An index theory for paths that are solutions of a class of strongly indefinite variational problems

Abstract.

We generalize the Morse index theorem of [12,15] and we apply the new result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of semi-Riemannian manifolds. More specifically, we consider semi-Riemannian manifolds $(M,\frak{g})$ admitting a smooth distribution spanned by commuting Killing vector fields and containing a maximal negative distribution for $\frak{g}$. In particular we obtain Morse relations for stationary semi-Riemannian manifolds (see [7]) and for the Gödel-type manifolds (see [3]).