Cooriented

Cooriented is a term used in differential topology to describe a property of a codimension-1 submanifold S of a smooth manifold M. S is said to be coorientable if its normal bundle νS is orientable, or equivalently if there exists a continuous everywhere nonvanishing normal vector field along S. Such a choice of normal vector field along S is called a coorientation, and it determines a preferred “side” of S within M.

In practical terms, a coorientation provides a global way to distinguish the two sides of a hypersurface. If M is oriented, the orientability of the ambient normal bundle νS is closely tied to the coorientability of S: νS being orientable allows one to fix a consistent normal direction along S, which in turn specifies a coorientation.

A common situation is when S is the boundary of a region in M; the outward normal along the boundary gives a natural coorientation. Conversely, not every hypersurface admits a global normal vector field, and such a hypersurface is not coorientable. A classical example is the Möbius band embedded in R^3, which is non-orientable and not coorientable as a hypersurface.

Coorientation plays a role in various areas, including the theory of integration over submanifolds, the definition of flux in physics, and in foliation theory, where a cooriented foliation has a transverse orientation. It is closely related to the notion of orientation and the concept of the normal bundle.