Existence of smooth charts in smooth manifolds

Question

Given a smooth manifold $M$ and any point $p \in M$, may I always take a smooth chart $(U, \varphi)$ (that is, $\varphi$ is a diffeomorfism between $U$ and $\varphi(U) \subset \mathbb{R}^n$) with $p \in U$? If so, why?

Answer 1

If $p\in M$, take any chart $(U,\varphi)$ around $p$. Then $\varphi$ is a homeomorphism between $U$ and $\varphi(U)$. To prove that $\varphi$ is a differmorphism, you just have to prove that $\varphi:U\to \varphi(U)$ and $\varphi^{-1}:\varphi(U)\to U$ are smooth, which means that when you read them in coordinate charts they are smooth. Now you have to identify the charts on $U$ and on $\varphi(U)$. The structures on these manifolds is the induced structure, so their structures are the maximal atlas generated by $(U,\varphi)$ and $(\varphi(U),id)$ respectively. Now take $q$ in $U$; $(U,\varphi)$ and $(\varphi(U),id)$ are charts around $q$ and $\varphi(q)$ and you have that the map $$id\circ\varphi\circ\varphi^{-1}:\varphi(U)\to\varphi(U)$$ is smooth (it is the identity). Therefore when you read $\varphi$ in coordinate charts, it is smooth. You can do the same thing for $\varphi^{-1}$, which proves what you were looking for.

I hope this helps !