Why is it necessary to consider taking the infimum over the lengths of all piece-wise smooth curves while defining the distance function on a Riemannian Manifold instead of just taking the infimum over all smooth curves ? Are there examples where these two processes give different answers ?
2026-05-05 20:20:12.1778012412
Metric on Riemannian manifolds
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The two infima are the same. Just for some proofs it is convenient to consider piecewise-smooth paths.