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

TL;DR

This paper investigates the polar distribution on jet spaces' integral elements, showing it as a prolongation of PDE systems called pasting conditions, and explores the structure of integral flags and their prolongations.

Contribution

It establishes that the polar distribution is the prolongation of pasting conditions on partial jet extensions and analyzes the structure of integral flags and their prolongations.

Findings

The polar distribution is the prolongation of pasting conditions.

Prolonging the polar distribution yields the space of integral flags.

The exterior differential system stabilizes after one prolongation for l>1.

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 $k$th 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.