Positive Compatibility of Charts on the Interior of a Smooth Manifold with Boundary

Let $M$ be a smooth manifold with boundary of dimension $n$, and let $(U,\varphi)$ and $(V,\psi)$ be charts in the chosen smooth atlas. We say that these charts are positively compatible if either

$$U\cap V\cap \operatorname{int}(M)=\varnothing,$$

or else the following condition holds.

For every point

$$x\in \varphi\bigl(U\cap V\cap \operatorname{int}(M)\bigr),$$

consider any local smooth extension

$$F:W\to W'$$

of the transition map $\psi\circ\varphi^{-1}$ at $x$ that is furnished by the smooth-compatibility condition from Smooth Compatibility of Charts Modeled on the Closed Upper Half-Space. Then

$$\det J_F(x)>0,$$

where the Jacobian determinant is the one from Jacobian Determinant of a Differentiable Map Between Euclidean Open Sets.

This condition is independent of the chosen local extension, because $x$ lies in the interior of the half-space chart image and any two such extensions agree on an open neighborhood of $x$ in $\mathbb{R}^n$.