We define an isometry as a bijection between surfaces whose differential is a linear isometry between tangents. I've read that this definition is equivalent to the following: $f$ preserves the inner distance between two points, i.e. the infimum of the length of the piecewise $C^1([0,1])$ curves that join them, where $$L(\gamma)=\int_0^1||\gamma'(t)||\ dt$$
I know that, using an approximation theorem by Whitney, the piecewise $C^1$ condition can be restricted to $C^\infty$. Also, we can use geodesics, which are always $C^\infty$. But also using geodesics I have some problems in proving that the definition "preserving distances" implies the definition "the differential is a linear isometry between tangents".
P.S. And in generic Riemannian manifolds? Is this true (I suppose it is; I know that the converse is true: an isometry in the "tangent" sense sends geodesics in geodesics, and preserves their length)? And how can it be proved?
Thank you in advance.
Yes, it is true for Riemannian manifolds of arbitrary dimension that a distance-preserving bijection is a Riemannian isometry. This is known as the Myers–Steenrod theorem. Proofs are somewhat complicated. Start by reading the Palais' paper mentioned in the wikipedia article.