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,
Pick a point . There is an such that is a diffeomorphism of the ball (the tangent space) into . Since is a homeomorphism, is an open set in and .
Now, since is an isometry, for all geodesics with , for the geodesic with and , we have that
This means exists (for all of the tangent space, as is complete); and since is complete, we can consider the map defined by This is a diffeomorphism, and inverse to the (restricted map) . So is a diffeomorphism.
Note. I do not see why this would not work if was only a local isometry.
Note 2. Thin Lizzy - Whiskey in the jar
Sorry, stupid me - use the argument about to get just to be a diffeo. No need to make our lives harder by computing inverses. Sorry again, I was... checking my signature out and was carried away :D
I can see how the hypotheses on M can be relaxed, by following the same argument. But no completeness for N, well... There is no way I can see it happen, and my guess is there is no simple way to do this :( :o :confused: