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T₁, ..., T_n

that are each vector fields defined on U, which we should assume vary smoothly as a function of Q in U, and are also linearly independent vectors at each point Q (assume for simplicity M has dimension n everywhere).

In very general terms, such a moving frame is the requirement of an observer in general relativity, where there is no privileged way of continuing the choice of t_i, known at P, to nearby points. In contrast in special relativity M is taken to be a vector space V (of dimension four). In that case t_i can be translated from P to any other point Q in a well-defined way.

In relativity and in Riemannian geometry, the most important kind of moving frames are the orthogonal and orthonormal frames, that is, frames comprising ordered sets of (unit) normal vectors at each point. At a given point P a general frame may be made orthonormal by orthogonalisation; in fact this can be done smoothly, so that the existence of a moving frame implies the existence of a moving orthonormal frame.

The existence of a moving frame is clear, locally on M; but global existence on M requires topological conditions. For example when M is a circleSee The Circle for the distributed file storage system, and see Ring (diacritic) for the diacritic mark. In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius from a fixed point, called the centre ., or more generally a torusSee also torus (nuclear physics). In geometry, a torus (pl. tori is a doughnut shaped surface of revolution generated by revolving a circle about an axis coplanar with the circle. The sphere is a special case of the torus obtained when the axis of rotatio, such frames exist; but not when M is a 2- sphereFor other uses, see sphere (disambiguation). A sphere is, roughly speaking, a ball-shaped object. In non-mathematical usage a sphere is often considered to be solid (which mathematicians call ball . But in mathematics, a sphere is the boundary of a ball,. A manifold that has a global moving frame is called parallelizable. Note for example how the unit directions of latitudeLatitude denoted φ, gives the location of a place on Earth north or south of the Equator. Latitude is an angular measurement ranging from 0° at the Equator to 90° at the poles. Usually, the difference in latitude largely affects the climate and/or wea and longitudeMap of Earth showing vertical lines of longitude Longitude sometimes denoted λ, describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. Longitude is given as an angular measurement ranging from 0° at on the Earth's surface break down as a moving frame at the north and south poles.

The method of moving frames of Élie Cartanlie Joseph Cartan ( 9 April 1869 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. He was born in Dolomieu in Savoie, and became a student at the Ecole Normale Superieure in Pari is based on taking a moving frame that is adapted to the particular problem being studied. For example, given a curveThis article is about the term used in mathematics. There is also a magazine called Curve. Metric geometry Geometry Topology General topology In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and c in space, the first three derivative vectors of the curve can in general give a frame at a point of it (cf. torsionDifferential geometry Riemannian geometry In mathematics, torsion has several meanings, mostly unrelated to each other. Differential geometry of curves In elementary differential geometry in three dimensions, the torsion of a curve measures how sharply it for this in quantitative form - it assumes the torsion is not zero). More generally, the abstract meaning of a moving frame is as a section of the principal bundle for GL_n that is the associated bundle to the tangent bundle as vector bundle. The general Cartan method exploits this, and is discussed at Cartan connection.