Cartesian product of $\mathbb{S}^1$ is symplectic

Question

Prove that the Cartesian products of $\mathbb{S}^1$ for $2n$ times is a symplectic manifold. I have just studied the concepts of symplectic manifold in the class of analytical mechanics. I have not studied any courses about geometry. So I want to know whether I can find a symplectic form directly. Thanks!

Answer 1

Hint:

You probably know how to find a symplectic form on $\mathbb{R}^{2n}$. This symplectic form is invariant under translations by $\mathbb{Z}^{2n}$. But $\mathbb{R}^{2n}/{\mathbb{Z}^{2n}=T^{2n}}$. Can you use this to construct a symplectic form on the torus?