2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/129296We 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.55 pages, published versionDifferential Geometry53C44; 53C22; 58E10text