A Morse index theorem for perturbed geodesics on semi-Riemannian manifolds

Abstract

Perturbed geodesics are trajectories of particles moving on
a semi-Riemannian manifold in the presence of a potential. Our
purpose here is to extend to perturbed geodesics on
semi-Riemannian manifolds the well known Morse Index Theorem. When
the metric is indefinite, the Morse index of the energy
functional becomes infinite and hence, in order to obtain a
meaningful statement, we substitute the Morse index by its
relative form, given by the spectral flow of an associated family
of index forms. We also introduce a new counting for conjugate
points, which need not to be isolated in this context, and prove
that our generalized Morse index equals the total number of
conjugate points. Finally we study the relation with the Maslov
index of the flow induced on the Lagrangian Grassmannian.