Closed geodesics and the first Betti number

Abstract We prove that, on any closed manifold of dimension at least two with non‐zero first Betti number, a generic Riemannian metric has infinitely many closed geodesics, and indeed closed geodesics of arbitrarily large length. We derive this existence result combining a theorem of Mañé together with the following new theorem of independent interest: The existence of minimal closed geodesics, in the sense of Aubry–Mather theory, implies the existence of a transverse homoclinic, and thus of a horseshoe, for the geodesic flow of a suitable ‐close Riemannian metric.