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.
Vector fields on torus
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differential-geometry
mathematical-physics
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2It doesn't say what a vector field on the torus is. Until you have that piece of information, showing whether or not _anything_ is a vector field on the torus is impossible. So that's where I'd go. See how your source (be it internet or a book) would define a vector field on a torus, and see if $X$ and $Y$ can fit that role. – 2012-10-14
1 Answers
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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..