The Morse index theorem in the case of two variable endpoints in conic Finsler manifolds

TL;DR

This paper proves a Morse index theorem for geodesics connecting two submanifolds in conic Finsler manifolds, extending classical results to a broader geometric setting with less regularity.

Contribution

It establishes the Morse index theorem for geodesics in conic Finsler manifolds with two variable endpoints, under positive definiteness conditions.

Findings

Morse index theorem holds in conic Finsler manifolds with two endpoints.

The theorem applies to $C^7$ manifolds with $C^6$ pseudo-Finsler metrics.

Positive definiteness of the fundamental tensor is crucial for the result.

Abstract

In this note, we prove the Morse index theorem for a geodesic connecting two submanifolds in a $C^7$ manifold with a $C^6$ (conic) pseudo-Finsler metric provided that the fundamental tensor is positive definite along velocity curve of the geodesic.