Coorientations

Coorientations are a geometric notion used to describe a consistent choice of a side or a transverse direction for codimension-one objects in differential geometry, including submanifolds and foliations. Roughly speaking, a coorientation amounts to selecting a global normal direction along the object.

Formally, let M be a smooth manifold and S ⊂ M a submanifold of codimension one. A coorientation of S is a global choice of a nonvanishing normal covector field along S, defined up to multiplication by a positive function. Equivalently, it is a nowhere-vanishing section of the normal bundle dual, N_S^*. If such a global section exists, S is called coorientable. This is the same as requiring the normal bundle N_S to be orientable, or, in orientable ambient manifolds, that S is two-sided—there is a neighborhood of S diffeomorphic to S × (-ε, ε) with a well-defined pair of sides.

In foliation theory, a codimension-one foliation F on M is coorientable if there exists a global nowhere-vanishing 1-form ω whose kernel is tangent to the leaves; equivalently, the foliation admits a transverse orientation. When coorientable, the coorientation provides a consistent choice of transverse normal direction to the leaves.

Coorientations are useful in gluing constructions, flux computations, and in the study of boundaries and transverse structures. They are distinct from orienting the submanifold itself: a coorientation concerns the normal directions, while the orientation of S concerns its own manifold orientation.