Moving frame
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.
In a geometrical setting, this problem was solved in the mid 19th century by Jean Frédéric Frenet and Joseph Alfred Serret.
[2] The Frenet–Serret frame plays a key role in the differential geometry of curves, ultimately leading to a more or less complete classification of smooth curves in Euclidean space up to congruence.
[3] The Frenet–Serret formulas show that there is a pair of functions defined on the curve, the torsion and curvature, which are obtained by differentiating the frame, and which describe completely how the frame evolves in time along the curve.
A key feature of the general method is that a preferred moving frame, provided it can be found, gives a complete kinematic description of the curve.
In the late 19th century, Gaston Darboux studied the problem of constructing a preferred moving frame on a surface in Euclidean space instead of a curve, the Darboux frame (or the trièdre mobile as it was then called).
It turned out to be impossible in general to construct such a frame, and that there were integrability conditions which needed to be satisfied first.
This homogeneity, or symmetry, of the class of frames captures the geometrical features of the linear, affine, Euclidean, or projective landscape.
Formally, a frame on a homogeneous space G/H consists of a point in the tautological bundle G → G/H.
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 the pullback of the tautological bundle to M. Intrinsically[5] 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. One can extend the notion of frames to a more general case: one can "solder" a fiber bundle to a smooth manifold, in such a way that the fibers behave as if they were tangent.
When the fiber bundle is a homogenous space, this reduces to the above-described frame-field.
When the homogenous space is a quotient of special orthogonal groups, this reduces to the standard conception of a vierbein.
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.
The considerations are largely local, so the parameter domain is taken to be an open subset of Rλ.
In this case, a moving tangent frame on a manifold M consists of a collection of vector fields e1, e2, …, en forming a basis of the tangent space at each point of an open set U ⊂ M. If
This matrix is useful for finding the coordinate expression of the dual coframe, as explained in the next section.
This is a n-tuple of smooth 1-forms which are linearly independent at each point q in U. Conversely, given such a coframe, there is a unique moving frame e1, e2, …, en which is dual to it, i.e., satisfies the duality relation θi(ej) = δij, where δij is the Kronecker delta function on U.
is a coordinate system on U, as in the preceding section, then each covector field θi can be expressed as a linear combination of the coordinate covector fields
Intuitively, it relates the velocities of a mechanical system (given by vector fields on the tangent bundle of the coordinates) to the corresponding momenta of the system (given by vector fields in the cotangent bundle; i.e. given by forms).
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).
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.
More generally, moving frames may be viewed as sections of principal bundles over open sets U.
In many cases, it is impossible to define a single frame of reference that is valid globally.
In addition, it is often desirable to endow these atlases with a smooth structure, so that the resulting frame fields are differentiable.
Although this article constructs the frame fields as a coordinate system on the tangent bundle of a manifold, the general ideas move over easily to the concept of a vector bundle, which is a manifold endowed with a vector space at each point, that vector space being arbitrary, and not in general related to the tangent bundle.
