Editing Moving frame (section)

== Moving tangent frames ==
{{main|Frame bundle}}
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 ''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n''</sub> forming a basis of the [[tangent space]] at each point of an open set {{nowrap|''U'' ⊂ ''M''}}.

If <math>(x^1,x^2,\dots,x^n)</math> is a coordinate system on ''U'', then each vector field ''e<sub>j</sub>'' can be expressed as a [[linear combination]] of the coordinate vector fields <math display="inline">\frac{\partial}{\partial x^i}</math>:<math display="block">e_j = \sum_{i=1}^n A^i_j \frac{\partial}{\partial x^i},</math>where each <math>A^i_j</math> is a function on ''U''. These can be seen as the components of a matrix <math>A</math>. This matrix is useful for finding the coordinate expression of the dual coframe, as explained in the next section.

===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
:''θ''<sup>1</sup>, ''θ''<sup>2</sup>, …, ''θ''<sup>''n''</sup>

which are linearly independent at each point ''q'' in ''U''. Conversely, given such a coframe, there is a unique moving frame ''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n''</sub> which is dual to it, i.e., satisfies the duality relation ''θ''<sup>''i''</sup>(''e''<sub>''j''</sub>) = ''δ''<sup>''i''</sup><sub>''j''</sub>, where ''δ''<sup>''i''</sup><sub>''j''</sub> is the [[Kronecker delta]] function on ''U''.

If <math>(x^1,x^2,\dots,x^n)</math> is a coordinate system on ''U'', as in the preceding section, then each covector field ''θ''<sup>i</sup> can be expressed as a linear combination of the coordinate covector fields <math>dx^i</math>:<math display="block">\theta^i = \sum_{j=1}^n B^i_j dx^j,</math>where each <math>B^i_j</math> is a function on ''U.'' Since <math display="inline">dx^i \left(\frac{\partial}{\partial x^j}\right) = \delta^i_j</math>, the two coordinate expressions above combine to yield <math display="inline"> \sum_{k=1}^n B^i_k A^k_j = \delta^i_j </math>; in terms of matrices, this just says that <math>A</math> and <math>B</math> are [[Matrix inverse|inverses]] of each other.

In the setting of [[classical mechanics]], when working with [[canonical coordinates]], the canonical coframe is given by the [[tautological one-form]]. 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). The tautological one-form is a special case of the more general [[solder form]], which provides a (co-)frame field on a general [[fiber bundle]].

===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 frame of reference|inertial observers]].

In relativity and in [[Riemannian geometry]], the most useful kind of moving frames are the '''orthogonal''' and '''[[orthonormal frame]]s''', 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 define a frame at a point of it (cf. [[torsion tensor]] for a quantitative description – it is assumed here that 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 bundle]]s over open sets ''U''. The general Cartan method exploits this abstraction using the notion of a [[Cartan connection]].