Anti - Holonomic Jets and the Lie Bracket

Second order anti-holonomic jets as anti-symmetric parts of second order semi-holonomic jets are introduced. The anti-holonomic nature of the Lie bracket is shown. A general result on universality of the Lie bracket is proved. 1. Introduction The concepts of non-holonomic (or iterated) and semi-holonomic jets, rst introduced by Ehresmann in 1], are commonly used in diierential geometry. In this paper, we use the concept of semi-holonomic jet to construct second order anti-holonomic jets as the anti-symmetric part of second order semi-holonomic jets. Further we introduce three diierential operators between some holonomic, semi-holonomic, and anti-holonomic jets, namely the prolongation, torsion, and curvature operators. Finally, using these operators, we show a close relation between the Lie bracket and anti-holonomic jets and prove some universal property of the Lie bracket. The deenition of anti-holonomic jets has many similarities with the deenition of diierence tensor from 3], and of dissym etrie from 9] (in fact, the manifold antiJ