Conformal geometry of surfaces in the Lagrangian--Grassmannian and second order PDE

TL;DR

This paper explores the conformal geometry of surfaces in the Lagrangian--Grassmannian LG(2,4), classifies hyperbolic surfaces and second order PDEs using Cartan's method, and introduces new geometric invariants and examples.

Contribution

It provides a conformal geometric framework for classifying hyperbolic surfaces and second order PDEs in LG(2,4), including invariants and conjugate PDEs, with novel examples like Lorentzian Dupin cyclides.

Findings

Classification of hyperbolic surfaces in LG(2,4) using Cartan's method.

Geometric invariants for hyperbolic Monge--Ampère equations.

First example of a Dupin cyclide in Lorentzian space.

Abstract

Of all real Lagrangian--Grassmannians $LG(n,2n)$, only $LG(2,4)$ admits a distinguished (Lorentzian) conformal structure and hence is identified with the indefinite M\"obius space $S^{1,2}$. Using Cartan's method of moving frames, we study hyperbolic (timelike) surfaces in $LG(2,4)$ modulo the conformal symplectic group $CSp(4,R)$. This $CSp(4,R)$-invariant classification is also a contact-invariant classification of (in general, highly non-linear) second order scalar hyperbolic PDE in the plane. Via $LG(2,4)$, we give a simple geometric argument for the invariance of the general hyperbolic Monge--Amp\`ere equation and the relative invariants which characterize it. For hyperbolic PDE of non-Monge--Amp\`ere type, we demonstrate the existence of a geometrically associated ``conjugate'' PDE. Finally, we give the first known example of a Dupin cyclide in a Lorentzian space.