On geodesics in low regularity

TL;DR

This paper investigates the existence, regularity, and uniqueness of geodesics in Riemannian and Lorentzian manifolds with low regularity metrics, extending classical results to less smooth settings.

Contribution

It generalizes recent results on geodesic solutions to locally Lipschitz continuous metrics and explores the subtle relations between extremal curves and geodesic equations in low regularity.

Findings

Existence of extremal curves for continuous metrics.

Generalization of Filippov solutions for geodesics.

Examples illustrating the interplay between extremal curves and geodesic equations.

Abstract

We consider geodesics in both Riemannian and Lorentzian manifolds with metrics of low regularity. We discuss existence of extremal curves for continuous metrics and present several old and new examples that highlight their subtle interrelation with solutions of the geodesic equations. Then we turn to the initial value problem for geodesics for locally Lipschitz continuous metrics and generalize recent results on existence, regularity and uniqueness of solutions in the sense of Filippov.