# Tangent space of the space of smooth sections of a bundle

Let $E\to M$ be a real vector bundle of finite rank over a closed differentiable manifold $M$. Let $C^{\infty}(E)$ denote the space of smooth sections of $E$ and let $e\in C^{\infty}(E)$ be a section. I often see statements of the type:

"The tangent space to $C^{\infty}(E)$ at any given section $e\in C^{\infty}(E)$ is isomorphic to $C^{\infty}(E)$ itself, namely $T_{e}C^{\infty}(E)\simeq C^{\infty}(E)$."

I wonder what is the precise formulation of the statement above.

I know that if we only take smooth sections, $C^{\infty}(E)$ admits the structure of an infinite dimensional Frechet manifold with respect to the appropriate topology. The statement should be then that the isomorphism $T_{e}C^{\infty}(E)\simeq C^{\infty}(E)$ holds understanding $C^{\infty}(E)$ as a Frechet manifold and taking $T_{e}C^{\infty}(E)$ to be the tangent space of $C^{\infty}(E)$ as a Frechet manifold? Or rather, should it be understood by assuming that we have implicitly Sobolev-completed $C^{\infty}(E)$ into $H_{s}(E)$ using some appropriate Sobolev norm as to make $H_{s}(E)$ into a smooth Hilbert manifold and then the isomorphism that actually holds is $T_{e}H_{s}(E)\simeq H_{s}(E)$?

My second question is: can one make sense of an isomorphism of the type $T_{e}C^{\infty}(E)\simeq C^{\infty}(E)$ if the base is non-compact?

And lastly, let $Q\to M$ be a smooth fiber bundle over a closed manifold $M$ with typical fiber $F$ given by a smooth manifold. How should be understood the tangent space at a point of the space of sections of $Q$?

References are welcome.

• Concerning the last question: a tangent vector to a given section $s$ would probably be a vertical vector field in $Q$ along the section $s$ (vertical wrt the projection $Q\to M$ ). Unfortunately I don't have a reference. – Michael Bächtold Jul 28 '17 at 20:39

$$C^\infty(E)$$ is a Frechet VECTOR space. Thus its tangent space at each point equals $$C^\infty(E)$$ via its affine structure.

This is also true if $$M$$ is not compact. However, for a fiber bundle $$Q\to M$$ one has to be more careful with the topology (if $$M$$ is not compact). See 10.10 of

• Peter W. Michor: Manifolds of differentiable mappings. Shiva Mathematics Series 3, Shiva Publ., Orpington, (1980) (pdf)

for an answer. In principle, the tangent space is the space of sections of the vertical bundle of $$Q$$ restricted to the the image of a section. There is also Sections 42, 43, .. of

• Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997 (pdf)

where a more easily handable notion of differentiability is developped and then used.

These methods also work for Sobolev spaces of sections. See

• MR3135704 Reviewed Inci, H.; Kappeler, T.; Topalov, P. On the regularity of the composition of diffeomorphisms. (English summary) Mem. Amer. Math. Soc. 226 (2013), no. 1062, vi+60 pp.

for the basics of these.

• Thanks a lot for the answer. If you can add some remarks about the other points in the question, I would appreciate it. In particular, is your remark true if $M$ is non-compact? – Bilateral Jul 28 '17 at 22:17
• @Peter having finally bought a physical copy of your book (one I ordered first was lost in transit!) I do wonder whether you'd consider making it available via print on demand publishing. – David Roberts Jul 30 '17 at 1:30