Simple closed geodesics in dimensions ≥ 3

We show that for a generic Riemannian or reversible Finsler metric on a compact differentiable manifold M of dimension at least three all closed geodesics are simple and do not intersect each other. Using results by Contreras (Ann Math 2(172):761-808, 2010; in: Proceedings of International Congress Mathematicians (ICM 2010) Hyderabad, India, pp 1729-1739, 2011) this shows that for a generic Riemannian metric on a compact and simply-connected manifold all closed geodesics are simple and the number N(t) of geometrically distinct closed geodesics of length <= t grows exponentially.