I want to know what constant vector fields on $T^{2n}$ look like.
We have obvious coordinates $(\theta_1,\phi_1,....,\theta_n,\phi_n)$ on $T^{2n}$. Let $X$ be a constant vector field on $T^{2n}$, i.e. $X(\theta_1,\phi_1,....,\theta_n,\phi_n)=v$ for some $v\in \mathbb R^{2n}$. It seems that we must have $X$ as some linear combination of $\partial/\partial \theta_i$'s and $\partial/\partial \phi_j$'s.
But It also feels that this not need be the cause because we could have chosendi some different coordinate system.