# Fréchet manifold structure on space of sections

I know that the space $$\mathsf{C}^\infty(M;N)$$ of smooth maps from a closed (smooth) manifold $$M$$ to a (smooth) manifold $$N$$ is a Fréchet manifold. I have been looking for a more general version of this statement along the following lines:

Let $$p: E \to B$$ be a smooth fiber bundle, where $$E$$ and $$B$$ are manifolds with corners (with some additional assumptions on $$p:E \to B$$?). Then $$\Gamma^\infty(B;E) := \{ s: B \to E \mid s$$ smooth, $$p \circ s = \mathsf{id}_B \}$$ is a Fréchet manifold (with corners?)

but I can't seem to find a precise statement or proof of something like this anywhere. I've tried looking in "A Convenient Setting for Global Analysis," but that book seems to work in a very large amount of generality that is a bit beyond what I would need. The only generalizations I am looking for are:

• instead of functions $$f: M \to N$$, we consider sections $$s: B \to E$$ of a fiber bundle $$p: E \to B$$,
• the manifolds in question can have boundary (or maybe even corners).

Then the original result for $$\mathsf{C}^\infty(M;N)$$ would then be recovered by taking $$M$$ and $$N$$ without corners and considering the trivial bundle $$M \times N \to M$$.

I would really appreciate it if anyone could suggest a reference where a result like this is stated/proven, or if someone could explain how I could formulate/prove this (namely, what are the charts on $$\Gamma^\infty(B;E)$$, what assumptions would we need on $$p: E \to B$$, and do we need a notion of "Fréchet manifold with corners"?). Thanks very much!

• I feel like you can use the local trivialization of the bundle to locally write $s$ as a smooth function (that preserves the fibres) from $U\subseteq B$ to $U \times F$ where $F$ is the fibre. You can then use a partition of unity for $B$ to write the section as a finite sum of compactly supported functions, from there the space of smooth sections on each of the open neighborhoods $U_\alpha$ is a Frechet manifold, what remains to be shown is that these glue together to make a global Frechet manifold which sounds possible? Anyway I'm just throwing ideas let me know what you think. Jul 6, 2020 at 18:30
• Thanks for your response! It does sound like a good idea to try to glue $\Gamma(E;B)$ together from the $\Gamma(U_\alpha;E|_{U_\alpha})$'s, though I'm not sure it'll work in this case; since $E$ isn't a vector bundle, it doesn't make sense to "multiply" sections by elements of a partition of unity to make them compactly supported, so I'm not sure how to obtain global sections of $E$ from local ones. Also, I'm also not clear on what to do if a given chart $U_\alpha$ is a boundary/corner chart, as in, what structure does $\Gamma(U_\alpha;E|_{U_\alpha})$ have in such a case?
– Alec
Jul 7, 2020 at 4:32
• Ah I see, yes that would be a problem I come mostly from a vector bundle background so that was what I thought of at first glance. The boundary and corner stuff also sounds like it could be a bit of a problem haha. I'll have to think about this a bit more, definitely an interesting problem though. Jul 7, 2020 at 19:46
• I think that this is asking too much. For general fiber bundles, there is no direct relation between local and global sections. Just think about examples like the sphere bundle associated to a vector bundle, whose sections correspond to nowhere vanishing sections of the vector bundle. Jul 21, 2020 at 12:45
• Thank you for your response – I don't understand the issue you're bringing up, however. If I choose some cover $\{U_\alpha\}_\alpha$ of $B$ together with trivializations $E|_{U_\alpha} \cong U_\alpha \times F$ which I fix, then it seems like I should be able to obtain a bijection between sections $s$ of $E$ and collections $\{s_\alpha\}_\alpha$ of sections of the $E|_{U_\alpha}$'s which agree on overlaps, and I can moreover use my trivializations to identify the $s_\alpha$'s with smooth $F$-valued functions on $U_\alpha$'s. Could this be an approach to defining the structure of $\Gamma(B;E)$?
– Alec
Jul 23, 2020 at 3:48