Hyperbolic Monge-Ampère systems with $S_1=0$
Yuhao Hu
TL;DR
This work analyzes hyperbolic Monge–Ampère systems on a 5-manifold through the Cartan–BGG invariant tensors $S_1$ and $S_2$. It proves that in the $S_1=0$ regime, $S_2$ must be degenerate, reducing to a rank-1 normal form with two relative invariants $Q_1,Q_2$ and three subcases I–III, whose local generalities are governed by 1 or 2 arbitrary functions of 3 variables. Notably, Case I yields a linear Monge–Ampère PDE with an explicit associated 3D geometry, while Case II and Case III exhibit progressively fewer symmetries and higher cohomogeneity, with Case II giving 1 function of 3 variables and Case III giving 2 such functions. A symmetry analysis shows that high symmetry occurs only in Case I, and the paper provides a mechanism to lift associated-geometry symmetries to the full Monge–Ampère system. Collectively, the results map the landscape of S1=0 hyperbolic Monge–Ampère systems, contrasting them with the Euler–Lagrange class and highlighting how invariant geometry constrains local generality and linearity aspects.
Abstract
For hyperbolic Monge-Ampère 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.