I am struggling to see the difference between the $n$-sphere and the $n$-torus. We define $\mathbb{T}^n = S^1 \times S^1 \times \cdots \times S^1$, where the Cartesian product is taken $n$ times.
I guess we should start with low dimensions and work our way up. If $n=1$, then the two manifolds are not just topologically equivalent, but they are geometrically equivalent. They are the same mathematical object.
Now let $n=2$. Both the 2-sphere and the 2-torus may be charted out by two angular coordinates. The difference to me is that the torus has both angles going from $0$ to $2\pi$ while the sphere has one angle going from $0$ to $2\pi$ and another going from $0$ to $\pi$. Is there an example where one is more beneficial than the other?
In dynamics, if I have a 2-segment pendulum in the plane parametrized by two angles, then I may represent the configuration manifold as a 2-torus since both angles range up to $2\pi$. Would a single segment pendulum with two degrees of freedom be an example of a configuration manifld that is a 2-sphere?
Now let's investigate $n = 3$. Consider the topological group $SO(3)$. We may chart out $SO(3)$ with Euler angles. The first and last angles range from $0$ to $2\pi$ while the middle angle ranges from $0$ to $\pi$. What sort of a topological object would this be associated with?
I can see that if we make a cut around the 2-torus and throw out a ring from the middle, we may identify edges and get a 2-sphere. Can a similar thing be done with the 3-torus and the 3-sphere?
Thanks.