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. 1. Introduction If M is a surface with a Riemannian metric g then closed geodesics on (M, g) are critical points of the length functional L(γ) = |γ (x)|dx defined on the space of unparametrized C.
