Moving frame

In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. Given such a manifold M and a point P in it, a frame at P means a vector space basis of the tangent space to M at P. That is, if M has dimension n, we are given n tangent vectors t1, ..., tn to M at P that are linearly independent. A moving frame in some neighborhood U of P requires that we are given

T1, ..., Tn

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 ti, 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 ti 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 circle, or more generally a torus, such frames exist; but not when M is a 2-sphere. A manifold that has 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 give a frame at a point of it (cf. torsion 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 GLn that is the associated bundle to the tangent bundle as vector bundle. The general Cartan method exploits this, and is discussed at Cartan connection.

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