# Riemannian Geometry

#### sroeyz

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.

Roey

Related Differential Geometry News on Phys.org

#### Hurkyl

Staff Emeritus
Gold Member
My instinct is to be lowbrow and just compute the derivative. Limit of ratios of distances, and all that.