I am reading the book "Convex Integrtion Theory by D.Spring" and need some help.
Let $\pi:X \rightarrow V$ be a smooth fiber bundle over a smooth base manifold $V$. Let $h\in\Gamma^{0}(X)$ and $\phi\in\Gamma^{0}(X^{(1)})$ be an extension of $h$ in the space of 1-jets of $X$. Let $N$ be a neighborhood of $\phi(V)$ in $X^{(1)}$. One identifies a small neighborhood $U$ of $h(V)$ in $X$ with a vertical disk bundle with respect to $\pi:X \rightarrow V$. Since $p^{1}_{0}:X^{(1)}\rightarrow X$ is a Serre fibration, there is a continuous lift $\nu:U\rightarrow N$ such that:(i) $p^{1}_{0}\circ\nu=id_{U}$; (ii)$\nu\circ h=\phi\in\Gamma^{0}(X^{(1)})$.
My question starts from here.
(1) How to identify such a small neighborhood of $h(V)$ with a vertical disk bundle?
(2) How to use the property of Serre fibration to obtain the desired lift.
Thank you!!!