For the following we will work in the smooth category. (But examples in the topological category is also welcome.)
The usual definition of a fibre bundle is
Def A fibre bundle is the quadruple $(E,B,\pi,F)$ where $E,B,F$ are differentiable manifolds, $\pi: E\to B$ is a surjection which is locally trivial. Locally trivial means that for every $e\in E$, there exists an open neighbourhood $B\supset U\ni \pi(e)$ such that $\pi^{-1}(U)$ is diffeomorphic to $U\times F$.
One also encounters the following definition of a fibred manifold in the literature
Def A fibred manifold is the triple $(E,B,\pi)$ where $E,B$ are differentiable manifolds and $\pi$ is a surjection from $E\to B$ with maximal rank such that the dimension of $E$ at $e\in E$ is greater than or equal to the dimension of $B$ at $\pi(e) \in B$.
One also would encounter the remark that a fibre bundle is in particular an example of a fibred manifold.
Questions
- I hope I am not mistaken in thinking that the definition of fibred manifold, with $\pi$ being surjective, maximal rank, and with the dimension condition, is equivalent to $\pi$ being a surjective submersion of $E\to B$. Is that correct or is there a counterexample (going either way)?
The more important question is, what does locally trivial actually do for us?
a. There is a trivial example of fibred manifold that is not a fibre bundle if we allow our differential manifolds to be disconnected with connected components of different dimensions. Then the local fibres $\pi^{-1}(b_1)$ and $\pi^{-1}(b_2)$ for $b_1\neq b_2\in B$ may have different dimensions and cannot be the same manifold $F$.
b. We also have Ehresmann's fibration theorem, which states that $f:M\to N$ smooth, surjective, submersive, and proper implies that $(M,N,f)$ is a locally trivial fibration.
In particular, I am looking to gain some intuition on what fibred manifolds that aren't fibre bundles can look like, aside from case (a) above. It would also be nice to see a non-proper counterexample to Ehresmann's theorem.