Orientability

:This article discusses orientability and orientation on surfaces and manifolds. For orientation of vector spaces see orientation (mathematics). For alternate uses, see orientation.

Orientation by top-dimensional forms

Another way of thinking about orientability is thinking of it as a choice of "right handedness" vs. "left handedness" at each point in the manifold.

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Formally, a n-dimensional differentiable manifold is called orientable if it possesses a differential form omega of degree n which is nonzero at every point on the manifold. Conversely, given such a form omega, we say that the manifold is oriented by omega.

Related Topics:
Manifold - Differential form

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The crucial point to observe here is that such a differential form gives a choice of "right handed" basis at each point. A traveler in an orientable manifold will never change his/her handedness by going on a round trip.

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