Transition Maps of Induced Boundary Charts of an Oriented Atlas are Orientation-Preserving

Let $n\in$ $\mathbb{N}$ with $n\ge 2$, and let $M$ be an oriented smooth manifold with boundary of dimension $n$ with nonempty boundary $\partial M$, the orientation being given by the chosen oriented smooth atlas. Then the following hold.

1. Let $(U,\varphi)$ and $(V,\psi)$ be charts of the oriented atlas with $U\cap V\ne\varnothing$, let $a\in\varphi(U\cap V)$, and let $F$ be any local smooth extension of the transition map $\psi\circ\varphi^{-1}$ at $a$ furnished by Smooth Compatibility of Charts Modeled on the Closed Upper Half-Space. Then the determinant of the Jacobian matrix $DF(a)$ satisfies $\det DF(a)>0$. (For $a$ in the image of the interior of $M$ this is the positive compatibility required of the oriented atlas; the content of this claim is that positivity extends to all points of $\varphi(U\cap V)$, including boundary points.)

2. In the setting of claim 1, suppose additionally that $a$ lies in the boundary hyperplane of the closed upper half-space $H^n$, and write $F=(F_1,\dots,F_n)$ in coordinates. Then the partial derivatives of the last coordinate function satisfy

$$\frac{\partial F_n}{\partial x_j}(a)=0 \quad\text{for } j\in\{1,\dots,n-1\},\qquad \frac{\partial F_n}{\partial x_n}(a)>0.$$

3. Let $\psi_1$ and $\psi_2$ be induced boundary charts of $\partial M$ in the sense of Boundary of a Smooth Manifold with Boundary as a Smooth Manifold of Dimension n-1, arising from charts of the oriented atlas, with overlapping domains. Then the transition map $\psi_2\circ\psi_1^{-1}$ between the corresponding images is a smooth diffeomorphism of admissible domains in Euclidean space $\mathbb{R}^{n-1}$, and it is orientation-preserving in the sense of that definition; that is, the determinant of its Jacobian matrix is positive at every point of its domain.