Cut and conjugate points of the exponential map, with applications

TL;DR

This paper investigates the singularities and structure of the exponential map's cut locus in Riemannian and Finsler manifolds, with applications to geometric conjectures and differential equations.

Contribution

It provides new insights into the structure of the cut locus and explores applications to the Ambrose conjecture and Hamilton-Jacobi equations.

Findings

Enhanced understanding of the singularities of the exponential map.

New results on the structure of the cut locus.

Applications to geometric and differential equation problems.

Abstract

The goal of this thesis is to study the singularities of the exponential map of Riemannian and Finsler manifolds (a concept related to caustics and catastrophes), and the object known as the cut locus (aka ridge, medial axis or skeleton), to improve existing results about its structure, to look at it in new ways, and to derive applications to the Ambrose conjecture and the Hamilton-Jacobi equations.