If $M$ is a metric space than it is a topological space and if it is locally homeomorphic to to $R^n$ we say that it is a manifold and if we equipped this manifold with a inner product $g_p$ on tangent space $T_pM$ we can define a "Riemannian distance" $$L=\int \sqrt {g(v(t),v(t))}dt$$.

In $R^n$ the metric distance and this "Riemannian distance" have the same value.

Are there others examples of this ? What is the conditions for this to be true?

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    $\begingroup$ The Riemannian distance does not depend on the initial metric. It only depends on topology/differential structure. So this makes this question a bit invalid. There is only one metric equal to the Riemannian distance on $M$ and it is the Riemannian distance itself. And lots of equivalent metrics are different. What kind of condition you are looking for? $\endgroup$ – freakish Apr 18 at 12:45
  • $\begingroup$ So it is coincidence that in $R^n$ they are equal . Are there others coincidence ? $\endgroup$ – amilton moreira Apr 18 at 12:48

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