Geodesic distance is the length of the shortest path between two points on a Riemannian manifold, where the path is a geodesic. It connects the concept of distance in ordinary Euclidean space to the more complex setting of curved spaces, providing a way to measure the 'straightest' distance in a curved environment. Understanding geodesic distance is crucial as it reveals how geometry behaves under different Riemannian metrics and emphasizes the intrinsic nature of distances on manifolds.
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