The Bott partial connection on the normal bundle of a foliation

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $F$ be a codimension-$q$ regular foliation of a smooth manifold $M$, with tangent distribution $E=TF$, and let $\nu :=TM\mathrm{/}E$ be the normal bundle of $F$ (Quotient vector bundles by a subbundle, A vector bundle quotient by a subbundle is a smooth vector bundle). For a leaf-tangent vector field $X\in \mathrm{\Gamma }(E)$ and a section $s\in \mathrm{\Gamma }(\nu )$ the Bott partial connection is

${\mathrm{\nabla }}_X^{B}s:=\pi [X,\tilde{s}],$

where on each bundle chart $\tilde{s}$ is a smooth local representative of $s$ and $\pi :TM\to \nu$ is the quotient map (The canonical map to a quotient bundle is a smooth bundle map). Local representatives exist by lifting the components of $s$ in a quotient-bundle frame; their projected brackets agree on overlaps by the next lemma and therefore define a global section. The derivation is along leaf directions only: $X$ is a section of the involutive distribution $E$ (Regular foliation atlases).

This defines a map ${\mathrm{\nabla }}^B:\mathrm{\Gamma }(E)\times \mathrm{\Gamma }(\nu )\to \mathrm{\Gamma }(\nu )$ that is $C^{\mathrm{\infty }}(M)$-linear in the vector-field variable $X$, $\mathbb{R}$-linear in $s$, and satisfies the Leibniz rule ${\mathrm{\nabla }}_X^B(fs)=X(f)s+f{\mathrm{\nabla }}_X^{B}s$ for $f\in C^{\mathrm{\infty }}(M)$. The well-definedness of the formula and its flatness along each leaf are established in the next result. In the terminology of this page ${\mathrm{\nabla }}^B$ is a partial connection along the leaves; it is not a connection on all of $TM$, and no splitting of $TM\to \nu$ is chosen.