Orientation (mathematics)

:See also Orientation (rigid body).

Alternate viewpoints

We present two alternate (and more abstract) ways of understanding orientations:

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1. For any real vector space V we can form the kth-exterior power of V, denoted ΛkV. This is a real vector space of dimension n-choose-k. The vector space ΛnV (called the top exterior power) therefore has dimension 1. That is, ΛnV is just a real line. There is no a priori choice of which direction on this line is positive. An orientation is just such a choice. Any nonzero element ω of ΛnV determines an orientation of V by declaring ω to be in the positive direction. To connect with the basis point of view we say that the positively oriented bases are those on which ω evaluates to a positive number (since ω is a n-form we can evaluate it on a ordered set of n vectors, giving an element of R). The form ω is called an orientation form. If {ei} is a privileged basis for V then the orientation form giving the standard orientation is e1∧e2∧…∧en.

Related Topics:
Exterior power - ''n''-choose-''k''

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2. Let B be the set of all ordered bases for V. Then the general linear group GL(V) acts freely and transitively on B. (In fancy language, B is a GL(V)-torsor). This means that as a manifold, B is (noncanonically) homeomorphic to GL(V). Note that the group GL(V) is not connected, but rather has two connected components according to whether the determinant of the transformation is positive or negative. The identity component of GL(V) is denoted GL+(V) and consists of those transformations with positive determinant. The action of GL+(V) on B is not transitive: there are two orbits which correspond to the connected components of B. These orbits are precisely the equivalence classes referred to above. Since B does not have a distinguished element (i.e. a privileged basis) there is no natural choice of which component is positive. Contrast this with GL(V) which does have a privileged component: the component of the identity. A specific choice of homeomorphism between B and GL(V) is equivalent to a choice of a privileged basis and therefore determines an orientation.

Related Topics:
General linear group - Acts - Torsor - Manifold - Homeomorphic - Connected - Connected component - Identity component

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