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Of all real Lagrangian Grassmannians LG( n , 2 n ), only LG(2, 4) admits a distinguished Lorentzian conformal structure and hence is identified with the indefinite Möbius 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). The CSp(4,ℝ)‐invariant classification obtained is simultaneously a contact‐invariant classification of in general, highly non‐linear second‐order scalar hyperbolic partial differential equations (PDE) in the plane. From the LG(2, 4) perspective, the trichotomy into elliptic, parabolic, and hyperbolic PDE in the plane is a manifestation of the special isomorphism Sp(4,ℝ) ≌ Spin(2, 3). We give a simple geometric argument for contact‐invariance of the general Monge—Ampère equation based on ‘spheres’ in LG(2, 4), and in the hyperbolic case we derive simple expressions for the relative invariants characterizing such PDE. For hyperbolic PDE of non‐Monge—Ampère 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 conformal space.

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