Can anyone help me showing the following: Let $E$ be a smooth vector bundle over $M$ and $F\subseteq TM$ a smooth distribution. A $F$-connection is a $\mathbb R$-bilinear aplication $$\nabla:\Gamma(F)\times \Gamma(E)\rightarrow \Gamma(E),$$ where $\Gamma(E)$ is the set of all smooth sections of $E$, $(X, e)\mapsto \nabla_Xe$, such that for every $f\in C^\infty(M)$, $X\in \Gamma(F)$ and $e\in \Gamma(E)$:
(i) $\nabla_{fX}e=f\nabla_Xe$.
(ii) $\nabla_X(fe)=f\nabla_Xe+X(f)e$.
If $F=\{F_p; p\in M\}\subseteq TM$ is an involutive distribution show that the map:
$$\nabla:\Gamma(F)\times \Gamma(\nu(F))\rightarrow \nabla(\nu(F)),\ (X, \overline{Y})\mapsto \overline{[X, Y]},$$
is a $F$-conection in the normal bundle $\displaystyle\nu(F)=\bigcup_{p\in M}\nu(F_p)$ where $\nu(F_p)=T_pM/F_p$. The map $\nabla$ is called Bott connection.
Obs: $[X, Y]$ is the Lie Bracket.