To the Theory of Semi { Holonomic Jets

To Ivan Koll a r on the occasion of his 60th birthday. The usual jets were introduced by C. Ehresmann as a fundamental tool in Diierential Geometry. They permit to globalize the theory of diierential systems and to give a formulation of the \innnite groups" of E. Cartan; this leads to the theory of Lie pseudogroups; initiated by C. Ehresmann, this theory was studied by many mathematicians (the author, introduced cohomological methods. When studying the prolongations of a diierential system or higher order connections (for instance the iteration of a linear connection on the tangent bundle) C. Ehresmann was led to introduce what he called, using the terminology of Mechanics , non holonomic and semi-holonomic jets; the ordinary jets are called holonomic jets. While the theory of holonomic jets is now classical, the theory of non holo-nomic and semi-holonomic jets seems \mysterious" to many mathematicians. The purpose of this paper is to explain how semi-holonomic jets occur naturally in Diierential Geometry and to serve as an elementary introduction to the works devoted to semi-holonomic jets; we leave to the reader the task of studying these papers. Among the mathematicians who have investigated semi-holonomic jets are C. A very important contribution has been made by I. Koll a r and his co-workers, especially concerning natural transformations and higher order connections, as well as G. Virsik and M. Modugno. For instance I. Koll a r has introduced the notion of equivalence with respect to curves for semi-holonomic jets; in the case of holonomic jets, equivalent jets coincide. This is linked with the research of natural transformations 174 PAULETTE LIBERMANN existing in a bundle of semi-holonomic jets. On the subject of natural transformations , we refer to the book \Natural Operations in Diierential Geometry" by I. Koll a r, P. Michor, J. Slovv ak which contains a great list of references. The non holonomic jets are obtained by iteration of 1-jets; among them semi-holonomic jets are obtained while \forgetting" the condition of Schwarz symmetry in higher order derivatives; they correspond to an iteration of linear maps in the following sense; the projection J q E ! J q?1 E (where J q means the semi-holonomic prolongation of order q) is endowed with an aane bundle structure whose associated vector bundle is a bundle of multilinear maps from TM to the vertical bundle V E = ker T. Here denotes the projection E ! …