Partial extensions of jets and the polar distribution on Grassmannians of non-maximal integral elements

Abstract

We study an intrinsic distribution, called polar, on the space of l-dimensional integral elements of the higher order contact structure on jet spaces. The main result establishes that this exterior differential system is the prolongation of a natural system of PDEs, named pasting conditions, on sections of the bundle of partial jet extensions. Informally, a partial jet extension is a kth order jet with additional (k+1)st order information along l of the n possible directions. A choice of partial extensions of a jet into all possible l-directions satisfies the pasting conditions if the extensions coincide along pairwise intersecting l-directions.We further show that prolonging the polar distribution once more yields the space of (l,n)-dimensional integral flags with its double fibration distribution. When l>1 the exterior differential system is holonomic, stabilizing after one further prolongation.The proof starts form the space of integral flags, constructing the tower of prolongations by reduction.