Charts on a $\mathcal C^k$ manifold are $\mathcal C^k$-diffeomorphisms Let $M$ be a $\mathcal C^k$ manifold of dimension $d$ and $\mathcal A=\{(U_i,\varphi_i)_{i\in I}\}$ be an altas on $M$. Is it correct to assume that the charts $(\varphi_i)_{i\in I}$ are $\mathcal C^k$-diffeomorphisms? We can use the characterization that a function $f:M\longrightarrow\mathbb R^d$ is  $\mathcal C^k$ iff $\exists(V,\psi)\in\mathcal A$ such that $f\circ\psi^{-1}$ is a $\mathcal C^k$-diffeomorphism. Since we know that the charts are compatible, then for any charts $\varphi_i$,$ \varphi_j$, $\varphi_i\circ\varphi_j^{-1}$ is a $\mathcal C^k$-diffeomorphism. We thus find that any chart $\varphi_i$ is $\mathcal C^k$ and is by definition invertible and by a similar reasoning, we get that $\varphi_i^{-1}$ is also $\mathcal C^k$. Is this reasoning correct?
 A: 
Let $M$ be a $\mathcal C^k$ manifold of dimension $d$ and $\mathcal A=\{(U_i,\varphi_i)_{i\in I}\}$ be an atlas on $M$. Is it correct to assume that the charts $(\varphi_i)_{i\in I}$ are $\mathcal C^k$-diffeomorphisms?

That charts are $\mathcal C^k$-diffeomorphisms cannot be an assumption in the definition of charts. This would not make any sense at the beginning because it is not yet clear what $\mathcal C^k$ means in this general context.
In a comment you write

The reason why I asked this question is because in the course I am studying, charts are supposed only to be homeomorphisms.

Correct, in the first instance a chart is only a homeomorphism. But in a $\mathcal C^k$ atlas $\mathcal A$ the charts satisfy certain compatibility conditions which allow to extend the concept of $\mathcal C^k$ maps in the classical sense (dealing only with maps between open subsets of Euclidean spaces) to maps between $\mathcal C^k$  manifolds. With this extended concept charts in $\mathcal A$ can in fact be proved to be diffeomorphisms (as you do). Let us emphasize

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*Not every chart on $M$ becomes a diffeomorphism; which charts depends on the given $\mathcal C^k$ atlas $\mathcal A$.


*It requires a carefully considered concept of $\mathcal C^k$ for maps between manifolds with $\mathcal C^k$ atlases before it makes sense to say that charts in  $\mathcal A$ are diffeomorphisms. Such a concept is not available at the time when a $\mathcal C^k$ atlas $\mathcal A$ is defined.
Let me try to explain it in  more detail.
In multivariable calculus one defines what it means that a function $f : U \to V$ between open subsets $U \subset \mathbb R^n$ and $V \subset \mathbb R^m$  is $\mathcal C^k$. This allows to define the concept of a $\mathcal C^k$ diffeomorphism, but this concept is of course limited to functions between open subsets of Euclidean spaces.
An $n$-dimensional topological manifold $M$ is a topological space which looks locally like $\mathbb R^n$ (and satisfies some other technical conditions). Looking locally like $\mathbb R^n$ means that each $p \in M$ has an open neighborhood which is homeomorphic to an open subset of $\mathbb R^n$. This gives rise to introduce the  concept of a chart on $M$: This is a pair $(U,\phi)$ consisting of an open $U \subset M$ and a homeomorphism $\phi : U \to U'$ to an open subset $U' \subset \mathbb R^n$. A topological atlas on $M$ is a collection of charts whose domains cover $M$.  Taking all charts gives the maximal topological atlas on $M$.
Given a map $f : M \to \mathbb R^m$, we can say that $f$ is $\mathcal C^k$ with respect to a chart $(U,\phi)$ if $f \circ \phi^{-1} : U' \to \mathbb R^m$ is $\mathcal C^k$ in the sense of multivariable calculus. Let us emphasize that this depends on the chart; for some charts $f \circ \phi^{-1}$ may be $\mathcal C^k$, for other charts this may fail.
Similarly, given a map $\mu : W' \to M$ defined on an open $W' \subset \mathbb R^m$, we can say that $\mu$ is $\mathcal C^k$ with respect to a chart $(U,\phi)$ if $\phi \circ \mu : W' \cap \mu^{-1}(U) \to U'$ is $\mathcal C^k$ in the sense of multivariable calculus. Again this depends on the chart.
Thus for a topological manifold $M$ we are not able to give an "absolute definition" for maps $f : M \to \mathbb R^m$ and $\mu : W' \to M$ being $\mathcal C^k$. We can only define $f$ and $\mu$ to be $\mathcal C^k$ with respect to a given topological atlas $\mathcal A$ on $M$ in the sense that $f \circ \phi^{-1} : U' \to \mathbb R^m$ and $\phi \circ \mu : W' \cap \mu^{-1}(U) \to U'$ are $\mathcal C^k$ for all charts in $\mathcal A$.
It is moreover desirable to work with a fixed topological atlas $\mathcal A$ on $M$ which determines whether any given $f : M \to \mathbb R^m$ is $\mathcal C^k$ and whether any given $\mu: W' \to M$ is $\mathcal C^k$; and more generally, whether any given $g : W \to \mathbb R^m$ defined on an open subset $W \subset M$ and any given $\mu : W' \to W$ are $\mathcal C^k$ with respect to the induced atlas $\mathcal A \cap W  = \{(U \cap W, \phi \mid_{U \cap W}) \mid (U,\phi) \in \mathcal A \}$.
The atlas $\mathcal A$ has the property that for each chart $(U,\phi) \in \mathcal A$ the maps $\phi : U \to U'$ and $\phi^{-1} : U' \to U$ are $\mathcal C^k$ with respect to $(U,\phi)$: Simply observe that $\phi \circ \phi^{-1} = id_{U'}$. But this does not mean that they are $\mathcal C^k$ with respect to $\mathcal A$ since this depends also on the other charts in $\mathcal A$.
An obvious requirement is that $\phi$ and $ \phi^{-1}$ should be $\mathcal C^k$ with respect to $\mathcal A$. It is easy to see that this requirement is equivalent to the following:
For any two charts $(U,\phi)$ and $(V,\psi)$, the transition functions
$$\psi \circ \phi^{-1} : \phi(U \cap V ) \to  U \cap V \to \psi (U \cap V)$$
$$\phi \circ \psi^{-1} : \psi(U \cap V ) \to  U \cap V \to \phi(U \cap V)$$
are $\mathcal C^k$  in the sense of multivariable calculus. If  $\mathcal A$ has this property, it is called a $\mathcal C^k$ atlas on $M$.
For a $\mathcal C^k$ atlas $\mathcal A$ all maps $\phi$ and $\phi^{-1}$ determined by the charts in  $\mathcal A$ are $\mathcal C^k$ diffeomorphisms - but not in an absolute sense, it depends on the atlas $\mathcal A$. With respect to another $\mathcal C^k$ atlas on $M$ they may be no longer $\mathcal C^k$.
In that sense it is tautological to say that $\phi$ and $\phi^{-1}$ are $\mathcal C^k$ diffeomorphisms. But we must be aware that this simple statement is based on the preceding development of a whole theory of $\mathcal C^k$ manifolds and $\mathcal C^k$ maps.
