Differential invariants of generic parabolic Monge-Ampere equations

TL;DR

This paper investigates the geometry of parabolic Monge-Ampère equations, classifying nonintegrable cases via directing distributions and constructing invariants to solve the equivalence problem.

Contribution

It introduces a classification of nonintegrable PMAs using directing distributions and develops contact invariants based on projective curve bundles.

Findings

All integrable PMAs are locally equivalent to u_{xx}=0.

Nonintegrable PMAs are classified into three classes based on their directing distributions.

Constructed invariants precisely measure the nonlinearity of PMAs.

Abstract

Some new results on geometry of classical parabolic Monge-Amp\`ere equations (PMA) are presented. PMAs are either \emph{integrable}, or \emph{nonintegrable} according to integrability of its characteristic distribution. All integrable PMAs are locally equivalent to the equation $u_{xx}=0$. We study nonintegrable PMAs by associating with each of them a 1-dimensional distribution on the corresponding first order jet manifold, called the \emph{directing distribution}. According to some property of this distribution, nonintegrable PMAs are subdivided into three classes, one \emph{generic} and two \emph{special} ones. Generic PMAs are completely characterized by their directing distributions, and we study canonical models of the latters, \emph{projective curve bundles} (PCB). A PCB is a 1-dimensional subbundle of the projectivized cotangent bundle of a 4-dimensional manifold. Differential invariants of projective curves composing such a bundle are used to construct a series of contact differential invariants for corresponding PMAs. These give a solution of the equivalence problem for generic PMAs with respect to contact transformations. The introduced invariants measure in an exact manner nonlinearity of PMAs.