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I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following:

Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set $C^k(M,N)$ of $C^k$-maps from $M$ to $N$, $k\in\mathbb{N}$, $k<\infty$. Then there exsists the structure of a smooth banach manifold on $C^k(M,N)$ and the underlying topology is the compact-open $C^k$-topology (the weak topology).

So far, I was only able to find references that prove more general results (and thus the proofs get harder and use things I don't really need for now), e.g. if $N$ is a banach manifold itself. These stronger results I found in:

  • Eliasson, Geometry of Manifolds of Maps

  • Abraham, Lectures of Smale on Differential Topology

Other sources I found deal only with $C^\infty(M,N)$ or $C^0(M,N)$. In this context the books of Lang (e.g. Fundamentals of Differential Geometry) are mentioned (e.g. here), but I'm not able to find these parts in his books.

I'd really appreciate it if someone could name me a reference where the above statement is proven.

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  • $\begingroup$ You can find proofs of these statements in the notes of John Moore found at the following link: math.ucsb.edu/~moore/globalanalysisshort.pdf. $\endgroup$
    – Andy Sanders
    Commented Jul 28, 2015 at 16:10
  • $\begingroup$ Let me refine that by saying that I don't think there is any mention of the compact-open topology there, just so I don't send you on a wild goose chase. Nonetheless, you'll find the proof that it is a smooth Banach manifold. $\endgroup$
    – Andy Sanders
    Commented Jul 28, 2015 at 16:23
  • $\begingroup$ Thanks, that helped a lot! I have a follow-up question, maybe you (or someone else) can help me with it: Can the tangent bundle of the smooth banach manifold $C^k(M,N)$ be realised as $\bigcup_{f\in C^k(M,N)}C^k(f^*TN)\rightarrow C^k(M,N)$ where a $C^k$-section $X\in C^k(f^*TN)$ is mapped to $f$? If so, what's the idea to prove that? $\endgroup$
    – uro
    Commented Jul 29, 2015 at 13:29
  • $\begingroup$ See this paper for the formal details arxiv.org/pdf/1403.3111v2.pdf. Then, just note that if $f_t:M\rightarrow N$ is a variation of $f,$ then the first variation is a section of $f^{*}TN.$ $\endgroup$
    – Andy Sanders
    Commented Jul 30, 2015 at 3:53
  • $\begingroup$ Thanks again for your help. Maybe I can ask one last question. In the notes you linked me, the $C^\infty$ structure on $C^k(M,N)$ is consctructed like that: Pick $f\in C^\infty(M,N)$, then use the exponential map to construct a chart $x_f:U_f\rightarrow V_f$ where $U_f$ is a neighborhood of $f$ in $C^k(M,N)$. Then it is shown that for another $g\in C^\infty (M,N)$, the transition maps $x_f\circ x_g^{-1}$ are smooth. Other sources I found seem to indicate that this works aswell for $f,g \in C^k(M,N)$, i.e. for $f,g \in C^k(M,N)$ the transtion map $x_f\circ x_g^{-1}$ is smooth. Is that true? $\endgroup$
    – uro
    Commented Jul 30, 2015 at 9:50

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