Closing geodesics in $C^1$ topology

TL;DR

This paper demonstrates how to modify the metric of a closed Riemannian manifold slightly in the $C^1$ topology to turn a geodesic orbit into a closed geodesic, providing a method for closing geodesics.

Contribution

It introduces a technique to close geodesics via small $C^1$ perturbations of the metric on closed Riemannian manifolds.

Findings

Any geodesic orbit can be closed by a small $C^1$ perturbation of the metric.

The method applies to arbitrary closed Riemannian manifolds.

The perturbation preserves the overall geometric structure.

Abstract

Given a closed Riemannian manifold, we show how to close an orbit of the geodesic flow by a small perturbation of the metric in the $C^1$ topology.