What is a $C^{n}$ surface? I know the definition of a regular surface (a set) and a continous/differentiable/regular parametrized surface (a map). But what is a $C^{n}$ surface?
Is a $C^{n}$ surface the image of a $C^{n}$ paramterized surface? I've a theorem where they introducea $C^{2}$ surface and use its Gaussian curvature. But Gaussian curvature is only defined for regular surfaces?
Can someone help me clarify this notation?
 A: Recall that for each point $x$ in our surface $S$ there is an open neighborhood $U$ of $S$ around $x$ and a homeomorphism $\phi : U \to \tilde{U}$ where $\tilde{U}$ is some open subset of $\Bbb R^2$. This is the "locally Euclidean" property of manifolds (and in particular, surfaces). We call such a neighborhood and homeomorphism a chart.
Now suppose we have two charts $(U, \phi)$ and $(V, \psi)$ such that $U\cap V \neq \emptyset$. We can consider the "transition function" $\psi \circ \phi^{-1}$ which is a homeomorphism from $\phi(U\cap V)$ to $\psi(U\cap V)$. As it is a map between open subsets of $\Bbb R^2$, we know what it means for it to be $C^n$ for $n = 1, 2, 3, \ldots, \infty$.
A $C^n$ surface is a surface on which we have chosen charts such that all of the transition functions between overlapping charts are $C^n$ maps. Thus, we do not include all possible charts, but instead choose a subset for which all the transition functions are $C^n$. An additional assumption is that the domains of the charts we choose must cover the entire surface.
You can check out this Wikipedia article for more information.
