Morse Theory for Geodesics in Conical Manifolds

TL;DR

This paper extends Morse theory to conical manifolds with singularities, defining geodesics and indices despite nonsmooth energy, and establishes Morse relations and a geometric multiplicity for geodesics.

Contribution

It introduces a Morse theory framework for conical manifolds, including a new definition of geodesics, indices, and multiplicity in the presence of singularities.

Findings

Morse relations hold for conical manifolds

A continuous retraction of energy sublevels exists without critical points

A geometrically meaningful multiplicity of geodesics is defined

Abstract

The aim of this paper is to extend the Morse theory for geodesics to the conical manifolds. We define these manifolds as submanifolds of $\R^n$ with a finite number of conical singularities. To formulate a good Morse theory we must use an appropriate definition of geodesic. The main theorem of this paper claims that, although the energy is nonsmooth, we can find a continuous retraction of its sublevels in absence of critical points. So, we can give a good definition of index for isolated critical values and for isolated critical points. We prove that Morse relations hold and, at last, we give a definition of multiplicity of geodesics which is geometrical meaningful.