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

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.