Editing Moving frame (section)

===Further details===

A moving frame always exists ''locally'', i.e., in some neighbourhood ''U'' of any point ''p'' in ''M''; however, the existence of a moving frame globally 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 does have 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 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 of [[principal bundle]]s over open sets ''U''. The general Cartan method exploits this abstraction using the notion of a [[Cartan connection]].