Locally extremal geodesic loops on Riemannian manifold

TL;DR

This paper proves that locally extremal, non-self-conjugate geodesic loops in Riemannian manifolds are closed geodesics and explores conditions under which such geodesics exist in complete, non-contractible manifolds.

Contribution

It establishes a link between locally extremal geodesic loops and closed geodesics, providing new existence results under specific geometric conditions.

Findings

Locally extremal, non-self-conjugate geodesic loops are closed geodesics.

Complete, non-contractible manifolds with diverging injectivity radii have minimizing closed geodesics.

Abstract

This note proves that any locally extremal non-self-conjugate geodesic loop in a Riemannian manifold is a closed geodesic. As a consequence, any complete and non-contractible Riemannian manifold with diverging injectivity radii along diverging sequences and without points conjugate to themselves, possesses a minimizing closed geodesic.