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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