Reference: Richard Millman - Elements of differential geometry
I'm reading this book and there is a really wierd definition in this book:
To descrive the theorem, here are some definitions in this book
Definition1
Let $U$ be open in $\mathbb{R}^2$ and $x:U\rightarrow \mathbb{R}^3$ be an injective $C^k$ function.
Then, $x$ is said to be a $C^k$ coordinate patch iff $\frac{\partial x}{\partial u_1}\times\frac{\partial x}{\partial u_2}\neq 0$.
Moreover, we call $x$ is proper if $x$ is a topological embedding.
Definition2
Let $U$ and $V$ be two open sets in $\mathbb{R}^2$.
Let $f:U\rightarrow V$ be a $C^k$ -bijective function.
Then, $f$ is a $C^k$ coordinate transformation iff its inverse is $C^k$
And below is the definition I don't get:
A $C^k$ surface is a subset $M\subset \mathbb{R}^3$ such that for every point $P\in M$ there is a proper $C^k$ coordinate patch whose image is in $M$ and which contains an $\epsilon$ neighborhood of $P$ for some $\epsilon>0$. Furthermore, if both $x:U\rightarrow \mathbb{R}^3$ and $y:V\rightarrow \mathbb{R}^3$ are such coordinate patches with $U'=x(U)$, $V'=y(V)$, them $y^{-1}\circ x: (x^{-1}(U'\cap V'))\rightarrow (y^{-1}(U'\cap V'))$ is a $C^k$ coordinate transformation.
This definition is super strange since this is quite opposite view of the standard definition of atlases and charts.
I especially don't get the last sentence.
The definition starts with "If $p\in M$, then there exists a $C^k$-coordinate patch $x$ such that ...." -(1), however I think it should be "There exists a collection $\{(U,x)\}$ such that for each $p\in M$..." - (2)
Am I right?
And below is an example which comes after the definition and in this example, argument is based on (2).
Which one is he talking about?
Typically, one does not have an atlas at hand. For this reason it is not beneficial to describe a surface as being the image of an atlas, only that one exists (which is what he is claiming by saying that $M$ is contained in the images of such charts or, there is a chart for each $p\in M$).
They are actually not different definitions. Suppose for every $p$ there is such a chart. Then the collection of these charts forms an atlas. Conversely, if one has such an atlas, then at every point $p\in M$ there is a chart.
The definition given is the typical definition for a $C^k$ differentiable structure on $M$.