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

• A more interesting example (and perhaps a source for the definition) is when $M={ℝ}^2-\{(0,0)\},$ 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 ${ℝ}^2$.

• 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.