Curve shortening and the topology of closed geodesics on surfaces

Abstract

We study “flat knot types” of geodesics on compact surfaces $M^2$. 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 $M^{2}$. We conclude existence of closed geodesics with prescribed flat knot types, provided the associated Conley index is nontrivial.