Moving frames and prolongation algebras

Abstract

We consider differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms. By setting up a Cartan–Ehresmann connection, in a fiber bundle over a base space in which the 2-forms live, one finds an incomplete Lie algebra of vector fields in the fibers. Conversely, given this algebra (a prolongation algebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg–de Vries and Harrison–Ernst systems.