Let $M$ be a \reftext{def:smooth-manifold-with-boundary-2026a}{smooth manifold with boundary}. An oriented smooth atlas on $M$ is a smooth atlas

such that for every $\alpha,\beta\in A$, the charts $(U_\alpha,\varphi_\alpha)$ and $(U_\beta,\varphi_\beta)$ are positively compatible on the interior of $M$ in the sense of \ref{def:positive-compatibility-charts-interior-manifold-boundary-2026a}.