Restriction of a Smooth Differential Form to the Boundary

Let $n\in$ $\mathbb{N}$ with $n\ge 2$, let $M$ be a smooth manifold with boundary of dimension $n$ with nonempty boundary $\partial M$, and equip $\partial M$ with the smooth manifold structure of dimension $n-1$ from Boundary of a Smooth Manifold with Boundary as a Smooth Manifold of Dimension n-1 (if $M$ is oriented, one may equally use the atlas of the induced orientation, whose chart maps differ from induced boundary charts at most by composition with the reflection $\rho$ appearing there). Let $k\in\mathbb{N}\cup\{0\}$ and let $\omega=(\omega_\alpha)_{\alpha\in A}$ be a smooth differential $k$-form on $M$, relative to the chosen atlas $((U_\alpha,\varphi_\alpha))_{\alpha\in A}$ of $M$.

Let $(U'\cap\partial M,\psi)$ be a chart of the chosen atlas of $\partial M$, induced as in Boundary of a Smooth Manifold with Boundary as a Smooth Manifold of Dimension n-1 from a chart $(U_\alpha,\varphi_\alpha)$ of $M$ (possibly composed with the reflection $\rho$), and let $\Omega_\psi$ denote its image. Between the Euclidean spaces $\mathbb{R}^{n-1}$ and $\mathbb{R}^n$, let

$$j:\mathbb{R}^{n-1}\to\mathbb{R}^n$$

be the unique affine map satisfying $j(y)=\varphi_\alpha(\psi^{-1}(y))$ for every $y\in\Omega_\psi$; it is the composition of the inverse translation (and, where applicable, the reflection $\rho$) with the inclusion of $\mathbb{R}^{n-1}$ into the boundary hyperplane of the closed upper half-space $H^n$. Let $L$ be the real matrix with $n$ rows and $n-1$ columns whose $r$th column, for $r\in\{1,\dots,n-1\}$, is $j(e_r)-j(0)$, where $e_r$ denotes the $r$th standard basis vector of $\mathbb{R}^{n-1}$.

The restriction of $\omega$ to $\partial M$, denoted $\iota^{*}\omega$ where $\iota:\partial M\to M$ is the inclusion map, is the family whose representative in the chart $(U'\cap\partial M,\psi)$ assigns to each $y\in\Omega_\psi$ the alternating $k$-linear form on $\mathbb{R}^{n-1}$ given by

$$(\iota^{*}\omega)_{\psi,y}(v_1,\dots,v_k)=\omega_{\alpha,\,j(y)}\bigl(L v_1,\dots,L v_k\bigr)$$

for all vectors $v_1,\dots,v_k\in\mathbb{R}^{n-1}$, where $L v_r$ is the matrix-vector product. For $k=0$ the family is given by $(\iota^{*}\omega)_{\psi}(y)=\omega_{\alpha}(j(y))$ for $y\in\Omega_\psi$.