Hyperbolic Monge-Amp\`ere systems with $S_1=0$

TL;DR

This paper studies hyperbolic Monge-Ampère systems with the invariant tensor ${S}_1=0$, classifies low-cohomogeneity cases, and finds their generality and linearity properties under contact transformations.

Contribution

It analyzes the ${S}_1=0$ case of hyperbolic Monge-Ampère systems, providing classification and generality results that were previously unexplored.

Findings

${S}_1=0$ systems have local generality of two functions of three variables.

All low-cohomogeneity ${S}_1=0$ systems are linear up to contact transformations.

The paper characterizes the structure of these systems and their invariants.

Abstract

For hyperbolic Monge-Amp\`ere systems, a well-known solution of the equivalence problem yields two invariant tensors, ${S}_1$ and ${S}_2$, defined on the underlying $5$-manifold, where ${S}_2=0$ characterizes systems that are Euler-Lagrange. In this article, we consider the `opposite' case, ${S}_1 = 0$, and show that the local generality of such systems is `$2$ arbitrary functions of $3$ variables'. In addition, we classify all $S_1=0$ systems with cohomogeneity at most one, which turn out to be linear up to contact transformations.