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](https://mathoverflow.net/questions/201670/ehresmann-fibration-theorem-for-manifolds-with-boundary) 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.



On the definition of smooth fiber bundle and smooth manifolds with boundary
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 $\partial M$.


