Differential invariants of [ Restructures

Introduction A differential invariant of a (^-structure is a function which depends on the r-jet of the (^-structure and such that it is invariant under the natural action of the pseudogroup of diffeomorphisms of the base manifold. The importance of these objects is clear, since they seem to be the natural obstructions for the equivalence of (2-structures. Hopefully, if all the differential invariants coincide over two r-jets of O-structure then they are equivalent under the action of the pseudogroup. If all the differential invariants coincide for every r it is hoped that the (?-structures are formally equivalent, and so equivalent in the analytic case. This is the equivalence problem of E. Cartan. In this paper we deal with the problem of finding differential invariants on the bundles of (Restructures, following the program pointed out in [3]. There are several reasons that justify the study of this type of (^-structures. The first one is that it is a non-complicated example that helps to understand the ^-structures with the property for the group G of having a vanishing first prolongation (i.e. of type 1). The simplicity comes from the fact that the algebraic invariants of U* are very simple. The differential geometry of this type of structure, however, has much in common with general G-structures of type 1. Also, IR*-S t,ructures are objects of geometrical interest. They can be interpreted as 'projective parallelisms' of the base manifold and they can also be interpreted as a generalization of Blaschke's notion of a web. The structure of the paper is as follows. In the first section we define the notion of an [R*-st, ruc ture, justifying the geometrical interpretations mentioned above. Then a result is proved giving the existence of a functorial connection associated to some (?-structures, and in particular to [R*-S tructures. The computation of differential invariants will be done from these functorial connections. In the second section a result is given proving that it is reasonable to believe that these invariants will solve the equivalence problem. In the third section the notion of a differential invariant is rigorously defined, proving that a certain space where these invariants are defined is the natural space to work with when the first prolongation of the Lie algebra of the group vanishes. The reason for this assertion is that this space is the desingularization space for the distribution given by the lifts of vector fields of …