Let $T^m = \mathbb{R}^m / \mathbb{Z}^m$ be the $m$-dimensional torus equipped with quotient topology, so the canonical surjection $\pi: \mathbb{R}^m \to T^m$ is a covering map.
- Show that there is an atlas on $T^m$ such that $\pi$ is a smooth immersion.
- Show that there are unique smooth vector fields $\xi_1, \dotsc, \xi_m$ on $T^m$ such that for every $p \in \mathbb{R}^m$, $$d\pi(p)\left( \frac{\partial}{\partial x_i} (p) \right) = \xi_i(\pi(p)).$$
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Let $X_1, \dotsc, X_m$ be commuting vector fields on a compact $m$-manifold $M$, and assume that $X_1(p), \dotsc, X_m(p)$ are linearly independent in $T_p(M)$ for every $p\in M$. Show that there is a diffeomorphism $f:M\to T^m=\mathbb{R}^m/\mathbb{Z}^m$ such that for each $i$,
$$f_*(X_i)= \sum_j a_j \xi_j,$$
where $a_j$ is a constant function on $T^m$.
My attempt:
Anyway, for #1, given $p\in T^m$ I would like to choose a chart $(U, \varphi)$ which is a ball of radius $1/2$. Then we can construct a diffeomorphism to some subset of the cube $[-1,2]^m$, by just taking the component of $\pi^{-1}(B)$ which contains the preimage of $p$ in $[0,1]^m$. Then $\pi$ is a local diffeomorphism (the transition maps are just translation), so it is a smooth immersion.
For #2, we must show that $d\pi$ agrees on fibers of $\pi$. But about the points in $\pi^{-1}(q)$ there are disjoint neighborhoods (components of the inverse image of a small neighborhood of $q$) on which $\pi$ acts identically. Therefore the function has the same derivative at these points.
For #3, we know that each point $p$ has a neighborhood $U$ in $M$ such that there is a diffeomorphism $f:U\to \mathbb{R}^m$ such that $f_*(X_i) = \frac{\partial}{\partial x^i}$. Now my idea is to show that the compactness of $M$ allows us to glue the little $f$'s together on all of $M$. Is this the right path?
Please let me know if anything is unclear, or if I should explain certain points more.
