On the definition of smooth fiber bundle and smooth manifolds with boundary

Question

On page 268 of Prof. John M Lee's book "Introduction to Smooth Manifolds" (second edition), it says if $E$, $M$ and $F$ are smooth manifolds with or without boundary, $\pi:E\to M$ is a smooth map and all the local trivializations $\Phi:\pi^{-1}(U)\to U\times F$ are diffeomorphisms, where $U$ is an open subset of $M$, then $\pi:E\to M$ is called a smooth fiber bundle.

My questions are

1. is there a problem if some or all of the smooth manifolds $E$, $M$ and $F$ have nonempty boundary? For example, if $E$ has empty boundary but $M$ and $F$ have nonempty boundary, and you happen to take $U$ in above to be an open submanifold with boundary, then $U\times F$ would be a smooth manifold with corner. But $\pi^{-1}(U)\subseteq E$ should not have a boundary?

2. From this post we know that Ehresmann fibration theorem, which states if "$E$ and $M$ are smooth manifolds without boundary and $f:E\to M$ is a smooth surjective submersion which is proper, then it is a smooth fiber bundle", should work when $E$ and $M$ are smooth manifolds with boundary by adding the extra assumption that the restriction $f|_{\partial M}:\partial M\to\partial N$ is a (proper?) submersion. If it works, am I correct that in this case the fiber is still a smooth manifold without boundary?

3. I would like to know which case is impossible and what extra assumptions need to be imposed. For example, if all the smooth manifolds $E$, $M$ and $F$ have empty boundary, the definition of a smooth fiber bundle is fine; while if all the smooth manifolds $E$, $M$ and $F$ has nonempty boundary, it does not seem fine to me unless more assumption(s) is added.

4. Besides Prof. Lee's book and Prof. Liviu I Nicolaescu's book "Lectures on the Geometry of Manifolds" (third edition) (in Definition 3.4.50, for which $E$ is a smooth manifold with boundary and $M$ is a smooth manifold with empty boundary) I have not seen a definition of smooth fiber bundle where some or all of the spaces are smooth manifolds with boundary. Is there a reference (paper, monograph, or even textbooks) which has such a definition.

Thanks in advance.

Answer 1

Allowing boundaries makes no big difference. Figure out, for example, the simple case of the first projection $\pi:D^2\times I\to D^2$ where $F=I$ is the compact interval and $M=D^2$ is the compact $2$-disk; the general case is not much more complicated.

First, there is nothing to change in the definition of a fibre bundle in the case where $M$, $E$, $F$ have boundaries. You should still take $U$ to be an open subset of $M$ in the sense of general topology, so it can meet $\partial M$, but this makes no difference.

Second, one has

1. $\partial M=\partial F=\emptyset$ implies $\partial E=\emptyset$;

2. $\partial M\neq\emptyset$ and $\partial F=\emptyset$ implies that $E$ has a smooth boundary $\partial E=\pi^{-1}(\partial M)$, which is "vertical" i.e. the total space of a locally trivial fibre bundle of fibre $F$ over $\partial M$;

3. $\partial M=\emptyset$ and $\partial F\neq\emptyset$ implies that $E$ has a smooth boundary $\partial E$ which is "horizontal" i.e. the total space of a locally trivial fibre bundle of fibre $\partial F$ over $M$;

4. $\partial M\neq\emptyset$ and $\partial F\neq\emptyset$ implies that $E$ has a boundary $\partial E$ which is the union of the two precedent ones, and they intersect on the corners, which form the total space of a locally trivial fibre bundle of fibre $\partial F$ over $\partial M$.

Third, Eheresmann's theorem:

a) it goes without change in the case $\partial E=\pi^{-1}(\partial M)$.

b) If $M$ has no boundary, it takes the form "Assume $E$ is a smooth manifold with smooth boundary and $M$ is a smooth manifold without boundary and $\pi:→$ is a smooth surjective submersion which is proper, also assume that $\pi\vert\partial E:\partial E\to M$ also is a submersion, then $\pi$ is a smooth fiber bundle".

c) If $M$ has a boundary and $\partial E$ is more than $\pi^{-1}(\partial M)$, it's a little more subtle, you have to control what happens close to the corners of $E$ by assuming that the corners $\partial^2E$ of $E$ split $\partial E$ into two components $\partial_vE=\pi^{-1}(\partial M)$ and $\partial_hE$ such that $\pi\vert\partial_hE$ is a submersion onto $M$, and that $\pi\vert\partial^2E$ is a submersion onto $\partial M$.