arXiv subjects
Publications and source records attributed to Antonio Kumpera.
It is presently our aim to undertake the discussion, of the Parts I and II, on the infinitesimal level and outline as well the transition from infinitesimal to finite, the main reason for this being, of course, the well known fact that arguments and calculation on the infinitesimal level are far simpler that those on the finite level.
We start discussing basic properties of Lie groupoids and Lie pseudo-groups in view of applying these techniques to the analysis of Jordan-Hölder resolutions and the subsequent integration of partial differential equations which is the summit of Lie and Cartan's work. Next, we discuss the integration problem for systems of partial differential equations in one unknown function and special attention is given to the first order systems. The Grassmannian contact structures are the basic setting for our discussion and the major part of our considerations inquires on the nature of the Cauchy characteristics in view of obtaining the necessary criteria that assure the existence of solutions. In all the practical applications of partial differential equations, what is mostly needed and what is in fact hardest to obtains are the solutions of the system or, occasionally, some specific solutions. We continue our discussion by examining the local equivalence problem for partial differential equations, illustrating it with some examples, since almost any integration process or method is actually a local equivalence problem involving a suitable model. We terminate the discussion by inquiring on non-integrable Pfaffian systems and their integral manifolds of maximal dimension. This work is based on four most enlightening Mémoires written by Élie Cartan in the beginning of the last century.
We discuss the local and global problems for the equivalence of geometric structures of an arbitrary order and, in later sections, attention is given to what really matters, namely the equivalence with respect to transformations belonging to a given pseudo-group of transformations. We first give attention to general prolongation spaces and thereafter insert the structures in their most appropriate ambient namely, as specific solutions of partial differential equations where the equivalence problem is then discussed. In the second part, we discuss applications of all this abstract nonsense and take considerable advantage in exploring Élie Cartan's magical trump called transformations et prolongements mériédriques that somehow seem absent in present day geometry.
This paper is a continuation of Part I where the general setup was developed. Here we discuss the general equivalence problem for geometric structures and provide criteria for the equivalence, local and global, of transitive structures. Cartan's Flag Systems illustrate the theory as a major example and, finally, some attention though little is given to non-transitive structures with regular orbits i.e., intransitivity classes.