Curve Shortening and the Topology of Closed Geodesics on Surfaces

We study “flat knot types” of geodesics on compact surfaces M2. For every flat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening flow on the space of immersed curves on M2. We conclude existence of closed geodesics with prescribed flat knot types, provided the associated Conley index is nontrivial.