Remarks on some variational problems on non-complete manifolds

We shall review recent results obtained in the study of some periodic variational problems on Riemannian and Lorentzian manifolds with boundary. Firstly we shall analyze the existence. of closed geodesics on a Riemannian manifold (M, < (.),(.)> (R))Then we shall deal respectively with periodic trajectories and periodic trajectories under a vectorial potential on stationary Lorentz manifolds. Finally, we discuss the different hypotheses on the boundary, and state some open questions.