> [!definition] > > Let $X$ be a set, and $E$ be a [[Banach Space|Banach space]]. A **boundary chart** is a [[Bounded Linear Functional|functional]] $\mu \in E^*$ and a bijection $\varphi: U \to E_\mu^+$. Two boundary [[Atlas|charts]] $\varphi: U \to E_\mu^+$ and $\psi: V \to F_\nu^+$ are **compatible** if the composition $\varphi \circ \psi^{-1}$ is a $C^p$-[[Boundary Derivative|isomorphism]]. An **atlas** is a collection of boundary charts that are mutually compatible. A [[Maximal Atlas|maximal atlas]] determines the manifold structure of $X$, making it a **manifold with boundary**. > [!definition] > > For any $p \in X$, if there exists a chart $\varphi: U \to E_\mu^+$ such that $\varphi(p) \in E_\mu^{0}$, then $p \in \partial X$ is a **boundary point** of $X$. If $p \in \partial X$ is a boundary point, then for any chart $\psi: U \to F_\nu^+$, $\psi(p) \in F_\nu^0$ by the [[Boundary Derivative|invariance of boundary]]. > > For any boundary chart $\varphi: U \to E_\mu^+$, $\varphi|_{\partial X}: \partial U \to E_\mu^0$ is a slice chart for $\partial X$, making the boundary a [[Submanifold|submanifold]]. > [!theorem] > > Let $X$ be a manifold with boundary and $p \in \partial X$. A tangent vector $p \in T_pX$ is **inward-pointing** if there exists $\eps > 0$ and a $C^1$ [[Curve|curve]] $\gamma: [0, \eps) \to X$ such that $\gamma(0) = p$ and $\gamma'(0) = v$, and **outward-pointing** if there exists a $C^1$ curve $\gamma: (-\eps, 0] \to X$ such that $\gamma (0) = p$ and $\gamma'(0) = v$. > > A tangent vector is both inward and outward pointing if and only if it is in the tangent space of $T_p\partial X$.