Differential invariants of R ∗-structures

Abstract

A differential invariant of a G-structure is a function which depends on the r-jet of the G-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 G-structures. Hopefully, if all the differential invariants coincide over two r–jets of G-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 G-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 *-structures, following the program pointed out in [3]. There are several reasons that justify the study of this type of G-structures. The first one is that it is a non-complicated example that helps to understand the G-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 * are very simple. The differential geometry of this type of structure, however, has much in common with general G-structures of type 1. Also, *-structures 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.