### 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.

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.

#### Keywords

Perturbed geodecics; semi-Riemannian manifold; spectral flow; conjugate points; generalized Morse index

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