moving frameMovingFrame22006-12-08 17:13:192007-01-11 23:50:33Definitionframeorthonormal frameparallelizable% this is the default PlanetMath preamble. as your knowledge
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Let $M$ be a smooth manifold. A \emph{moving frame} (sometimes just a \emph{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 $M$.
\subsection*{Examples and Remarks}
\begin{itemize}
\item If $M=\mathbb{R}^n$, then any basis of $\mathbb{R}^n$ trivially gives a frame as well.
\item 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 $\mathbb{R}^2$.
\item Similar to the previous example, one can show that the 2-sphere admits no frames. A manifold which admits a (global) frame is called \emph{parallelizable.}
\item A key example of a frame is the Frenet frame.
\item One places adjective in front of "moving frame" if that adjective pertains to each basis, e.g. an \emph{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.
\item Frames arise in general relativity as a formalization of the observation that there is no ``preferred'' observer standpoint.
\end{itemize}
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\bibitem {wikibinom} Wikipedia's \PMlinkexternal{entry on moving frame}{http://en.wikipedia.org/wiki/Moving_frame}
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