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.