Question

Let $M$ and $N$ be closed manifolds. Is it true that $C^{k}(N,M)$, which is the space of functions $f: N\to M$ such that $f\in C^{k}$, is a $C^{\infty}$ Banach manifold? If so, can you help me to write charts?

Answer 1

Yes, $C^k(M,N)$ is a smooth Banach manifold, when $M$ and $N$ are smooth closed manifolds.

In fact, you can consider manifolds of maps with more general domains and regularities, this lies in the foundations of "Global Analysis". The classic work in the area is mostly due to Eells, Palais, Eliasson and others. I also particularly like this nice short note by Tausk, that considers Banach manifolds of maps from sets to manifolds, with minimum regularity requirements.

In the previous links you can find explicit descriptions of how charts are constructed; but, very roughly (in the following, I am omitting all regularities for the sake of simplicity), you can regard a neighborhood of a map $f\colon M\to N$ as consisting of maps $g\colon M\to N$ that can be written as $g(p)=\exp_{f(p)} X(p)$, where $X\colon M\to TN$ is a vector field along $f$, and the exponential map is regarded w.r.t. some choice of background metric $h$ on $N$. Of course $X$ has to be small enough, e.g., its norm (that depends of what regularity we're talking about) has to be less than the injectivity radius of $h$ along $f(M)$. The correspondence given by such chart is then $g\leftrightarrow X$, which tells you that the tangent space at $f$ to $Maps(M,N)$ consists of vector fields along $f$ (of the same regularity as the maps considered). Details can be found in any of the links above, but I hope this brief description at least gives you some intuition...

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Edit: Another good reference that I forgot to mention is this book by Hirsch, see Chap 2.