Induced Orientation on the Boundary of an Oriented Smooth Manifold with Boundary

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 $\partial M$. 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, built from the induced boundary charts arising from charts of the chosen oriented atlas.

Let $\rho:\mathbb{R}^{n-1}\to\mathbb{R}^{n-1}$ denote the reflection of Euclidean space given by

$$\rho(y_1,y_2,\dots,y_{n-1})=(-y_1,y_2,\dots,y_{n-1}).$$

When $n$ is odd we have $n-1\ge 2$, so $\rho$ leaves the last coordinate unchanged and maps the region of points of $H^{n-1}$ with positive last coordinate into itself; hence composing a chart map with $\rho$ again yields a chart.

The induced orientation on $\partial M$ is the orientation of $\partial M$, in the sense of Orientable and Oriented Smooth Manifold with Boundary, given by the following oriented smooth atlas, where even and odd refer to the parity of $n$.

1. If $n$ is even: the atlas of all induced boundary charts $(U'\cap\partial M,\psi)$ from Boundary of a Smooth Manifold with Boundary as a Smooth Manifold of Dimension n-1 arising from charts of the chosen oriented atlas of $M$. This is an oriented smooth atlas by Transition Maps of Induced Boundary Charts of an Oriented Atlas are Orientation-Preserving.

2. If $n$ is odd: the atlas of all charts $(U'\cap\partial M,\rho\circ\psi)$, where $(U'\cap\partial M,\psi)$ ranges over the induced boundary charts as in case 1. This is an oriented smooth atlas because each transition map of this atlas is the conjugate by $\rho$ of the corresponding transition map of the atlas in case 1, and conjugation by $\rho$ does not change the Jacobian determinant of the transition, which is positive by Transition Maps of Induced Boundary Charts of an Oriented Atlas are Orientation-Preserving.

This sign convention is chosen so that the Stokes identity holds without an extraneous sign; it corresponds to the outward-normal-first convention for orienting the boundary.