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.