In mathematics, a moving frame is a flexible generalization of the notion of a coordinate frame (an ordered basis of a vector space, in conjunction with an origin) often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
Introduction In lay terms, a frame of reference is a system of measuring rods used by an observer to measure the surrounding space by providing 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 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. The Frenet–Serret frame is a moving frame defined on a curve which can be constructed purely from the velocity and acceleration of the curve. 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. 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. 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 geometries). Some examples of frames are:
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 vectors (an orthonormal basis). An affine frame of an affine space consists of a choice of origin along with an ordered basis of vectors in the associated difference space. A Cartesian 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. Frame fields in general relativity are four-dimensional frames, or vierbeins, in German. 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. 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 of the tautological bundle to M. Intrinsically 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.
Method of the moving frame Cartan (1937) formulated the general definition of a moving frame and the method of the moving frame, as elaborated by Weyl (1938). The elements of the theory are
A Lie group G. A Klein space X whose group of geometric automorphisms is G. A smooth manifold Σ which serves as a space of (generalized) coordinates for X. A collection of frames ƒ each of which determines a coordinate function from X 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:
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