# Oriented manifold

From Diffgeom

*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
- An
*orientation*on the tangent space at each point of , 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*.