Connections for a Class of Pseudogroup Structures

Affine connections have long been familiar in differential geometry, among other things for providing a measure of deviation from flatness for manifolds; here flatness means the existence of a system of local coordinates such that the transition functions between any two coordinate systems are affine transformations. The intention of the present note is to show how similar connections can be introduced, in real C ∞ or in complex analytic manifolds, associated to some general classes of pseudogroup structures, for the purpose of investigating the existence of systems of local coordinates with transition functions lying in the pseudogroups. The point of view adopted here is analytical rather than geometrical, in the sense that connections are envisaged as arising from the formal properties of certain systems of partial differential equations rather than as deriving from the splittings of vector bundles into vertical and horizontal components. The classical affine and projective connections of course appear again, and the familiar relationship between them seems perhaps simpler in this context.