Morse Theory for geodesics in singular conformal metrics

TL;DR

This paper develops a Morse theory framework for geodesics in conformal metrics that degenerate on a hypersurface, motivated by applications to brake orbits and homoclinics in dynamical systems.

Contribution

It introduces a Morse theory approach for geodesics in singular conformal metrics with vanishing conformal factors on hypersurfaces.

Findings

Establishes Morse index theory for degenerate geodesics

Provides tools for studying brake orbits and homoclinics

Extends classical Morse theory to singular metric settings

Abstract

Motivated by the use of degenerate Jacobi metrics for the study of brake orbits and homoclinics, we develop a Morse theory for geodesics in conformal metrics having conformal factors vanishing on a regular hypersurface of a Riemannian manifold.