Vector fields on torus

Question

Let $A=d/dx$ and $B=d/dy$ be vector fields on $\mathbb{R}^2$, prove that they induce vector fields $X$ and $Y$ on the torus $T$ by $X(f)=A(f(\pi)), Y(f)=B(f(\pi))$ where $\pi$ is quotient map from $\mathbb{R}^2$ to torus.

Answer 1

Well, the torus can be defined as the quotient space $T=\Bbb R^2/\sim$, where $(u,v)\sim(u',v')$ iff $(u'-u),(v'-v)\in\Bbb Z$. Best is to draw on a squared paper.. Vectors of $A$ go right, $B$ goes up, with unit speed.

In particular, it is represented by the unit square $[0,1]^2$.

Practically, all we have to prove is that if points $U\sim V$ then $AU = AV$ and $BU=BV$. In this case, all smooth functions $f$ on $T$ can be 'lifted' to $\Bbb R^2$, i.e. can be viewed as a $\Bbb Z$-periodic function $f:\Bbb R^2\to\Bbb R$, that is, $fU\sim fV$ if $U\sim V$. And so..