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We consider geodesics in both Riemannian and Lorentzian manifolds withmetrics of low regularity. We discuss existence of extremal curves forcontinuous metrics and present several old and new examples that highlighttheir subtle interrelation with solutions of the geodesic equations. Then weturn to the initial value problem for geodesics for locally Lipschitzcontinuous metrics and generalize recent results on existence, regularity anduniqueness of solutions in the sense of Filippov.

On geodesics in low regularity