A pseudoorientation of a submanifold is an orientation of how the ambient manifold lies around the submanifold.
This contrasts with an orientation of the submanifold itself: neither a pseudoorientation nor an orientation for a submanifold necessarily provides the other, unless one has an orientation for the ambient manifold.
An orientation of may be visualized with a set of arrows lying within . A pseudoorientation is properly visualized instead with a set of arrows passing through , or circulating around it.
A pseudoorientation is sometimes also called a transverse orientation, and the term is also spelled pseudo-orientation or pseudorientation.
For a submanifold of a manifold , a pseudoorientation of is a map that, for each point on , takes a local orientation of at to a local orientation of at , continuously in and taking opposite orientations to opposite orientations.
An equivalent, more explicitly geometrical definition justifies the synonym transverse orientation:
For a submanifold of a manifold , a pseudoorientation of is an orientation of the vector bundle , a bundle on whose fiber at each is the quotient of the tangent space to at by the tangent space to at .
When one has an orientation of the containing manifold , that gives a correspondence between orientations and pseudoorientations for any submanifold . As a result, many treatments conflate orientations with pseudoorientations.
A pseudoorientation is precisely the added structure one needs in order to integrate over a -pseudoform on . This is much like how if one has instead an (untwisted) -form on and wants to integrate it over , one needs an orientation of . For details, see integration of differential forms.
When , a local orientation of is the same thing as a local orientation of . This correspondence makes a canonical pseudoorientation for .
This canonical pseudoorientation is what makes the integration of -pseudoforms the most fundamental kind of integration on a manifold, as discussed at integration of differential forms.
Many useful explanations by Toby Bartels and John Baez in this long Usenet thread.
Last revised on May 15, 2023 at 05:50:14. See the history of this page for a list of all contributions to it.