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Let $p:E\to B$ be a locally trivial fibration of connected, non-compact smooth manifolds. Let $U\subset E$ be a connected open subset and $p|_U:U\to p(U)$ has diffeomorphic fibers.

Can we conclude that $p|_U$ is again a locally trivial fibration?

This question is related to my other question I posted before, but I find this general version interesting in its own right.

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    $\begingroup$ It's probably best to accept Idrissi's answer as it did answer your question, at the time. You could then open a new question. Generally speaking its frowned upon to change a question after answers start appearing. Ideally questions are not edited at all. $\endgroup$
    – Ryan Budney
    Commented Jun 30, 2023 at 17:57
  • $\begingroup$ Ryan, if someone can edit his/her wrong answer then a question also can be edited for the discussion to continue in the desired direction. $\endgroup$
    – RKS
    Commented Jul 1, 2023 at 10:01

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As far as I can tell, the answer is no. OP added after I answered the requirement that the fibers have to be connected.

Let $S^1 = \mathbb{R} \cup {\infty}$. The projection on the first coordinate $S^1 \times \mathbb{R} \to S^1$ is a locally trivial fibration of connected non-compact smooth manifolds. But the restriction to the following open subset of $S^1 \times \mathbb{R}$:

enter image description here

has diffeomorphic fibers (they're all diffeomorphic to $(0,1) \cup (2,3)$) but it isn't a locally trivial fibration. Here the green part is glued with the green part, the red with the red.

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    $\begingroup$ @OP. I'm not going to try hitting a moving target. $\endgroup$
    – Najib Idrissi
    Commented Jun 30, 2023 at 11:38

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