Local extension and differential geometry

[Hyperreality]

I have to prove the map g:torus --> (S^2 3 dimensional sphere of radius 1) is a C infinity map in my assignment.

The torus is parameterized as

x(u,v)=(3+cos u)(cos v)
y(u,v)=(3+cos u)(sin v)
z(u,v)=sin u

The map g is given by

g=[6yz/(x^2+y^2+z^2+8), 3-sqrt(x^2+y^2), -xz/sqrt(x^2+y^2)]

I have done that by using suitable coordinate maps. But I am terribly confuse in how to prove this by choosing suitable local extensions between open subsets of R^3. To be honest I don't really know the definition of local extension of a coordinate map!

All I know is about local extension is that for f:u1 -> u2, local extension of f is F:U1 -> U2 such that u1 is a subset of U1 and u2 is a subset of U2, I am not sure if f has to equal F (f=F).

I think this can be proven easily if I just local extend g to G:R^3 -> R^3 where g=G and it is obvious that G is C-infinity then so does g. But how do I do this by choosing local subset of R^3, I don't really have a clue.

 

[Almsco]

That's a great question! It's definitely tricky to prove a map is C infinity, but the good news is that there are some steps you can take to make it easier. First, you'll need to define a coordinate map that takes the torus to R^3. Once you've done that, you'll need to create local extensions of your coordinate map between open subsets of R^3. To do this, you can use a partition of unity to construct local extensions of your coordinate map that are defined on open sets in R^3. After you have your local extensions in place, you can then use them to show that the original map is C infinity. Hopefully this helps and good luck with your assignment!