While learning the basics about smooth manifolds with boundary in this semesters' course about analysis on manifolds, there's a seemingly basic property I didn't find anywhere.
Namely I want to classify which type of mapping $f$ between a smooth manifold with boundary $M$ and $N = f(M)$ passes on the structure of $M$ onto $N$. Hence, for which family of mappings $f: M \rightarrow N$ with $f(M)=N$ is $N$ still a manifold with boundary.
My naive approach was to start with $dim(M) = n = dim(N)$ and an already known smooth manifold structure on $N$ (nothing about its boundary yet).
Taking the $n$-ball $B^n:=\{x \in \mathbb{R}^n : ||x||_2 \leq 1 \}$ as an example for a smooth manifold with boundary, one can show that an $n$-ellispoid also is a smooth manifold with boundary. While this can be achieved using regular value sets, I would be interested in said deformations of manifolds with boundary.
Given $a_i \in \mathbb{R}$ ($1\leq i \leq n$) with $a_i > 0$ we can deform the ball to an ellipsoid using $$f(x) := \sum_{i=1}^n x_i/a_i^2 \ \ \ \ \Rightarrow \ \ \ \ \mathcal{E}^n = f(B^n)$$ I'd be very interested to learn about structure preserving mappings between manifolds with boundary. My intuition tells me that said mappings should exist, but I'm still very new to the field and sometimes find myself having trouble being precise enough. Any thoughts on the topic would be much appreciated!