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moving frame (Definition)

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_pM$ 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.

Bibliography

1
Wikipedia's entry on moving frame



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"moving frame" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: Frenet frame

Other names:  frame
Also defines:  frame, orthonormal frame, parallelizable
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Cross-references: inner product, orthogonal, places, similar, point, vectors, source, principal bundle, section, smooth, tangent space, basis, smooth manifold
There are 10 references to this entry.

This is version 3 of moving frame, born on 2006-12-08, modified 2007-01-11.
Object id is 8606, canonical name is MovingFrame2.
Accessed 2122 times total.

Classification:
AMS MSC: 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)
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Not a definition either (moving frame) by CWoo on 2006-12-11 13:11:07
Thanks Lando47 for offering me this entry. But since I am not familiar with this, I have to decline your offer.

But what I don't understand is that you have accepted my correction notice nevertheless. You have made some changes in the wording of your entry, but the concern I have with the entry is still present. After reading this entry, I am no where close to understanding what a moving frame is. What is a "dimensional space"? What do you mean by moving a vector from one point to another with "very simple calculations"? The entry has a type of "definition", but I don't see where a moving frame is clearly defined. Can you please define this rigorously?
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