Orientation (mathematics)

:See also Orientation (rigid body).

Orientation on manifolds

One can also discuss orientation on manifolds. Each point p on an n-dimensional differentiable manifold has a tangent space TpM which is an n-dimensional real vector space. One can assign to each of these vector spaces an orientation. However, one would like to know whether or not one can choose the orientations so that they "vary smoothly" from point to point. One may not be able do this, there are certain topological restrictions. A manifold which admits a smooth choice of orientations for its tangents spaces is said to be orientable. See the article on orientability for more on orientations of manifolds.

Related Topics:
Manifold - Tangent space - Topological - Orientability

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