nLab pseudoorientation

Idea

A pseudoorientation of a submanifold $U \subset X$ is an orientation of how the ambient manifold $X$ 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 $U$ may be visualized with a set of arrows lying within $U$. A pseudoorientation is properly visualized instead with a set of arrows passing through $U$, 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 $U$ of a manifold $X$, a pseudoorientation of $U$ is a map that, for each point $a$ on $U$, takes a local orientation of $X$ at $a$ to a local orientation of $U$ at $a$, continuously in $a$ and taking opposite orientations to opposite orientations.

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

Definition

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

Properties

Relation to orientations

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

Integration

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

Examples

Canonical pseudoorientation for a top-dimension submanifold

When $k = n$, a local orientation of $X$ is the same thing as a local orientation of $U$. This correspondence makes a canonical pseudoorientation for $U$.

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

Related concepts

• An absolute differential form can be integrated on an entirely unoriented submanifold, with neither an orientation nor a pseudoorientation.

References

Many useful explanations by Toby Bartels and John Baez in this long Usenet thread.