Hello. Let M,N be a connected smooth riemannian manifolds. I define the metric as usuall, the infimum of lengths of curves between the two points. (the length is defined by the integral of the norm of the velocity vector of the curve). Suppose phi is a homeomorphism which is a metric isometry. I wish to prove phi is a diffeomorphism. Please, anyone who can help. Thanks in advance, Roey
My instinct is to be lowbrow and just compute the derivative. Limit of ratios of distances, and all that.
if you can embed them so that the metric is induced from that of euclidean space, wouldnt an isomoetry just be a restricted linearmap? that makes it seem as if klocally it is alkways true, and derivatives are local properties.