Periodic geodesics in singular spaces

TL;DR

This paper generalizes the existence of closed geodesics from smooth manifolds to certain singular spaces, showing that topological nontriviality guarantees periodic geodesics in these metric spaces.

Contribution

It extends classical geodesic existence results to singular spaces satisfying CAT(κ) conditions, broadening the scope of geodesic theory.

Findings

Existence of periodic geodesics in compact CAT(κ) spaces with nontrivial homotopy groups.

Applicable to locally CAT(κ) manifolds and locally uniquely geodesic spaces.

Generalizes Lyusternik-Fet theorem to singular metric spaces.

Abstract

We extend the classical result of Lyusternik and Fet on the existence of closed geodesics to singular spaces. We show that if $X$ is a compact geodesic metric space satisfying the CAT($\kappa $) condition for some fixed $\kappa >0$ and $\pi_n(X)\ne 0$ for some $n>0$ then $X$ has a periodic geodesic. This condition is satisfied for example by locally CAT($\kappa $) manifolds. Our result applies more generally to compact locally uniquely geodesic spaces.