1) I was told in class many years ago that, the tangent space if the sphere $\mathbb{S}^2$at a point $p$, i.e.
$T_p\mathbb{S}^2$ is isomorphic to $\mathbb{R}^2$. Could anyone give me a proof of this? Why?
2) Is it true in general, that is, does that always exist an isomorphism between a general manifold of dimension $n$ to $\mathbb{R}^n$?