector form. The relationship to the Cartan decomposition of a semisimple group is pointed out, and meanings attached to the assertion that the resulting flows act (pointwise) transitively in the bundle of frames. In §2 we recall the definition of horospheres and give growth estimates for certain functions associatz>d,~ith them. §3 is devoted to defining the horocycles. §4 is qualitative, pointing out the analogies to the Iwasawa and Bruhat decompositions connected with rank one symmetric spaces. In §S we give an application to ergodic theory of these ideas, namely, the Theorem. Let M be a compact Riemannian manifold with i/4-pinched negative curvature. Then any generalized geodesic flow in the bundle of orthonormal frames is ergodic.