moving frame

Let M be a smooth manifoldMathworldPlanetmath. A moving frameMathworldPlanetmathPlanetmath (sometimes just a frame) on M is a choice, for every PM, of a basis for the tangent space TpM to M at P. More formally (and abstractly), a frame is a (smooth) sectionPlanetmathPlanetmath of the principal bundle for GLn over M.

Examples and Remarks

  • If M=n, then any basis of n trivially gives a frame as well.

  • A more interesting example (and perhaps a source for the definition) is when M=2-{(0,0)}, and we take the vectors r and θ at a point (r,θ). Note that this frame cannot be extended to a smooth frame on all of 2.

  • Similar to the previous example, one can show that the 2-sphere admits no frames. A manifold which admits a (global) frame is called parallelizable.

  • 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 on moving frame
Title moving frame
Canonical name MovingFrame
Date of creation 2013-03-22 16:27:01
Last modified on 2013-03-22 16:27:01
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Definition
Classification msc 53A04
Synonym frame
Related topic TNBFrame
Defines frame
Defines orthonormal frame
Defines parallelizable