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Let $M$ and $N$ be finite dimensional smooth manifolds and $p: E \rightarrow N$ a finite rank vector bundle. $M$ can be assumed to be compact if necessary but I would prefer to work without this assumption.

As described e.g. in Kriegl's and Michor's book, $C^\infty(M,N)$ is an $\infty$-dimensional manifold. Now I would like to construct a vector bundle (of rank $\infty$) whose fibre at $\varphi \in C^\infty(M,N)$ is given by $\Gamma(\varphi^\ast E)$. This should be possible because the fibres $\Gamma(\varphi_1^\ast E)$, $\Gamma(\varphi_2^\ast E)$ at homotopic maps $\varphi_1$, $\varphi_2$ are isomorphic. I have more or less an idea how to do the construction but have not worked out all the details needed to prove smoothness (of transition functions). However, the construction looks quite natural and I would almost expect that it has already appeared somewhere in the literature. Does anybody know a reference ? (I did some research but could not find anything ...)

1. Update(06.May 13)

As discussed below in the answer by Peter Michor, the total space of the bundle I am looking for is $C^\infty(M,E)$ and the (smooth) projection is $p_\ast : C^\infty(M,E) \rightarrow C^\infty(M,N)$.

However, I do not see that it actually defines vector bundle over the base $C^\infty(M,N)$, probably I am overlooking a piece of the argument. As far as I understand the proof of theorem 42.17 in the "Convenient Setting", it is shown that there exists a diffeomorphism $\Phi: TC^\infty(N,M) \cong C^\infty(M,TN)$ (I just neglect compact supports here) and $C^\infty(M,TN)$ inherits the structure of a vector bundle. In my case, I don't see that $C^\infty(M,E)$ is diffeomorphic to $T(something)$ and I did not find a "general" proof of such a vector bundle property for $C^\infty(M,E)$ in the Convenient Setting. Please let me know if there is any.

One idea to construct local trivializations for $C^\infty(M,E)$ might be given by identifying spaces $\Gamma(\varphi_1^\ast E)$ and $\Gamma(\varphi_2^\ast E)$ (for $\varphi_1,\varphi_2$ homotopic) using parallel transport w.r.t. some connection on $E$. I think this should be possible but proving that this construction depends smoothly on the basepoints $\varphi \in C^\infty(M,N)$ and does not depend on the choice of the connection seems to be technically complicated and I would like to avoid it if possible.

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2 Answers

up vote 6 down vote accepted

In "Convenient Setting" the construction that you want is done in 42.17.

There is also the old book, where 10.10 is relevant and your construction (for the special case $E=TN$) is done in (too many) details in 10.12:

  • Peter W. Michor: Manifolds of differentiable mappings. Shiva Mathematics Series 3, Shiva Publ., Orpington, (1980), iv+158 pp., MR 83g:58009, ZM 433.58001, (pdf)

If $M$ is not compact you have to use a very fine topology where the set maps which differ from a given map only on a compact set are open. Otherwise the space might be not locally contractible and you cannot find open charts.

Then you just take $C^\infty(M,E)$ as total space with $C^\infty(M,p)=p_*:C^\infty(M,E)\to C^\infty(M,N)$ as projection, or an open subspace therein inf $M$ is not compact. The fiber over $\phi:M\to N$ are the maps $M\to E$ which project to $\phi$, i.e., the sections (with compact support if $M$ is not compact) of $\phi^*E$.

Edit:

Okay, here is a proof of local triviality ($U$ will always denote a neighborhood of the zero section): We need:

  • A Riemann metric on $N$ and its Riemannian exponential mapping $\exp^N: TN\supset U \to N$ such that $(\pi_N,\exp^N):TN\supset U \to N\times N$ is a diffeomorphism onto a neighborhood of the diagonal. If $f\in C^\infty(M,N)$ and if $g$ is near enough to $f$ (in a neighborhood $U_f$ of $f$ in $C^\infty(M,N)$), then $$u_f(g)(x) = (\pi_N,\exp^N)^{-1}(f(x),g(x))$$ defines a chart which maps $g\in U_f$ to $u_f(g)\in\Gamma(f^*TN)$. Note that $$ t\mapsto c_{f,g,x}(t) := u_f^{-1}(t.u_f(g))(x) = \exp^N_{f(x)}(t.u_f(g)(x)) $$ is a smooth curve from $f(x)$ to $g(x)$ in $N$ which depends smoothly on $x$ and on $g$.

  • A linear connection on $E$ with its parallel transport $Pt(c,t):E_{c(0)}\to E_{c(t)}$ along each smooth curve $c$ in $N$. The parallel transport is smooth in $c$ (easy to proof with convenient calculus) and in $t$. In detail, the following mapping is smooth by convenient calculus and usual results about linear ODEs in finite dimension:

$$Pt: C^\infty(\mathbb R,N)\times_{ev_0,p} E \times \mathbb R \to E,\quad (c,v_{c(0)},t)\mapsto Pt(c,t)v_{c(0)}$$

  • To describe the chart structure on $C^\infty(M,E)$ we need also a Riemannian metric on $E$. The best one now is a fiber metric on the vector bundle $E$ which is respected by the connection on $E$, and then we pull the metric from $N$ to the horizontal bundle in $TE$ and lift the fiber metric to the vertical bundle in $TE$ and declare them to be orthogonal. I will not use this explicitly now.

For $s\in C^\infty(M,E)$ with $p\circ s$ near $f$ consider the mapping $$ \Phi(s)(x) := Pt(c_{f,p\circ s,x},1)^{-1}(s(x))\in E_{f(x)}. $$
Then $$ s\mapsto (p\circ s, \Phi(s))\in U_f\times \Gamma(f^*E) $$ is a smooth local trivialization of $p_\star:C^\infty(M,E)\to C^\infty(M,N)$.

You can play with the topology on $C^\infty(M,E)$ by requiring compact support in $C^\infty(M,N)$ but using the compact $C^\infty$-topology along the fibers of $E$; or also compact support there. If you need more details here, tell me.

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Thanks a lot for the useful answer. Unfortunately, I am still stuck at one point of the problem and I edited my original question to clarify that. –  fhanisch May 6 '13 at 18:06
    
So it is necessary to do the explicit construction of bundle charts ... good to know! Thanks for the very detailed instruction how to do this, I really appreciate this. I had a rough plan how to do it but I did not realize that smoothness of the parallel transport w.r.t the "transportation curve" can be used in this way to obtain smoothness of the charts. Concerning the topology on the fibres: Thanks for offering help, I think I have enough information for the time being. If I can not work it out, I'll post again. –  fhanisch May 10 '13 at 11:51
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I believe that the book, "The Structure of Classical Diffeomorphism Groups" by Augustin Banyaga had a very clear construction of the sought after tangent space. Also, "The Inverse Function Theorem of Nash and Moser" by Richard S. Hamilton develops a few of the tool need in order to work with Frechet manifolds.

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