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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 ?

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The two infima are the same. Just for some proofs it is convenient to consider piecewise-smooth paths.

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    $\begingroup$ For example, to establish the triangle inequality. $\endgroup$ – ronno Sep 19 '13 at 9:34
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    $\begingroup$ The distance function defined by infimum of paths lenght can be introduced for connected Riemannian manifolds.It is easy to see connectness is equivalent to arcwise connectness. Therefore we know that for any two points, say $ p,q \in M $, there exists a CONTINUOUS path joining them . A priori this path is not smooth and not piecewise smooth. Using local coordinates it is always possible to construct a piecewise smooth path joining $ p,q $ starting by the original continuous path. The question is: is it always possible to construct a smooth one? $\endgroup$ – user55449 Sep 21 '13 at 7:20
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    $\begingroup$ @user55449: Yes, it is possible and not too hard. The point is that can construct a map of an interval having prescribed values and values of all its derivatives at end/points of the interval. $\endgroup$ – Moishe Kohan Sep 21 '13 at 9:07

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