coorientable

Coorientable is a term used in differential geometry and topology to describe a relation between a submanifold and its ambient space. A submanifold N of a manifold M is coorientable if its normal bundle ν(N) is orientable. In particular, when N has codimension one, ν(N) is a real line bundle, and coorientability means ν(N) is orientable, or equivalently that there exists a global nowhere-vanishing normal vector field along N.

For hypersurfaces (codimension one submanifolds), coorientability is often characterized by the existence of a global defining function. There exists an open neighborhood U of N in M and a smooth function f: U → R such that N = U ∩ f^{-1}(0) and df is nowhere zero on N. If the ambient manifold M is orientable, coorientability of N is also equivalent to N being two-sided, meaning there is a globally defined unit normal vector field along N.

Examples and related notions: In Euclidean space, a smooth embedded hypersurface is coorientable exactly when it is two-sided. In foliations, a foliation of codimension q is coorientable if its normal bundle is orientable; equivalently there exists a transverse orientation, or a globally defined family of q independent 1-forms whose common kernel is the tangent distribution of the foliation. In contact geometry, a hyperplane field ξ is coorientable if there exists a global 1-form α with ξ = ker α.

See also: orientation, normal bundle, hypersurface, foliation, transverse orientation.