Editing Moving frame (section)

==Introduction==

In lay terms, a ''[[frame of reference]]'' is a system of [[measuring rod]]s used by an [[observation|observer]] to measure the surrounding space by providing [[Cartesian coordinate system|coordinates]].  A '''moving frame''' is then a frame of reference which moves with the observer along a [[trajectory]] (a [[curve]]).  The method of the moving frame, in this simple example, seeks to produce a "preferred" moving frame out of the [[kinematics|kinematic]] properties of the observer.  In a geometrical setting, this problem was solved in the mid 19th century by [[Jean Frédéric Frenet]] and [[Joseph Alfred Serret]].<ref name="Chern">{{harvnb|Chern|1985}}</ref>  The [[Frenet–Serret formulas|Frenet–Serret frame]] is a moving frame defined on a curve which can be constructed purely from the [[velocity]] and [[acceleration]] of the curve.<ref>D. J. Struik, ''Lectures on classical differential geometry'', p. 18</ref>

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 (geometry)|congruence]].<ref name="Griffiths">{{harvnb|Griffiths|1974}}</ref> The [[Frenet–Serret formulas]] show that there is a pair of functions defined on the curve, the [[torsion of a curve|torsion]] and [[curvature]], which are obtained by [[derivative|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.

[[File:Darboux trihedron.svg|thumb|right|Darboux trihedron, consisting of a point ''P'', and a triple of [[orthogonality|orthogonal]] [[unit vector]]s '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, and '''e'''<sub>3</sub> which is ''adapted to a surface'' in the sense that ''P'' lies on the surface, and '''e'''<sub>3</sub> is perpendicular to the surface.]]

In the late 19th century, [[Gaston Darboux]] studied the problem of constructing a preferred moving frame on a [[surface (mathematics)|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 for differential systems|integrability conditions]] which needed to be satisfied first.<ref name="Chern" />

Later, moving frames were developed extensively by [[Élie Cartan]] and others in the study of submanifolds of more general [[homogeneous spaces]] (such as [[projective space]]). In this setting, a '''frame''' carries the geometric idea of a basis of a vector space over to other sorts of geometrical spaces ([[Klein geometry|Klein geometries]]).  Some examples of frames are:<ref name="Griffiths" />

* A '''[[linear frame]]''' is an [[ordered basis]] of a [[vector space]].
* An '''[[orthonormal frame]]''' of a vector space is an ordered basis consisting of [[orthogonal]] [[unit vector]]s (an [[orthonormal basis]]).
* An '''[[affine frame]]''' of an affine space consists of a choice of [[affine space|origin]] along with an ordered basis of vectors in the associated [[affine space|difference space]].<ref>[http://www.proofwiki.org/wiki/Definition:Affine_Frame "Affine frame" Proofwiki.org]</ref>
* A '''[[Euclidean frame]]''' of an affine space is a choice of origin along with an orthonormal basis of the difference space.
* A '''[[projective frame]]''' on ''n''-dimensional [[projective space]] is an ordered collection of ''n''+2 points such that any subset of ''n''+1 points is [[linearly independent]]. <!--Could do more examples probably, e.g. [[conformal frame]]?-->
* [[Frame fields in general relativity]] are four-dimensional frames, or [[vierbein]]s, in German.

In each of these examples, the collection of all frames is [[homogeneous space|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 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 the [[pullback bundle|pullback]] of the tautological bundle to ''M''.  Intrinsically<ref>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 principal bundle of a homogeneous space.</ref> 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 form|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 group]]s, 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 (differential geometry)|pullback]] the [[Maurer-Cartan form]] of ''G'' to ''M'' (or ''P''), and thus obtain a complete set of structural invariants for the manifold.<ref name="Griffiths" />