Moving frame

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In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space. Moving frames were first introduced by Gaston Darboux in the 19th century, through his studies of the Frenet-Serret frame of a curve embedded in Euclidean space.[citation needed] Later, they were brought to maturity by Élie Cartan and others in the study of submanifolds of more general homogeneous spaces (such as projective space).

A frame carries the geometric idea of a basis of a vector space over to other sorts of geometrical spaces (Klein geometries). Some examples of frames are:

In each of these examples, the collection of all frames is homogeneous in a certain sense. In the case of linear frames, for instance, any two frames are related by an element of the general linear group. Projective frames are related by the projective linear group. This homogeneity, or symmetry, of the class of frames captures the geometrical features of the linear, affine, Euclidean, or projective landscape. A moving frame, in these circumstances, is just that: a frame which varies from point to point.

Formally, a frame on a homogeneous space G/H consists of a point in the tautological bundle G → G/H. A moving frame is a section of the this bundle. It is moving in the sense that as the point of the base varies, the frame in the fibre changes by an element of the symmetry group G. A moving frame on a submanifold M of G/H is a section of pullback of the tautological bundle to M. Intrinsically[1] a moving frame can be defined on a principal bundle P over a manifold. In this case, a moving frame is given by a G-equivariant mapping φ : P → G, thus framing the manifold by elements of the Lie group G.

Although there is a substantial formal difference between extrinsic and intrinsic moving frames, they are both alike in the sense that a moving frame is always given by a mapping into G. The strategy in Cartan's method of moving frames, as outlined briefly in Cartan's equivalence method, is to find a natural moving frame on the manifold and then to take its Darboux derivative, in other words pullback the Maurer-Cartan form of G to M (or P), and thus obtain a complete set of structural invariants for the manifold.

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[edit] Moving tangent frames

The most commonly encountered case of a moving frame is for the bundle of tangent frames (also called the frame bundle) of a manifold. In this case, a moving tangent frame on a manifold M consists of a collection of vector fields X1, X2 , ..., Xn forming a basis of the tangent space at each point of an open set U ⊂ M.

[edit] Coframes

A moving frame determines a dual frame or coframe of the cotangent bundle over U, which is sometimes also called a moving frame. This is a n-tuple of smooth 1-forms

α1, α2 , ..., αn

which are linearly independent at each point q in U. Conversely, given such a coframe, there is a unique moving frame X1, X2 , ..., Xn which is dual to it, i.e., satisfies the duality relation αi(Xj) = δij, where δij is the Kronecker delta function on U.

[edit] Uses

Moving frames are important in general relativity, where there is no privileged way of extending a choice of frame at an event p (a point in spacetime, which is a manifold of dimension four) to nearby points, and so a choice must be made. In contrast in special relativity, M is taken to be a vector space V (of dimension four). In that case a frame at a point p can be translated from p to any other point q in a well-defined way. Broadly speaking, a moving frame corresponds to an observer, and the distinguished frames in special relativity represent inertial observers.

In relativity and in Riemannian geometry, the most useful kind of moving frames are the orthogonal and orthonormal frames, that is, frames consisting of orthogonal (unit) vectors at each point. At a given point p a general frame may be made orthonormal by orthonormalization; in fact this can be done smoothly, so that the existence of a moving frame implies the existence of a moving orthonormal frame.

[edit] Further details

A moving frame always exists locally, i.e., in some neighbourhood U of any point p in M; however, the existence of a moving frame globally on M requires topological conditions. For example when M is a circle, or more generally a torus, such frames exist; but not when M is a 2-sphere. A manifold that does have a global moving frame is called parallelizable. Note for example how the unit directions of latitude and longitude on the Earth's surface break down as a moving frame at the north and south poles.

The method of moving frames of Élie Cartan is based on taking a moving frame that is adapted to the particular problem being studied. For example, given a curve in space, the first three derivative vectors of the curve can in general give a frame at a point of it (cf. torsion for this in quantitative form - it assumes the torsion is not zero). In fact, in the method of moving frames, one more often works with coframes rather than frames. More generally, moving frames may be viewed as sections of principal bundles over open sets U. The general Cartan method exploits this abstraction using the notion of a Cartan connection.

[edit] See also

[edit] Notes

  1. ^ See Cartan (1983) 9.I; Appendix 2 (by Hermann) for the bundle of tangent frames. Fels and Olver (1998) for the case of more general fibrations. Griffiths (1974) for the case of frames on the tautological bundle of a homogeneous space.

[edit] References

  • Griffiths, Philip (1974). "On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry". Duke Math. J. 41 (4): 775-814. 
  • Fels, M., Olver, P.J. (1999). "Moving coframes II: Regularization and Theoretical Foundations". Acta Applicandae Mathematicae: An International Survey 55: 127. Springer. doi:10.1023/A:1006195823000. 
  • Cartan, Elie (1983). Geometry of Riemannian Spaces. Math Sci Press, Massachusetts. 
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