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Is the following fact known? If yes - what is the reference?

Let $\phi:X\to Y$ be a submersion of smooth manifolds with connected fibers. Let $s_0,s_1:Y\to X$ be its (smooth) sections. Then, for any $y\in Y$, there exists an open neighborhood $U$ of $y$ and a smooth homotopy $h:[0,1] \times U\to X$ such that $h|_{\{0\}\times U}=s_0$, $h|_{\{1\}\times U}=s_1$, and $h|_{\{t\}\times U}$ is a section of $\phi$ for any $t$.

We seem to have a proof, but we prefer to give a reference.

Improve this question
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    $\begingroup$ I would not look for a ref. Instead I would say that the statement can be reduced to the case of projection map $\mathbb R^{n+m}\to \mathbb R^{n}$ where it is obvious. $\endgroup$
    – Anton Petrunin
    Jan 28, 2016 at 14:27
  • $\begingroup$ Yes it should be $X \to Y$ $\endgroup$
    – Rami
    Jan 28, 2016 at 15:52
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    $\begingroup$ The reduction to the linear case is not obvious. Connectivity is not a local property. $\endgroup$
    – Rami
    Jan 28, 2016 at 15:53
  • $\begingroup$ then add "since a given point can be mapped to any other point of connected manifold by a diffeomorphism". $\endgroup$
    – Anton Petrunin
    Jan 28, 2016 at 18:14

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