Tangent Space for Torus

Question

I'm trying to find the basis for $T_{p}M$ for the torus $T^{2}= S^{1}$x $S^{1}$ so, my atlas for the manifold is \begin{equation} \varphi(u,v) = ((r\cos{u}+a)\cos{v},(r\cos{u}+a)\sin{v},r\sin{u}) \end{equation} But I don't know how to apply the definition of a bases for For a ${\displaystyle C^{\infty }}$ manifold ${\displaystyle M}$ , if a chart ${\displaystyle \varphi =(x^{1},\ldots ,x^{n}):U\to \mathbb {R} ^{n}}$ is given with ${\displaystyle p\in U}$ , then one can define an ordered basis ${\displaystyle \left(\left({\frac {\partial }{\partial x^{i}}}\right)_{p}\right)_{i=1}^{n}}$ of ${\displaystyle T_{p}M}$ by ${\displaystyle \forall i\in \{1,\ldots ,n\},~\forall f\in {C^{\infty }}(M):\qquad {\left({\frac {\partial }{\partial x^{i}}}\right)_{p}}(f)~{\stackrel {\text{df}}{=}}~({\partial _{i}}(f\circ \varphi ^{-1}))(\varphi (p)).}$

Thanks

Answer 1

$\phi:\mathbb{R}^2\to M$ is neither a chart nor an atlas. It is, however, a universal cover for $M$, and thus it is a local diffeomorphism. Therefore $$T_p\phi:T_p\mathbb{R}^2\to T_pM$$ is an isomorphism for each point $p:\mathbb{R}^2$, so the tangent spaces $T_pM$ are generated by the image $$\left\{\frac{\partial}{\partial u},\frac{\partial}{\partial v}\right\}$$ under $T_p\phi$.

To get an actual atlas from $\phi$, you need to find three contractible $U_0,U_1,U_2\subseteq \mathbb{R}^2$ such that

1. the $U_k$ are each homeomorphic to $\mathbb{R}^2$,
2. $\phi$ induces homeomorphisms $\phi(U_k)\cong U_k$, and
3. The $\phi(U_k)$ are an open cover of $M$.

For example, consider $$U_k=I_k\times I_k\subseteq \mathbb{R}^2$$ $$I_k= \left(\tfrac{2\pi k}{3}-\pi, \tfrac{2\pi k}{3}+\pi\right)$$