On the Geodesical Connectedness for a Class of Semi-Riemannian Manifolds1

We prove a variational principle for geodesics on a semi-Riemannian manifold M of arbitrary index k and possessing k linearly independent Killing vector fields that generate a timelike distribution on M. Using such a principle and a suitable completeness condition for M, we prove some existence and multiplicity results for geodesics joining two fixed points of M