An Index Theory for Paths that are Solutions of a Class of Strongly Indefinite Variational Problems

TL;DR

This paper extends Morse index theory to semi-Riemannian manifolds with indefinite metrics, providing lower bounds on geodesic counts and Morse relations for specific classes of manifolds.

Contribution

It develops a generalized Morse index theorem for semi-Riemannian geodesics and applies it to estimate the number of geodesics in certain indefinite metric manifolds.

Findings

Generalized Morse index theorem for semi-Riemannian geodesics

Lower bounds on the number of geodesics between points

Morse relations for stationary and G"odel-type manifolds

Abstract

We prove a generalized version of the Morse index theorem for geodesics endowed with a non positive definite metric tensor (semi-Riemannian manifolds). We apply the result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of manifolds. More specifically, we consider semi-Riemannian manifolds $(M,\mathfrak g)$ admitting a smooth distribution spanned by commuting Killing vector fields and containing a maximal negative distribution for $\mathfrak$. In particular we obtain Morse relations for stationary semi-Riemannian manifolds and for the {\em G\"odel-type} manifolds.