Riemannian Manifolds as Dynamical Systems: the Geodesic Flow and Periodic Geodesics

Abstract

Following the transition presented in chapter 8, it is now quite natural to study the geodesic behaviour of a Riemannian manifold. For local metric geometry it is natural because geodesics are locally the shortest paths. This point of view was treated in §6.5. But geodesics of any length are of interest for the geometer. Another strong motivation for the study of geodesic dynamics comes from mechanics. Since Riemannian manifolds provide a very general setting for Hamiltonian mechanics, with their geodesics being the desired Hamiltonian trajectories, we are of course interested in their behaviour for any interval of time (any length). This perspective mixes dynamics and geometry and is extremely popular today. People always want to predict the future, more or less exactly. We will comment more on this below. Dynamics plays an ever larger role in geometry, even in very simple contexts such as the study of pentagons and Pappus theorems (see Schwartz 1993,1998 [1114, 1115] and the important work d’Ambra & Gromov 1991 [426]. Note also that Gromov 1987 [622] introduced dynamical systems into the study of discrete groups.