Prove that the map F:Rm----->Cm given by F: (x1,...,xm)----->(eix1,...,eixe ) is a parametrization of the m-dimensional torus Tm in Cm. Here C is stand for complex numbers. I have no idea really. Thank you for your helps.
The torus is created by taking the unit circle [in the complex domain] and creating Cartesian products of all of them to get T^m.
You have to show that by creating these Cartesian products you get the mapping listed above.
Hint - Try looking at an inductive proof by showing that T^m X T = T^[m+1].