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?