Prolongations, invariants, and fundamental identities of geometric structures

Abstract

Working in the framework of nilpotent geometry, we give a unified scheme for the equivalence problem of geometric structures which extends and integrates the earlier works by Cartan, Singer-Sternberg, Tanaka, and Morimoto. By giving a new formulation of the higher order geometric structures and the universal frame bundles, we reconstruct the step prolongation of Singer-Sternberg and Tanaka. We then investigate the structure function $\gamma$ of the complete step prolongation of a normal geometric structure by expanding it into components $\gamma = \kappa + \tau + \sigma$ and establish the fundamental identities for $\kappa$, $\tau$, $\sigma$. This then enables us to study the equivalence problem of geometric structures in full generality and to extend applications largely to the geometric structures which have not necessarily Cartan connections. Among all we give an algorithm to construct a complete system of invariants for any higher order normal geometric structure of constant symbol by making use of generalized Spencer cohomology group associated to the symbol of the geometric structure. We then discuss thoroughly the equivalence problem for geometric structure in both cases of infinite and finite type. We also give a characterization of the Cartan connections by means of the structure function $\tau$ and make clear where the Cartan connections are placed in the perspective of the step prolongations.