Closed geodesics on complete surfaces

Abstract

The existence of closed geodesics on complete Riemannian manifolds depends to a large extent on the topology of the manifold. Generally, closed geodesics may become rare or at least hard to detect if the topology of the manifold is simple. For complete, non-compact surfaces M the situation is as follows: If M is not homeomorphic to a plane, a cylinder or a M6bius band there always exist infinitely many closed geodesics on M, cf. [-15]. They arise as minima of the energy functional on certain free homotopy classes of loops. Here the possibility that two of these closed geodesics are coverings of each other can be excluded by choosing the free homotopy classes appropriately. On a complete M6bius band these methods yield one closed geodesic, and, indeed, this may be the only one that exists. Obviously there exist complete cylinders and planes without any closed geodesics. In this paper we give conditions on the Riemannian structure which ensure the existence of infinitely many closed geodesics on the three exceptional surfaces as well. For the problem of closed geodesics these three surfaces are the most interesting non-compact ones since their topology is so simple. Combining two more specific theorems we obtain as main result