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.
To be selfcontained, note that the usual notation for the $n$-dimensional sphere is $S^n$. This is the surface of the $(n+1)$-dimensional ball and can be given as
$$S^n=\{(x_1,...,x_{n+1})\in\Bbb R^{n+1}\mid x_1^2+\cdots+x_{n+1}^2=1\}.$$
The product space $S^n\times S^m$ will give a surface, but no torus in general. Note that an $n$-dimensional torus is defined as
$$\underbrace{S^1\times\cdots\times S^1}_{n\text{ times}}.$$
Also, just because you can describe a space in some topological sense, this does not mean that there is the unique representation of it in a higher dimensional space. There are many ways how $S^n\times S^n$ can be embedded into higher dimensional Euclidean spaces.
Here is one example: the set
$$X:=\{(x_1,...,x_{n+1},y_1,...,y_{m+1})\in\Bbb R^{n+m+2}\mid x_1^2+\cdots+x_{n+1}^2=1\,\wedge\,x_{1}^2,...,x_{m+1}^2=1 \}$$
is topologically equivalent to $S^n\times S^m$ as a subspace of $\Bbb R^{n+m+2}$. This can be improved to lower dimensions for the embedding space. For example: $S^1\times S^1$ (the usual torus) is here embedded into $\Bbb R^4$. we know we can do better. One way to do this in general would be to see that $X$ is a (scaled) subset of $S^{n+m+1}$ and use stereographic projection to map this into $\Bbb R^{n+m+1}$.
For a visualization you could look at $\Bbb R^3$-slices. But I think this will give not much information on the shape for $n>3$.