User:Silly rabbit/Sandbox/Moving frame

In mathematics, a frame is a flexible generalization of the notion of an ordered basis of a vector space. Some examples of frames are considered as follows:

A linear frame is an ordered basis of a vector space.
An affine frame of a vector space V consists of a choice of origin for V along with an ordered basis of of vectors in V.
An orthonormal frame of a vector space is an ordered basis consisting of orthogonal unit vectors (an orthonormal basis).
A Euclidean frame of a vector space is a choice of origin along with an orthonormal basis for the vector space.
A projective frame on n-dimensional projective space is an ordered collection of n+1 linearly independent points in a the space.

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. More 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.

Moving frames may also be defined for classes of smooth manifolds besides homogeneous spaces. In the classical differential geometry of curves, the Frenet-Serret frame was a moving Euclidean frame which moved along a curve which was embedded in Euclidean space. In other words, it was a moving Euclidean frame which was defined over the subset of Euclidean space traced out by the curve. The Frenet-Serret frame is an invariant of the curve, in the sense that if the curve is transformed by a Euclidean motion, then the components of the frame also change equivariantly (or tensorially) under that same motion. This transformation property is the key to establishing a complete classification of curves up to Euclidean motions via the curvature and torsion of curves.

The success of the Frenet-Serret frame for Euclidean geometries was later^[1] adopted by other non-Euclidean geometries, notably projective geometry which was of great interest to the late 19th century differential geometers. Here moving projective frames played an important role in the classification of smooth projective curves. Moving frames also proved useful in the study of the extrinsic geometry of embedded surfaces and higher-dimensional manifolds into Euclidean and non-Euclidean spaces.

Each of these investigations of fin de siècle differential geometry can be understood, in general terms, as a problem to do with situating a manifold M inside a homogeneous space G/H. The fundamental problem was to construct, in some way, a natural frame on M: that is a section of the tautological bundle G over M. The Maurer-Cartan forms would then restrict to M and give a complete set invariants of the structure. Abstractly, the Maurer-Cartan forms, together with their structural equations, completely determined the integrability conditions for situating M into the tautological bundle. Hence, moving frame came to mean a certain special system of 1-forms satisfying a structural condition.

With the advent of Einstein's relativity theory, an intrinsic characterization of moving frames became desirable. Elie Cartan, motivated by his studies of contact conditions between moving frames on embedded spaces, observed that the fundamental structural conditions for a moving frame with a particular symmetry could be formulated intrinsically as well. These structural conditions, in Cartan's viewpoint, were formulated as an analog of the Maurer-Cartan equations on a system of differential forms defined on the manifold.

Formal definition

In order to formulate a suitable definition of a moving frame, we fix the following data:

A Lie group G
A vector space V carrying a representation of G, ρ : G → GL(V) and associated infinitesimal representation dρ : g → End(V).

In most applications of moving frames, V is either the adjoint representation of G, or a standard representation of G (realized, for instance, as a matrix group).

Let M be a manifold, and U and open set in M. A local moving frame on U is a non-singular V-valued 1-form θ_U on U. (Note in particular that as a result V must have dimension no smaller than the dimension of M.)

A pair of local moving frames θ_U and θ_V are G-related if there is a G-valued function g : U ∩ V → G such that

θ_V(x) = ρ(g(x)) θ_U(x), for all x ∈ U ∩ V.

Extrinsic moving frames

Example: The Frenet-Serret frame

Consider a curve γ embedded into 3-dimensional Euclidean space E³. One gives this curve its natural arclength parametrization γ : I → E³. The Frenet-Serret frame is given by

t(s) = γ′(s) (the unit tangent)
n(s) = t(s)/κ(s) (the principal normal), where κ(s) = |t′(s)| is the curvature of the curve,
b(s) = t(s) × n(s) (the binormal).

It is readily verified that t, n, and b are mutually orthogonal unit vectors. So the quadruple (γ(s), t(s), n(s), b(s)) determines a moving Euclidean frame on the curve. This frame is natural in the sense that it is uniquely constructed from the curve itself.

Linear frames

In the theory of smooth manifolds, a manifold M is equipped with a vector space at every point p in M, the tangent space to M at p, and the term frame is understood in terms meaning that it can vary from point to point. More precisely, given such a manifold M of dimension n and a point p in it, a frame at p is an ordered basis of the tangent space T_pM, that is, an n-tuple of tangent vectors to M at P which are linearly independent. A moving frame (or smooth frame) in some neighborhood U of p is then a n-tuple of vector fields

X₁, X₂ , ..., X_n

defined on U, which vary smoothly as a function of q in U (formally, they are smooth sections of the tangent bundle TM over U), and are also linearly independent at each point q in U. In more abstract terms, a moving frame is a section of the frame bundle of M over U, which is a principal bundle for GL_n.

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 , ..., X_n which is dual to it, i.e., satisfies the duality relation αⁱ(X_j) = δⁱ_j, where δⁱ_j is the Kronecker delta function on U.

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.

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.