Let be a smooth manifold. A moving frame (sometimes just a frame) on is a choice, for every , of a basis for the tangent space to at . More formally (and abstractly), a frame is a (smooth) section of the principal bundle for over .
Examples and Remarks
If , then any basis of trivially gives a frame as well.
A more interesting example (and perhaps a source for the definition) is when and we take the vectors and at a point . Note that this frame cannot be extended to a smooth frame on all of .
A key example of a frame is the Frenet frame.
One places adjective in front of ”moving frame” if that adjective pertains to each basis, e.g. an orthogonal frame is a frame for which each basis is orthogonal (with respect to a given inner product). Given any frame, one can always ”orthonormalize” it in a smooth manner to provide an orthonormal frame.
Frames arise in general relativity as a formalization of the observation that there is no “preferred” observer standpoint.
- 1 Wikipedia’s http://en.wikipedia.org/wiki/Moving_frameentry on moving frame
|Date of creation||2013-03-22 16:27:01|
|Last modified on||2013-03-22 16:27:01|
|Last modified by||mathcam (2727)|