Conjugate Points in Length Spaces

TL;DR

This paper extends the concept of conjugate points from Riemannian manifolds to complete length spaces, introduces new types of conjugate points, and generalizes key theorems like the homotopy lemma and injectivity radius estimate to this broader setting.

Contribution

It introduces symmetric and ultimate conjugate points in length spaces and generalizes classical Riemannian theorems, including comparison theorems and the homotopy lemma, to ${ m CBA}( ext{ extdegree})$ spaces.

Findings

Proved no ultimate conjugate points less than π apart in ${ m CBA}(1)$ spaces.

Established Rauch-type comparison theorems in length spaces.

Generalized the homotopy lemma and injectivity radius estimates to this setting.

Abstract

In this paper we extend the concept of a conjugate point in a Riemannian manifold to complete length spaces (also known as geodesic spaces). In particular, we introduce symmetric conjugate points and ultimate conjugate points. We then generalize the long homotopy lemma of Klingenberg to this setting as well as the injectivity radius estimate also due to Klingenberg which was used to produce closed geodesics or conjugate points on Riemannian manifolds. Our versions apply in this more general setting. We next focus on ${\rm CBA}(\kappa)$ spaces, proving Rauch-type comparison theorems. In particular, much like the Riemannian setting, we prove an Alexander-Bishop theorem stating that there are no ultimate conjugate points less than $\pi$ apart in a ${\rm CBA}(1)$ space. We also prove a relative Rauch comparison theorem to precisely estimate the distance between nearby geodesics. We close with applications and open problems.