I have some questions about differentiable manifolds with boundaries.
1- First of all, do you have good references of books about manifolds with boundaries ? I'm not really familiar with these objects, so it might be good for me to have a benchmark.
2- I read that the notions of differentiable map, tangent space, tangent map, vector field, etc., are not really different in the case of a manifold with boundary (compare to a classical differentiable manifold). However, I have the feeling that things are not always similar. For example, consider the map $f: [0,1] \rightarrow \mathbb{R}, x \mapsto 2x$.
2.a- Is $f$ differentiable at $0$ (or at $1$) and if yes, what is the tangent map of $f$ at $0$ (or at $1$) ?
2.b- A useful theorem for me is the following one:
"Let $M$ be a differentiable manifold (without boundary), $m \in M$ and $\phi: M \rightarrow \mathbb{R}$ a differentiable map. If $m$ is a local extremum of $\phi$, then $d\phi(m) = 0$." (where $d\phi(m)$ is the tangent map of $\phi$ at $m$).
The problem is that, with the manifold (with boundary) $[0,1]$, we obtain that $\mathrm{arg}\mathrm{max}_{x \in [0,1]} f(x) = \{1\}$, but that $df(1) \neq 0$ (I guess that for every $x \in [0,1]$ and every $y \in \mathbb{R}$, $df(x)(y) = f(y)$, but I maybe make a mistake), so this theorem does not work with manifolds with boundary...
2.c- In view of question 2.b: to which extend do we have similarity between manifolds without boundary and manifolds with boundary ?
3- A particular case where I need to be sure that we don't have so much differences between manifolds without boundary and manifolds with boundary is the following one: I have a differentiable manifold $\Sigma$ with boundary and I need to consider a vector field $X: \Sigma \rightarrow T\Sigma$. The aim for me is to study the ODE induced by $X$ and the eventual dynamical system that this ODE may induces and I am not really confident to go into this problem, due to the fact that my manifold has a boundary (and for the reasons explain before)...
Thank you for your help !
The main difference is the structure of a local chart for a boundary point. Let $M$ be a smooth manifold with boundary, and let $p\in \partial M$. Then the local chart around $p$ is some open set $U$ and a homeomorphism $\phi :U \rightarrow \mathbb{R}^{n}$ such that $\phi(U \cap \partial M)\subset \{x\in \mathbb{R}^{n} | x_n = 0 \}$. Here, $x_n$ is the n-th coordinate of the point $x$. If two of these charts overlap, then the transition functions have to be local diffeomorphism and respect this hyperplane condition. So, if $(U,\phi)$ and $(V,\psi)$ are two overlaping charts, then $\phi\circ \psi^{-1}$ is a diffeomorphism. This means that it is the restriction of a diffeomorphism on an open set containing $\psi(U\cap V)$ to an open set containing $\phi(U\cap V)$. Further, it has to respect the hyperplane condition: $$ \phi\circ \psi^{-1} |_{\psi(U \cap \partial M)}:{\psi(U \cap V \cap \partial M)}\rightarrow \phi(U \cap V \cap \partial M))$$
That should answer 2 after unpacking. For 1, I don't have a good book that deals just with manifolds with boundary, but most texts on smooth manifolds deals with the boundary case at some point. Id recommend Hirsch, Guillemin/Pollack, or Lee's. All say something about the boundary case (if my memory is correct).