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Anti-holonomic jets and the Lie bracket. (English) Zbl 0915.58005
The algebraic structure of the standard fibers of semi-holonomic jet prolongations has been well known in the literature. The whole fiber carries a natural action of the second order jet group, there is its part corresponding to the second order derivatives, and the kernel of the symmetrization of the projection onto this part is a natural subspace which gives rise to a first order natural bundle. Motivated by the existence of the composition analogous to the jet composition, the author uses the name ‘anti-holonomic’ jets. Links to earlier closely related papers [I. Kolář, Czechoslovak Math. J. 21, 124-136 (1971; Zbl 0216.18501); J. Pradines, C. R. Acad. Sci., Paris, Sér. A 278, 1523-1526 (1974; Zbl 0285.58002)] are also given. After a review of the basic notions of semi-holonomic jets, their anti-holonomic quotients are discussed. Then, by means of the new approach, the universality of the Lie bracket for a class of natural operators on regular $$m$$-dimensional velocities on $$n$$-dimensional manifolds ($$m\leq n$$) is proved.
Reviewer: J.Slovák (Brno)

##### MSC:
 58A20 Jets in global analysis 53A55 Differential invariants (local theory), geometric objects
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