orientability

Orientability is a property of a manifold (a space that locally resembles Euclidean space) that asks whether a consistent choice of orientation for tangent spaces can be made across the whole space. Equivalently, a manifold is orientable if there exists a nowhere-vanishing top-degree differential form on it, or if all coordinate transition maps have positive determinant. In short, an orientable manifold has a global notion of “handedness” or orientation.

Examples: The sphere S^2 and the torus T^2 are orientable, meaning one can consistently distinguish clockwise from counterclockwise directions everywhere on them. The Möbius strip, the Klein bottle, and the real projective plane RP^2 are non-orientable: following a path around certain loops reverses the chosen orientation.

Related concepts: A manifold is orientable if and only if its tangent bundle admits a global orientation, equivalently its first Stiefel-Whitney class vanishes. Nonorientable manifolds admit an orientation double cover that is orientable. An orientation on a manifold corresponds to a connected component of the frame bundle, or to a choice of a volume form up to positive scaling.

Notes: For spaces with boundary or in more advanced contexts, orientability can be defined analogously; orientability is a fundamental notion in geometry and topology with consequences for integration, homology, and the definition of certain invariants.