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Let $ F = C^{\infty}(M, N)$. I wish to give $F$ the structure of a Fréchet manifold. My plan was to emulate the construction of a smooth manifold. I know that for a finite dimensional smooth manifold M, $T_pM$ will be isomorphic to the model space (i.e if M is m dimensional, then $T_pM \cong \mathbb{R}^m$). Further more, I know that $T_pM$ can be identified with the equivalence classes of curves on M under the relation: For $\gamma, \, \gamma' \in C^\infty((-\epsilon,\epsilon), M)$ with \gamma(o) = p, $\quad$ $\gamma \sim \gamma'$ iff $\dot{\gamma(0)} = \dot{\gamma'(o)}.$ Define a path in $F$ to be a map $C:M\times[0,1] \rightarrow N$ with $C(x,t) = C_t(x)$ such that $C_0(x) = f, \, C_1(x) = g, \, C_{t_0}(x) \text{ is smooth for all } t_0 \in [0,1], \text{ and } C_{t}(x_0) \text{ is smooth for all } x_0 \in M.$ After a little work, ones sees that for some $f \in F, \, T_{f}F = \Gamma_f(M, TN)$ which I have proved is (set)isomorphic to $\Gamma(M, f^{*}TN)$ where $f^{*}TN$ denotes the pullback bundle.

Question(s): (i) What is a good candidate for a semi-norm (or family of semi-norms) on $\Gamma(M, f^{*}TN)\,$? (ii) Is the metric structure on $\Gamma(M, f^{*}TN)\,$ dependent upon the choice of semi-norm (or family of semi-norms)?

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1 Answer 1

Why do you want to emulate the construction of the tangent space to a manifold when defining a smooth (Fréchet-manifold) structure on $C=C^{\infty}(M,N)$? Am I missing something?

Anyways, here is a construction (from memory): we define charts for $C$ by taking charts $\Phi$ and $\Psi$ of $M$ and $N$, and defining the domain $U_{\Phi,\Psi}$ to be the set of smooth maps $f:M\to N$ such that $f(U_{\Phi})\subset U_{\Psi}$ and then sending $f\in U_{\Phi,\Psi}$ to $\Psi\circ f\circ\Phi^{-1}\in C^{\infty}(U_{\Phi},U_{\Psi})$. The latter space is already a Fréchet space. This gives a well-defined (this needs checking) Fréchet manifold structure on $C$.

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Thank you, this helps! I decided to emulate because I didn't know any better (I'm trying to learn these things on my own). Would you be able to provide a few (not too high level) references? –  Mathmonkey Sep 27 '12 at 4:53
    
You can check out Morris W. Hirsch's Differentiable Topology (I think it is called), he goes into detail about the topology of mapping spaces, and proves that it has the Baire Property, that immersions form an open dense subset etc... (again, from memory) –  Olivier Bégassat Sep 27 '12 at 5:17
    
This does not work. You don't have enough charts for the mapping space here: you only get maps whose image is contained in a single chart of N and there may be, for example, surjective maps. There are many places to find the construction, Kriegl and Michor's book A Convenient Setting for Global Analysis contains it, as does Michor's Manifolds of Differentiable Mappings. If you want an online reference, take a look at ncatlab.org/nlab/show/manifold+structure+of+mapping+spaces (there's one or two details lacking for the most general case but these are okay for this case). –  Loop Space Oct 1 '12 at 8:42
    
Oh, and you absolutely have to have compact source space. –  Loop Space Oct 1 '12 at 8:42
    
@AndrewStacey I didn't understand your objection at first, but I get it now. This only defines a topology (the weak topology) on the space of smooth maps, but not charts, if only because it sends two distinct functions that agree on a chart domain to the same place. –  Olivier Bégassat Oct 1 '12 at 16:00

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