On the closed geodesics problem (Q2390103)

The authors review and discuss the following `closed geodesic problem': find a (sharp) lower bound for the number of geometrically distinct closed geodesics on a compact Riemannian manifold. A celebrated result of Gromoll and Meyer states that the number of closed geodesics on a simply connected Riemannian manifold is infinite, if the topology of the manifold is `sufficiently complicated'. In the paper some immediate generalizations of the problem to semi-Riemannian and Finsler manifolds is also discussed.