Inmathematics, amoving frame is a flexible generalization of the notion of acoordinate frame (anordered basis of avector space, in conjunction with anorigin) often used to study theextrinsic differential geometry ofsmooth manifolds embedded in ahomogeneous space.

In lay terms, aframe of reference is a system ofmeasuring rods used by anobserver to measure the surrounding space by providingcoordinates. Amoving frame is then a frame of reference which moves with the observer along atrajectory (acurve). The method of the moving frame, in this simple example, seeks to produce a "preferred" moving frame out of thekinematic properties of the observer. In a geometrical setting, this problem was solved in the mid 19th century byJean Frédéric Frenet andJoseph Alfred Serret.^[1] TheFrenet–Serret frame is a moving frame defined on a curve which can be constructed purely from thevelocity andacceleration of the curve.^[2]

The Frenet–Serret frame plays a key role in thedifferential geometry of curves, ultimately leading to a more or less complete classification of smooth curves in Euclidean space up tocongruence.^[3] TheFrenet–Serret formulas show that there is a pair of functions defined on the curve, thetorsion andcurvature, which are obtained bydifferentiating 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 asurface in Euclidean space instead of a curve, theDarboux frame (or thetrièdre mobile as it was then called). It turned out to be impossible in general to construct such a frame, and that there wereintegrability conditions which needed to be satisfied first.^[1]

Later, moving frames were developed extensively byÉlie Cartan and others in the study of submanifolds of more generalhomogeneous spaces (such asprojective space). In this setting, aframe 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:^[3]

• Alinear frame is anordered basis of avector space.
• Anorthonormal frame of a vector space is an ordered basis consisting oforthogonalunit vectors (anorthonormal basis).
• Anaffine frame of an affine space consists of a choice oforigin along with an ordered basis of vectors in the associateddifference space.^[4]
• AEuclidean frame of an affine space is a choice of origin along with an orthonormal basis of the difference space.
• Aprojective frame onn-dimensionalprojective space is an ordered collection ofn+2 points such that any subset ofn+1 points islinearly independent.
• Frame fields in general relativity are four-dimensional frames, orvierbeins, in German.

In each of these examples, the collection of all frames ishomogeneous in a certain sense. In the case of linear frames, for instance, any two frames are related by an element of thegeneral linear group. Projective frames are related by theprojective 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 ahomogeneous spaceG/H consists of a point in the tautological bundleG →G/H. Amoving frame is a section of this bundle. It ismoving in the sense that as the point of the base varies, the frame in the fibre changes by an element of the symmetry groupG. A moving frame on a submanifoldM ofG/H is a section of thepullback of the tautological bundle toM. Intrinsically^[5] a moving frame can be defined on aprincipal bundleP over a manifold. In this case, a moving frame is given by aG-equivariant mapping φ :P →G, thusframing the manifold by elements of the Lie groupG.

One can extend the notion of frames to a more general case: one can "solder" afiber bundle to asmooth 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 ofspecial orthogonal groups, this reduces to the standard conception of avierbein.

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 intoG. The strategy in Cartan'smethod of moving frames, as outlined briefly inCartan's equivalence method, is to find anatural moving frame on the manifold and then to take itsDarboux derivative, in other wordspullback theMaurer-Cartan form ofG toM (orP), and thus obtain a complete set of structural invariants for the manifold.^[3]

Cartan (1937) formulated the general definition of a moving frame and the method of the moving frame, as elaborated byWeyl (1938). The elements of the theory are

• ALie groupG.
• AKlein spaceX whose group of geometric automorphisms isG.
• Asmooth manifold Σ which serves as a space of (generalized) coordinates forX.
• A collection offrames ƒ each of which determines a coordinate function fromX to Σ (the precise nature of the frame is left vague in the general axiomatization).

The following axioms are then assumed to hold between these elements:

• There is a free and transitivegroup action ofG on the collection of frames: it is aprincipal homogeneous space forG. In particular, for any pair of frames ƒ and ƒ′, there is a unique transition of frame (ƒ→ƒ′) inG determined by the requirement (ƒ→ƒ′)ƒ = ƒ′.
• Given a frame ƒ and a pointA ∈ X, there is associated a pointx = (A,ƒ) belonging to Σ. This mapping determined by the frame ƒ is a bijection from the points ofX to those of Σ. This bijection is compatible with the law of composition of frames in the sense that the coordinatex′ of the pointA in a different frame ƒ′ arises from (A,ƒ) by application of the transformation (ƒ→ƒ′). That is,$${\displaystyle (A,f^{\prime })=(f\to f^{\prime })\circ (A,f).}$$

Of interest to the method are parameterized submanifolds ofX. The considerations are largely local, so the parameter domain is taken to be an open subset ofR^λ. Slightly different techniques apply depending on whether one is interested in the submanifold along with its parameterization, or the submanifold up to reparameterization.

The most commonly encountered case of a moving frame is for the bundle of tangent frames (also called theframe bundle) of a manifold. In this case, a moving tangent frame on a manifoldM consists of a collection of vector fieldse₁,e₂, …,e_n forming a basis of thetangent space at each point of an open setU ⊂M.

If${\displaystyle (x^1,x^2,\dots ,x^n)}$ is a coordinate system onU, then each vector fielde_j can be expressed as alinear combination of the coordinate vector fields$\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}$:$${\displaystyle e_j=\sum _{i=1}^n{A}_j^i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i},}$$where each${\displaystyle A_j^i}$ is a function onU. These can be seen as the components of a matrix${\displaystyle A}$. This matrix is useful for finding the coordinate expression of the dual coframe, as explained in the next section.

