Smooth Banach vector bundle and section

Let $M$ be a smooth ($C^{\mathrm{\infty }}$) Banach manifold modelled on the real Banach space $E$ (Countable base Banach manifold and smooth map) and let $F$ be a real Banach space whose norm topology is second countable (Banach space). This countability hypothesis makes the product $U\times F$, with its product atlas, a Banach manifold in the library's second-countable convention.

A smooth Banach vector bundle over $M$ with fibre $F$ is a smooth Banach manifold $\mathcal{E}$ together with a surjective smooth map $\pi :\mathcal{E}\to M$ (Countable base Banach manifold and smooth map) such that:

1. every fibre ${\mathcal{E}}_p:={\pi }^{-1}(p)$, $p\in M$, is a real vector space;
2. for every $p\in M$ there is an open neighbourhood $U$ of $p$ and a local trivialization, a diffeomorphism $\mathrm{\Phi }:{\pi }^{-1}[U]\to U\times F$ satisfying ${\mathrm{p}\mathrm{r}}_1\circ \mathrm{\Phi }=\pi$, whose restriction ${\mathrm{\Phi }}_q:={\mathrm{p}\mathrm{r}}_2\circ \mathrm{\Phi }{\mathrm{\mid }}_{{\mathcal{E}}_q}:{\mathcal{E}}_q\to F$ is a linear isomorphism for every $q\in U$;
3. cocycle condition: if $\mathrm{\Phi }$ over $U$ and $\mathrm{\Psi }$ over $V$ are local trivializations, then on ${\pi }^{-1}[U\cap V]$, $\mathrm{\Psi }\circ {\mathrm{\Phi }}^{-1}(q,v)=(q,g(q)v)$ for a map $g:U\cap V\to \mathcal{B}(F)$ into the bounded operators on $F$ (The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators) whose representative in every base chart is $C^{\mathrm{\infty }}$ in the Banach-space sense, and $g(q)$ is invertible for every $q\in U\cap V$.

A smooth section of $\pi$ is a smooth map $s:M\to \mathcal{E}$ with $\pi \circ s={\mathrm{i}\mathrm{d}}_M$. Its zeros are the points $p\in M$ with $s(p)=0$, the zero of the vector space ${\mathcal{E}}_p$. In a local trivialization $\mathrm{\Phi }$ over $U$ the section corresponds to the smooth map $\sigma :={\mathrm{p}\mathrm{r}}_2\circ \mathrm{\Phi }\circ s:U\to F,$ and $s(p)=0$ if and only if $\sigma (p)=0$.

At a zero $p$ of $s$ the vertical derivative of $s$ at $p$ is

$D^{v}s(p):T_{p}M⟶{\mathcal{E}}_p,D^{v}s(p)(\xi ):={\mathrm{\Phi }}_p^{-1}(D\sigma (p)\xi ),$

where $\mathrm{\Phi }$ is any local trivialization around $p$ and $D\sigma (p)$ is the differential of the smooth manifold map $\sigma$ at $p$, the target $F$ being read with its single identity chart; $\xi$ denotes a tangent vector in $T_{p}M$.

• The vertical derivative is well defined. Let $\mathrm{\Phi }$ over $U$ and $\mathrm{\Psi }$ over $V$ be local trivializations around the zero $p$, with cocycle $g:U\cap V\to \mathcal{B}(F)$ as above, and let ${\sigma }_{\mathrm{\Phi }}={\mathrm{p}\mathrm{r}}_2\circ \mathrm{\Phi }\circ s$, ${\sigma }_{\mathrm{\Psi }}={\mathrm{p}\mathrm{r}}_2\circ \mathrm{\Psi }\circ s$ be the local representatives. Then ${\sigma }_{\mathrm{\Psi }}(q)=g(q){\sigma }_{\mathrm{\Phi }}(q)$ for $q\in U\cap V$, and at the zero $p$ the product rule (Chain sum product and composition rules for Banach derivatives) applied to the composition of $q\mapsto (g(q),{\sigma }_{\mathrm{\Phi }}(q))$ with the bounded bilinear evaluation map $\mathcal{B}(F)\times F\to F$, $(T,v)\mapsto Tv$ (A bounded bilinear map between normed spaces, For a bilinear map, boundedness is equivalent to joint continuity), gives

$D{\sigma }_{\mathrm{\Psi }}(p)=g(p)D{\sigma }_{\mathrm{\Phi }}(p)+Dg(p){\sigma }_{\mathrm{\Phi }}(p)=g(p)D{\sigma }_{\mathrm{\Phi }}(p),$

the last term vanishing because ${\sigma }_{\mathrm{\Phi }}(p)=0$. Since ${\mathrm{\Psi }}_p=g(p)\circ {\mathrm{\Phi }}_p$ as linear isomorphisms ${\mathcal{E}}_p\to F$, the two prescriptions give the same element of ${\mathcal{E}}_p$.

• Fibrewise linear structure is intrinsic. The linear structure on ${\mathcal{E}}_p$ is part of the data, and each trivialization restricts to a linear isomorphism on it. The transition maps are fibrewise bounded linear and depend smoothly on the base; the vector bundle axioms are not restated here as a list of identities because they are exactly the conditions 1–3 above.

• The zero section. The assignment $p\mapsto 0\in {\mathcal{E}}_p$ is a smooth section, the zero section, whose vertical derivative at every point is the zero operator. Transversality of a section $s$ to the zero section is the condition that at every zero $p$ the map $D^{v}s(p)$ is surjective with complemented kernel (A complemented closed subspace of a normed space); it is the hypothesis of the next theorem on this page, where the zero set is straightened.

• Ranks and dimension. Nothing is assumed about the dimension of $F$ or of $E$ beyond the second-countability convention above; the fibre may be infinite dimensional and second countable, which is exactly the case the infinite-dimensional transversality theorem below needs. When $\dim F<\mathrm{\infty }$ and $D^{v}s(p)$ is onto, its kernel has finite codimension in $T_{p}M$ and is therefore automatically complemented, so the local condition of transversality reduces to surjectivity (Closed finite-codimensional subspaces are complemented).