Smooth Surface Transformation for Gents: From One Metric to Another

[Neitrino]

Gents,

I'm quit new and have one question,

So is it possible with smooth and continious transformation ...from one surface (with given metric) to get another one (also its metric is given)?

Thks

 

[kleinwolf]

Well try to transform (with a diffeomorphism) the metric of the sphere into the flat metric...Since the sphere has a non-vanishing gaussian curvature, and the flat metric a vanishing one, and because the gaussian curvature is independent under reparametrization, then you cannot do that for example...Is that what you mean ?

 

[Neitrino]

yes, I mean exactly that, and what are the restrictions that alow/forbid me to make such transformations in generaly, or among what group of surface can i do that...

Thks very much

 

[kleinwolf]

The group of surfaces is the homeomorphy group (the group of all surfaces obtained via an homeomorphism out of your surface). It's precisely the goal of topology to find criteria to decide if 2 surfaces are homeomorphic (You should look somewhere at topology, fundamental group, homology groups, Betti numbers, aso...But I don't know the answer).