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

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If $M=\\mathbb{R}^{n}$ , then any basis of $\\mathbb{R}^{n}$ trivially gives a frame as well.
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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 $\\mathbb{R}^{2}$ .
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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.
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A key example of a frame is the Frenet frame.
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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.
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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

单词	MovingFrame
释义	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
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