A geodesic conjugacy between two closed manifolds is a homeomorphism between their unit tangent bundles that takes geodesic flow orbits of one to that of the other in a time-preserving manner. One of the central problems in Riemannian geometry is to understand the extent to which a geodesic conjugacy determines a closed Riemannian manifold itself. While an answer to the question in this generality has yet remained elusive, we give an overeview of results on closed surfaces – the most important illustrative case where a complete picture about questions of geodesic conjugacy rigidity is available.

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