Let $M$ be a \reftext{def:smooth-manifold-with-boundary-2026a}{smooth manifold with boundary}. We say that $M$ is orientable if it admits an oriented smooth atlas in the sense of \ref{def:oriented-smooth-atlas-manifold-boundary-2026a}.

An oriented smooth manifold with boundary is a smooth manifold with boundary together with a chosen oriented smooth atlas. Such a chosen oriented smooth atlas is called an orientation of $M$.