Setup: Let $B$ be a $C^r$ $n$-manifold ($r \geq 1$) and $M$ a closed $k$-dimensional $C^r$ submanifold of $B$. Assume there exists a smooth retraction $p:B \to M$ which is also a submersion, so that $p:B\to M$ is a $C^r$ fibered manifold. Assume that there exists an increasing sequence of open neighborhoods $M \subset U_1 \subset U_2 \subset U_3 \subset \ldots $ such that:
- $p|_{U_i}:U_i\to M$ is a $C^r$ fiber bundle with fiber $D^{n-k}$, the open ball of radius $1$ centered at the origin in $\mathbb{R}^{n-k}$.
- For any $i,j \in \mathbb{N}$, there exists a fiber-preserving $C^r$ diffeomorphism $h_{ij}:U_i\to U_j$ (i.e., $h_{ij}$ is a $C^r$ diffeomorphism and $p_j\circ h_{ij} = p_i$).
Question: Is $p:B\to M$ also a $C^r$ fiber bundle with fiber $D^{n-k}$? Furthermore, for any $i \geq 1$, is there a $C^r$ fiber-preserving diffeomorphism from $U_i$ to $B$ (a $C^r$ diffeomorphism $h_i:U_i\to B$ such that $p\circ h = p_i$)?
Motivation: It is true that the increasing union of $C^r$ disks is $C^r$-diffeomorphic to $\mathbb{R}^{n-k}$. The smooth case of this is proved in, e.g., Milnor's Chapter 6 of Lectures in Modern Mathematics, Vol. II by T. L. Saaty. The Monotone Union of Open n-Cells is an Open n-Cell by M. Brown contains a proof of the continuous case. My question is asking about a possible generalization of this result.
Note: I am primarily interested in answers for the $C^r$ case with $r\geq 1$, but I would also be interested in answers for the $C^0$ case.
