Horizontal lifts and integral curves

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:

1. $$\mathcal{H}$$ is complementary to the vertical bundle $$TE = \mathcal{H} \oplus \mathcal{V}\mathcal{E}$$
2. $$\mathcal{H}$$ 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.

• My first guess would be: use a local trivialization around $t_0$, so that $E\supset U=I\times \mathbb{R}^m$, where $I$ is an interval containing $t_0$. Since the $t$ component of the lift is non-zero at $t_0$, by continuity it shouldn't vanish in a neighborhood. What is $\partial \delta$? I don't know about the subject, so I apologize for any mistake. – user90189 Nov 22 '18 at 2:15
• @user90189 I managed to solve this exercise using another idea. If you have any thoughts on my recently posted question related to this, please tell me! – Dedalus Nov 24 '18 at 0:28
• Please tag this as exercise and, if you solved it, please post the answer, so we can learn from it and upvote it – magma Nov 24 '18 at 8:59