# Fréchet manifold structure of C(M, N)

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 $\gamma(o) = p$, $\gamma \sim \gamma'$ iff $\dot{\gamma(0)} = \dot{\gamma'(0)}.$ 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 $C_0(x) = f$, $C_1(x) = g$, $C_{t_0}(x)$ is smooth for all $t_0 \in [0,1]$, and $C_{t}(x_0)$ is smooth for all $x_0 \in M.$ After a little work, ones sees that for $f \in F$, $T_{f}F = \Gamma_f(M, TN)$ 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)? (iii) Does there exist any literature in which an explicit construction of even a trivial example of a Fréchet manifold can be found? (Hamilton's,"The inverse function theorem of Nash and Moser", I feel, comes closest to an example.)

(I should mention also that this is a copy of a question on the math stackexchange website. http://math.stackexchange.com/questions/202048/frechet-manifold-structure-of-cm-n)

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why doesn't the answer you already got help you? –  Mariano Suárez-Alvarez Sep 30 '12 at 10:19
For the family of seminorms on the section spaces, just choose a metric and take the Sobolev norms with respect to the Levi civita connection. This induces a Topology on M which does not depend on the Metric chosen if M is compact - otherwise, it does. –  Matthias Ludewig Sep 30 '12 at 17:16
@Mariano It did help me, but didn't answer my questions (or at least I could figure out how it did). –  Henry Zorrilla Sep 30 '12 at 22:20
@Mariano: Err, because it's wrong? Or at least misleading. –  Loop Space Oct 1 '12 at 18:07
@Kofi: Could you please elaborate a little more on you last sentence, e.g. why a change in the metric does not result in equivalent seminorms for non-compact base manifold. Any pointers to the literature are also welcome. –  Tobias Diez Apr 1 '13 at 21:17

## 1 Answer

See Theorem III.1.11 on page 76 of "Stable mappings and their singularities" by Golubitsky and Guillemin for a proof that $C^\infty(M,N)$ is a Fréchet manifold when $M$ is compact.

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And A Convenient Setting for Global Analysis (Kriegl and Michor), Manifolds of Differentiable Mappings (Michor), and the nlab page ncatlab.org/nlab/show/manifold+structure+of+mapping+spaces –  Loop Space Oct 1 '12 at 8:43