Remarks on non-maximal integral elements of the Cartan plane in jet spaces

TL;DR

This paper investigates the structure of integral elements in jet spaces, revealing an affine bundle structure and analyzing a Monge-Ampère type PDE's contact equivalence, contributing to geometric analysis of differential equations.

Contribution

It establishes an affine bundle structure for integral elements of Cartan planes in jet spaces and examines the contact equivalence of certain nonlinear PDEs.

Findings

Integral elements form an affine bundle over Grassmannian bundles.

A natural distribution associated with the bundle is studied.

A third-order nonlinear PDE of Monge-Ampère type is not contact-equivalent to a quasi-linear PDE.

Abstract

There is a natural filtration on the space of degree-$k$ homogeneous polynomials in $n$ independent variables with coefficients in the algebra of smooth functions on the Grassmannian $\mathrm{Gr}(n,s)$, determined by the tautological bundle. In this paper we show that the space of $s$-dimensional integral elements of a Cartan plane on $J^{k-1}(E,n)$, with $\mathrm{dim}\, E=n+m$, has an affine bundle structure modeled by the the so-obtained bundles over $\mathrm{Gr}(n,s)$, and we study a natural distribution associated with it. As an example, we show that a third-order nonlinear PDE of Monge-Amp\`ere type is not contact-equivalent to a quasi-linear one.