Coorientation

For a codimension-one tangent distribution, a coorientation is an orientation of its normal quotient line bundle. An oriented real line bundle is trivial: local positive sections can be combined with a partition of unity without cancellation. Hence a cooriented distribution is the kernel of a global nowhere-vanishing one-form. This choice is independent of the orientation of the ambient manifold, and reversing it does not change the Godbillon-Vey invariant.