In the book of Mathematical Analysis II by Zorich, In section 12.1, it gives the definition of a k-dimensional surface, and then giving some argument, he gives the definition of smooth k-dimensional surface, but I'm really confused by those arguments in between, so to make sure, is my understand of a smooth $k$-dimensional surface correct ?
My understanding:
A smooth $k$-dimensional surface in $\mathbb{R}^n$ is a subset $S \subseteq \mathbb{R}^k $ each point of which has a neighbourhood in $S$ that is homeomorphic to $\mathbb{R}^k$ and those homeomorphisms are in the class of $\mathbb{C}^m$ ($m\geq 1$) and are bijections.
Of course, the smoothness of the surface is the same as the degree of the smoothness of $m$ those homeomorphisms.
Further questions:
Do we need that those homeomorphism to be diffeomorphims ? i.e. do their inverse also have to be in the class of $C^m$ ?
$S$ must be locally homeomorphic to $\mathbb{R}^k$, not to $\mathbb{R}^n$. Suppose $U\subset S$ is open with $\phi\colon U\rightarrow \mathbb{R}^k$ a homeomorphism. The smoothness you are referring to then actually means that $i\circ\phi^{-1}$ is $C^m$, where $i:S\rightarrow \mathbb{R}^n$ is the inclusion map.