Manifolds with boundary.

We denote by $\mathbb{R}^n_+$ the half-space

$$\mathbb{R}^n_+ = \{ (x^1, \dots, x^n) \mid x^1 > 0\}.$$

We define the boundary of $\mathbb{R}^n_+$ to be

$$\partial \mathbb{R}^n_+ = \{ (x^1, \dots, x^n) \mid x^1 = 0\}.$$

and we identify it with $\mathbb{R}^{n-1}$. Recall that a set $U \subset \mathbb{R}^n_+$ is open if it is of the form $U = V \cap \mathbb{R}^n_+$ where $V$ is open in $\mathbb{R}^n$. If $U$ is open in $\mathbb{R}^n_+$ we say that $f \colon U \to \mathbb{R}$ is smooth if there is an open set $V \subset \mathbb{R}^n$ with $U = V \cap \mathbb{R}^n_+$ and a smooth map $F \colon V \to \mathbb{R}$ such that $F(x) = f(x)$ for all $x \in U$. If $V \subset \mathbb{R}^n$ we define $\partial V = V \cap \partial \mathbb{R}^n_+$.

Let $M$ be a set with a subset denoted by $\partial M$ that we call the boundary of $M$. We say that $(U, \psi)$ is a co-ordinate chart on $M$ if it is a co-ordinate chart as defined before but in addition $\psi(\partial U ) \subset \partial \psi(U)$, $\psi(\partial U )$ is open in $\partial \psi(U)$,and $\psi_{\vert\partial U } \colon \partial U \to \partial \psi(U)$ is a bijection. We define compatibility of charts in the usual way but with the extended notion of smoothness above. Once we have this we can define the idea of an atlas and the notion of a manifold $M$ with boundary $\partial M$. Notice that if we discard the boundary points $\partial M$ we immediately see that $M - \partial M$ is a manifold. Similarly $\partial M$ is a manifold. We can extend everything we have done so far to the case of manifolds with boundary.