# 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 $\operatorname{GL}_{n}$ 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 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.

## References

• 1 Wikipedia’s http://en.wikipedia.org/wiki/Moving_frameentry on moving frame
Title moving frame MovingFrame 2013-03-22 16:27:01 2013-03-22 16:27:01 mathcam (2727) mathcam (2727) 6 mathcam (2727) Definition msc 53A04 frame TNBFrame frame orthonormal frame parallelizable