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