Let d be the exterior differentiation operator and I a closed differential ideal (dI over a finite dimensional differentiable manifold, generated by sets of q-forms . An exterior differential system for a geometry generally is a closed differential ideal. The differential geometry of I is essentially reflected in the properties of the generators of I and in the structure equations. In the present paper, the authors study exterior differential ideals and sets of invariant generators for a number of four-dimensional Riemannian conformal and projective geometries determined as sub-bundles of higher frame bundles over the base M.
The final section contains some results on the Einstein-Maxwell ideals. The method consists of establishing exterior differential systems for the geometries under consideration, using canonical basic frames. For that purpose, the authors calculate the Cartan characters and the genus g (the maximum dimension of the regular integral manifolds of I).