moving frame
Let $M$ be a smooth manifold^{}. A moving frame^{} (sometimes just a frame) on $M$ is a choice, for every $P\in M$, of a basis for the tangent space ${T}_{p}M$ to $M$ at $P$. More formally (and abstractly), a frame is a (smooth) section^{} of the principal bundle for ${\mathrm{GL}}_{n}$ over $M$.
Examples and Remarks

•
If $M={\mathbb{R}}^{n}$, then any basis of ${\mathbb{R}}^{n}$ trivially gives a frame as well.

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

•
Similar to the previous example, one can show that the 2sphere 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.
References
 1 Wikipedia’s http://en.wikipedia.org/wiki/Moving_frameentry on moving frame
Title  moving frame 

Canonical name  MovingFrame 
Date of creation  20130322 16:27:01 
Last modified on  20130322 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 