Orientation (vector Space) - Orientation On Manifolds

Orientation On Manifolds

One can also discuss orientation on manifolds. Each point p on an n-dimensional differentiable manifold has a tangent space T_pM 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 it is possible to choose the orientations so that they "vary smoothly" from point to point. Due to certain topological restrictions, there are situations when this is impossible. 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.