Oriented manifold

This article defines a differential manifold with the following additional structure -- the structure group is reduced to: positive-determinant general linear group

Definition

An oriented manifold comprises the following data:

• A differential manifold $M$
• An orientation on the tangent space at each point of $M$, in such a way that the orientation varies smoothly with the point.

A differential manifold that can be given the structure of an oriented manifold is termed an orientable manifold.

For a connected orientable manifold, there are only two possible orientations (a given orientation and its negative). Hence, the additional structure of being oriented is not often clearly distinguished from the manifold property of being orientable.

Relation with other structures

Stronger structures

• Oriented Riemannian manifold
• Almost symplectic manifold
• Symplectic manifold
• Almost complex manifold
• Complex manifold