I am trying to show that the n-torus, $\mathbb{R}^{2n}/ \mathbb{Z}^{2n}$ always has a symplectic form.
By the Quotient Manifold Theorem, we have a smooth structure on the n-torus and a covering map $\pi \colon \mathbb{R}^{2n}\longrightarrow \mathbb{R}^{2n}/ \mathbb{Z}^{2n}$.
Then, we have an open covering of the n-torus and in each open we can define a symplectic form by passing the symplectic form of an open of $\mathbb{R}^{2n}$ by a diffeomorphism.
However, I am having troubles to understand why the symplectic form is defined globally; i.e, why it varies smoothly.
Thanks in advance.
Consider the symplectic form $ \omega=dx_1\wedge dx_2+...+dx_{2n-1}\wedge dx_{2n}$ it is preserved by the translations of $\mathbb{R}^{2n}$ so it defines a symplectic form $\Omega$ on $T^{2n}$ such that if $p:\mathbb{R}^{2n}\rightarrow T^{2n}$ is the quotient map, $p^*\Omega=\omega$.