Coorientability

Coorientability is a property of certain geometric objects in differential topology, notably hypersurfaces and foliations, that expresses the existence of a global transverse orientation. It is defined in terms of the normal or conormal bundle: a submanifold or foliation is coorientable if its normal bundle admits a global orientation.

For a submanifold S of codimension 1 (a hypersurface) in a manifold M, coorientability means that the normal line bundle N_S is trivial, i.e., there exists a globally defined nowhere vanishing normal vector field along S. Equivalently, S is two-sided in M. In terms of differential forms, this can be phrased as the existence of a global conormal form α that annihilates the tangent spaces TS, or, on a neighborhood of S, a global 1-form whose kernel along S equals TS. If M is orientable, coorientability of S is compatible with a global orientation of the ambient space and the tangent bundle of S.

For foliations, a foliation F of codimension q is coorientable if the normal bundle TM/F is orientable. Equivalently, there exists a globally defined system of q independent 1-forms α1,…,αq on M whose common kernel is the tangent distribution to F, i.e., F = ker α1 ∩ … ∩ ker αq. In codimension 1 this reduces to the existence of a single global nowhere-vanishing 1-form α with F = ker α.

An obstruction to coorientability is the first Stiefel-Whitney class of the normal bundle; it vanishes if and only if the object is coorientable. Examples include the standard horizontal foliation of R^2, which is coorientable, and the Möbius band embedded in R^3, which is not coorientable.