$C^k$ manifold : question about the definition In my book they define a manifold to be of class $C^k$ if all the map linking one chart to another are $C^k$ function.
But I don't really understand this as the fact that a manifold is of class $C^k$ should'nt depend on the charts we use.
So do I have to reformulate the definition as : "A manifold is a $C^k$ one if it exists one atlas for which all the map linking one chart to another are $C^k$ functions"
There are probably other way to define $C^k$ manifold with topology arguments but I am not very strong in topology so I would like if possible avoid too much topology arguments :)

Extra questions about vocabulary :
In $(U,\phi_U)$ do we call the chart either the function $\phi_U$ and the subset $U\in M$ (M is the manifold) ?? Or what we call chart is only the function (or the subset).
Also, is there a simple name to call the functions $\phi_U \circ \phi_V^{-1}$ that goes from one chart to another ? Or there is not conventional name for theses.
 A: A manifold is a set together with a collection of charts; it's called $C^k$ if the charts have the property you describe. 
The circle, for instance, can be given charts that make it a $C^k$ manifold for any $k$ you happen to like.
Just to give a for-instance, if you define coordinates on the $y > 0$ portion of the circle via $(x, y) \mapsto \frac{x}{\sqrt{x^2 + y^2}}$, and use the same definition on the $y < 0$ portion, and use  $(x, y) \mapsto \frac{y}{\sqrt{x^2 + y^2}}$ on the $x > 0$ and the $x < 0$ portions, then this four-chart atlas gives you transition functions that are $C^0$ but not $C^1$. 
A: Suppose $f: M \to \mathbb{R}^N$ is smooth. Since smoothness is a local property, we mean smooth w.r.t some chart $(U, \phi)$  i.e $f$ smooth implies $f \circ \phi^{-1}: \phi(U) \to \mathbb{R}^N$ is smooth. Now let $ (\psi, V)$ be any other chart with $U \cap V \not = \emptyset$ then,
$$ f \circ \phi^{-1} = (f \circ \psi^{-1}) \circ  \psi \circ \phi^{-1}$$
Hence if you want smoothness to be independent of charts then we require $ \psi \circ \phi^{-1}$ to be smooth diffeomorphisms for any charts $\psi, \phi$.
