I admit up front i'm a complete newbie not even close to being a geometer/topologist, but i need some advice about what the product of spheres would look like. A 1 sphere times a 1 sphere is a two dimensional torus, which is a closed surface, but what happens when we up the ante to an n sphere times an n sphere? In particular a 3 or 4 sphere times a 3 or 4 sphere? Is that also called a torus and is it also a closed surface in some higher dimensional space? And how would we set up a coordinate system to describe that object and how would we project those coordinates into three dimensions so we could see what it looked like? I know we could never see the whole thing. How would we do this in mathematica so we could make a pretty picture out of it?
And a related question: what would the product of an n sphere by an m sphere look like? Perhaps this is the question I should have asked first.