A moving frame determines adual frame orcoframe of thecotangent bundle overU, which is sometimes also called a moving frame. This is an-tuple of smooth1-forms

θ¹,θ², …,θⁿ

which are linearly independent at each pointq inU. Conversely, given such a coframe, there is a unique moving framee₁,e₂, …,e_n which is dual to it, i.e., satisfies the duality relationθⁱ(e_j) =δⁱ_j, whereδⁱ_j is theKronecker delta function onU.

If${\displaystyle (x^1,x^2,\dots ,x^n)}$ is a coordinate system onU, as in the preceding section, then each covector fieldθⁱ can be expressed as a linear combination of the coordinate covector fields${\displaystyle d{x}^i}$:$${\displaystyle {\theta }^i=\sum _{j=1}^n{B}_j^{i}d{x}^j,}$$where each${\displaystyle B_j^i}$ is a function onU. Since$d{x}^i\left(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^j}\right)={\delta }_j^i$, the two coordinate expressions above combine to yield$\sum _{k=1}^n{B}_k^i{A}_j^k={\delta }_j^i$; in terms of matrices, this just says that${\displaystyle A}$ and${\displaystyle B}$ areinverses of each other.

In the setting ofclassical mechanics, when working withcanonical coordinates, the canonical coframe is given by thetautological 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 generalsolder form, which provides a (co-)frame field on a generalfiber bundle.

Moving frames are important ingeneral relativity, where there is no privileged way of extending a choice of frame at an eventp (a point inspacetime, which is a manifold of dimension four) to nearby points, and so a choice must be made. In contrast inspecial relativity,M is taken to be a vector spaceV (of dimension four). In that case a frame at a pointp can be translated fromp to any other pointq in a well-defined way. Broadly speaking, a moving frame corresponds to an observer, and the distinguished frames in special relativity representinertial observers.

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

A moving frame always existslocally, i.e., in some neighbourhoodU of any pointp inM; however, the existence of a moving frame globally onM requirestopological conditions. For example whenM is acircle, or more generally atorus, such frames exist; but not whenM is a 2-sphere. A manifold that does have a global moving frame is calledparallelizable. Note for example how the unit directions oflatitude andlongitude on the Earth's surface break down as a moving frame at the north and south poles.

Themethod of moving frames ofÉlie Cartan is based on taking a moving frame that is adapted to the particular problem being studied. For example, given acurve 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 ofprincipal bundles over open setsU. The general Cartan method exploits this abstraction using the notion of aCartan connection.

In many cases, it is impossible to define a single frame of reference that is valid globally. To overcome this, frames are commonly pieced together to form anatlas, thus arriving at the notion of alocal frame. In addition, it is often desirable to endow these atlases with asmooth structure, so that the resulting frame fields are differentiable.

Although this article constructs the frame fields as a coordinate system on thetangent bundle of amanifold, the general ideas move over easily to the concept of avector 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.

Aircraft maneuvers can be expressed in terms of the moving frame (aircraft principal axes) when described by the pilot.

• Darboux frame
• Frenet–Serret formulas
• Turtle graphics
• Yaw, pitch, and roll

• Cartan, Élie (1937),La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile, Paris: Gauthier-Villars.
• Cartan, Élie (1983),Geometry of Riemannian Spaces, Math Sci Press, Massachusetts.
• Chern, S.-S. (1985), "Moving frames",Elie Cartan et les Mathematiques d'Aujourd'hui, Asterisque, numero hors serie, Soc. Math. France, pp. 67–77.
• Cotton, Émile (1905), "Genéralisation de la theorie du trièdre mobile",Bull. Soc. Math. France,33:1–23.
• Darboux, Gaston (1887),Leçons sur la théorie génerale des surfaces, vol. I, Gauthier-Villars.
• Darboux, Gaston (1915),Leçons sur la théorie génerale des surfaces, vol. II, Gauthier-Villars.
• Darboux, Gaston (1894),Leçons sur la théorie génerale des surfaces, vol. III, Gauthier-Villars.
• Darboux, Gaston (1896),Leçons sur la théorie génerale des surfaces, vol. IV, Gauthier-Villars.
• Ehresmann, C. (1950), "Les connexions infinitésimals dans un espace fibré differential",Colloque de Topologie, Bruxelles, pp. 29–55.
• Evtushik, E.L. (2001) [1994],"Moving-frame method",Encyclopedia of Mathematics,EMS Press.
• Fels, M.;Olver, P.J. (1999), "Moving coframes II: Regularization and Theoretical Foundations",Acta Applicandae Mathematicae,55 (2): 127,doi:10.1023/A:1006195823000,S2CID 826629.
• Green, M (1978), "The moving frame, differential invariants and rigidity theorem for curves in homogeneous spaces",Duke Mathematical Journal,45 (4):735–779,doi:10.1215/S0012-7094-78-04535-0,S2CID 120620785.
• Griffiths, Phillip (1974), "On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry",Duke Mathematical Journal,41 (4):775–814,doi:10.1215/S0012-7094-74-04180-5,S2CID 12966544
• Guggenheimer, Heinrich (1977),Differential Geometry, New York:Dover Publications.
• Sharpe, R. W. (1997),Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Berlin, New York:Springer-Verlag,ISBN 978-0-387-94732-7.
• Spivak, Michael (1999),A Comprehensive introduction to differential geometry, vol. 3, Houston, TX: Publish or Perish.
• Sternberg, Shlomo (1964),Lectures on Differential Geometry,Prentice Hall.
• Weyl, Hermann (1938),"Cartan on groups and differential geometry",Bulletin of the American Mathematical Society,44 (9):598–601,doi:10.1090/S0002-9904-1938-06789-4.

• Connection (mathematics)
• Differential geometry
• Frames of reference

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