Let $[0,b]$be an interval and let $t$ be the standard coordinate on this interval. Suppose that $\pi:E \rightarrow [0,b]$ is a vector bundle with a given Ehresmann connection. Recall that a Ehresmann connection in this situation is a smooth distribution $\mathcal{H}$ on the total space $E$ such that:
- $\mathcal{H}$ is complementary to the vertical bundle $$TE = \mathcal{H} \oplus \mathcal{V}\mathcal{E}$$
- $\mathcal{H}$ is homogeneous, that is, $$T_y \mu_r(\mathcal{H}_y) = \mathcal{H}_{ry}$$ for all $y \in E, r \in \mathbb{R},$ where $\mu_r:E \rightarrow E$ is the multiplication map
Let $\tilde{\partial}$ denote the horizontal lift of $\partial/\partial t$ and let $0 \leq t_0 < b.$ I want to show that there is a fixed $\epsilon >0$ that depends only on $t_0$ such that all maximal integral curves of $\tilde{\partial}$ originating in the fixed fiber $E_{t_0}$ are defined at least on $[t_0,e).$
The hint I have been given is that I should endow $E$ with a bundle metric and consider all integral curves originating in the unit sphere in $E_{t_0}$ and then use the second property of an Ehresmann connection, but I do not see how to proceed with this hint. Any further tips, or solutions are welcome.