nLab pseudoorientation

Contents

Contents

Idea

A pseudoorientation of a submanifold UXU \subset X is an orientation of how the ambient manifold XX 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 UU may be visualized with a set of arrows lying within UU. A pseudoorientation is properly visualized instead with a set of arrows passing through UU, or circulating around it.

A pseudoorientation is sometimes also called a transverse orientation, and the term is also spelled pseudo-orientation or pseudorientation.

Definition

Definition

For a submanifold UU of a manifold XX, a pseudoorientation of UU is a map that, for each point aa on UU, takes a local orientation of XX at aa to a local orientation of UU at aa, continuously in aa and taking opposite orientations to opposite orientations.

An equivalent, more explicitly geometrical definition justifies the synonym transverse orientation:

Definition

For a submanifold UU of a manifold XX, a pseudoorientation of UU is an orientation of the vector bundle T U(X)/T U(U)T_U(X) / T_U(U), a bundle on UU whose fiber at each xUx \in U is the quotient T x(X)/T x(U)T_x(X) / T_x(U) of the tangent space to XX at xx by the tangent space to UU at xx.

Properties

Relation to orientations

When one has an orientation of the containing manifold XX, that gives a correspondence between orientations and pseudoorientations for any submanifold UU. As a result, many treatments conflate orientations with pseudoorientations.

Integration

A pseudoorientation is precisely the added structure one needs in order to integrate over UU a kk-pseudoform on XX. This is much like how if one has instead an (untwisted) kk-form on XX and wants to integrate it over UU, one needs an orientation of UU. For details, see integration of differential forms.

Examples

Canonical pseudoorientation for a top-dimension submanifold

When k=nk = n, a local orientation of XX is the same thing as a local orientation of UU. This correspondence makes a canonical pseudoorientation for UU.

This canonical pseudoorientation is what makes the integration of nn-pseudoforms the most fundamental kind of integration on a manifold, as discussed at integration of differential forms.

References

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.