Moving Frame - Method of The Moving Frame

Method of The Moving Frame

Cartan (1937) formulated the general definition of a moving frame and the method of the moving frame, as elaborated by 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 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′, there is a unique transition of frame (ƒ→ƒ′) in G determined by the requirement (ƒ→ƒ′)ƒ = ƒ′.

• Given a frame ƒ and a point A ∈ X, there is associated a point x = (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′ of the point A in a different frame ƒ′ arises from (A,ƒ) by application of the transformation (ƒ→ƒ′). That is,

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 Rλ. Slightly different techniques apply depending on whether one is interested in the submanifold along with its parameterization, or the submanifold up to reparameterization.