Equicontinuous Geodesic Flows
نویسنده
چکیده
This article is about the interplay between topological dynamics and differential geometry. One could ask how many informations of the geometry are carried in the dynamic of the geodesic flow. M. Paternain proved in [6] that an expansive geodesic flow on a surface implies that there are no conjugate points. Instead of regarding notions that describe chaotic behavior (for example expansiveness) we regard a notion that describes the stability of orbits in dynamical systems, namely equicontinuity and distality. In this paper we give a new sufficient and necessary condition for a compact Riemannian surface to have all geodesics closed (P-manifold): (M, g) is a P-manifold iff the geodesic flow SM ×R → SM is equicontinuous. We also prove a weaker theorem for flows on manifolds of dimension 3. At the end we discuss some properties of equicontinuous geodesic flows on noncompact surfaces and higher dimensional manifolds.
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