Riemannian Geometry

by sroeyz
Tags: geometry, riemannian
sroeyz is offline
Aug23-06, 02:17 PM
P: 5
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,

Phys.Org News Partner Science news on Phys.org
Going nuts? Turkey looks to pistachios to heat new eco-city
Space-tested fluid flow concept advances infectious disease diagnoses
SpaceX launches supplies to space station (Update)
Hurkyl is offline
Aug23-06, 02:23 PM
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,101
My instinct is to be lowbrow and just compute the derivative. Limit of ratios of distances, and all that.
mathwonk is offline
Aug23-06, 09:06 PM
Sci Advisor
HW Helper
mathwonk's Avatar
P: 9,421
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.

Register to reply

Related Discussions
how do you prove immersion? (Basic Riemannian Geometry) Calculus & Beyond Homework 2
Coordinates in Riemannian Geometry Differential Geometry 1
Riemannian manifolds Differential Geometry 16
Some basic problems in Riemannian Geometry Differential Geometry 17
lorentzian vs riemannian Beyond the Standard Model 2