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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.

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  • $\begingroup$ 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. $\endgroup$ – user90189 Nov 22 '18 at 2:15
  • $\begingroup$ @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! $\endgroup$ – Dedalus Nov 24 '18 at 0:28
  • $\begingroup$ Please tag this as exercise and, if you solved it, please post the answer, so we can learn from it and upvote it $\endgroup$ – magma Nov 24 '18 at 8:59

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