Editing Moving frame (section)

== Method of the moving frame ==
{{harvtxt|Cartan|1937}} formulated the general definition of a moving frame and the method of the moving frame, as elaborated by {{harvtxt|Weyl|1938}}.  The elements of the theory are

* A [[Lie group]] ''G''.
* A [[Klein space]] ''X'' whose group of geometric automorphisms is ''G''.
* A [[smooth manifold]] Σ which serves as a space of (generalized) coordinates for ''X''.
* A collection of ''frames'' ƒ each of which determines a coordinate function from ''X'' to Σ (the precise nature of the frame is left vague in the general axiomatization).

The following axioms are then assumed to hold between these elements:

* There is a free and transitive [[Group action (mathematics)|group action]] of ''G'' on the collection of frames: it is a [[principal homogeneous space]] for ''G''.  In particular, for any pair of frames ƒ and ƒ&prime;, there is a unique transition of frame (ƒ→ƒ&prime;) in ''G'' determined by the requirement (ƒ→ƒ&prime;)ƒ&nbsp;=&nbsp;ƒ&prime;.
* Given a frame ƒ and a point ''A''&nbsp;∈&nbsp;''X'', there is associated a point ''x''&nbsp;=&nbsp;(''A'',ƒ) belonging to Σ.  This mapping determined by the frame ƒ is a bijection from the points of ''X'' to those of Σ.  This bijection is compatible with the law of composition of frames in the sense that the coordinate ''x''&prime; of the point ''A'' in a different frame ƒ&prime; arises from (''A'',ƒ) by application of the transformation (ƒ→ƒ&prime;).  That is, <math display="block">(A,f') = (f\to f')\circ(A,f).</math>

Of interest to the method are parameterized submanifolds of ''X''.  The considerations are largely local, so the parameter domain is taken to be an open subset of '''R'''<sup>λ</sup>.  Slightly different techniques apply depending on whether one is interested in the submanifold along with its parameterization, or the submanifold up to reparameterization.