Consider the subset $T^{2}$ of $\mathbb{R}^{3}$ constructed as follows: for each $\vec{r_\theta}=r exp(iθ)$ $\in\mathbb{R}^{2}$ $\approx$ $\mathbb{R}^{2} X ${0}$ , r>1$ consider the plane $P_\theta$ $\subset \mathbb{R}^{3}$ generated by $\vec{r_\theta}$ and $e_3=(0,0,1)$ . In $P_\theta$ consider the unit circle centered on $\vec{r_\theta}$ .Being $S_\theta^{1}$ $\subset P_\theta$ such a circle, provide an atlas for $T^{2}$= $\cup_{\theta \in [0,\pi]}$ $S_\theta^{1}$ Tell me if this atlas is orientable. Finally, describe the tangent space of $T^{2}$.
My biggest doubt is how exactly he started to realize this question, because I initially thought to assemble the atlas formed by 2 cards in the would have a $\phi$ to cover the inner circle and the same $\phi$ defined in another way would cover the larger circle. But I couldn't assemble this $\phi$ so if you have any tips I would be very grateful.