Before I ask my question, I would like to build up the setting.
Let $\pi:E \rightarrow M$ be a smooth fiber bundle. Since $\pi$ is a smooth submersion, there is an induced foliation $\mathcal{F}(\pi)$ on E, the leaves of which are the connected components of the fibers $\pi^{-1}(x), ~x\in M$. Considering the local trivialization of E, the leaves of this foliation are, in a sense, vertical.
The next step is to construct a fiber bundle $\pi:E \rightarrow M$ and a foliation on the bundle space $E$, the leaves of which are horizontal. This construction is done in some steps.
Consider the universal covering $\tilde{M}$ of a smooth manifold $M$ and a group $G$ acting freely and properly on $\tilde{M}$ such that $\tilde{M}/G \cong M$. Such a group always exists, for example take $G=\pi_1(M,x_0)$.
Choose a $G$-manifold $F$, $G$ acts on the left, and consider the product manifold $\tilde{M} \times F$. This is also a $G$-manifold: $(\tilde{m},f)\cdot g = (\tilde{m} \cdot g, g^{-1} \cdot f)$, and the action is free and proper.
3.The orbit space $E=\tilde{M}\times F /G$ is a smooth (Hausdorff) manifold and there is an induced map $\pi:E \rightarrow M , \pi([\tilde{m},f])=[\tilde{m}]$ which turns $E$ into a fiber bundle over M.
- The foliation $\tilde{M}\times {f}, ~f \in F$ on $\tilde{M}\times F$ induces a foliation $\mathcal{F}$ on $E$, through the quotient foliation construction, with leaves diffeomorphic to $\tilde{M}/G_f$.
I am trying to understand the fibers of $\pi:E \rightarrow M $ , in order to understand the "horizontal" nature of the leaves of $\mathcal{F}$. In this direction, my calculation give that the fibers are diffeomorphic to $[\tilde{m}]\times F$. In addition, by studying the fibers we can understand the vertical bundle $Vec(E)$. Regaring this issue, I have the feeling that $T\mathcal{F}$ determines a horizontal bundle, hence a connection on $E$, but I have trouble showing this. I cannot write explicitly the above vector bundles in order to get the desired results.
Any insights, ideas or advices are more than welcome. Thanks in advance