Exceptionally simple PDE

Abstract

We give local descriptions of parabolic contact structures and show how their flat models yield explicit PDE having symmetry algebras isomorphic to all complex simple Lie algebras except ${\mathfrak{sl}}_2$. This yields a remarkably uniform generalization of the Cartan-Engel models from 1893 in the $G_2$ case. We give a formula for the harmonic curvature of a $G_2$-contact structure and describe submaximally symmetric models for general $G$-contact structures.

Abstract and introduction revised; exposition improved and text reorganized; Section 4.1 on G2-contact structures is new

References in corpus (2)

• New relations between $G_2$-geometries in dimensions 5 and 7
• Lowest degree invariant 2nd order PDEs over rational homogeneous contact manifolds

Cited by in corpus (4)

• Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems
• Lowest degree invariant 2nd order PDEs over rational homogeneous contact manifolds
• Exceptionally simple super-PDE for $F(4)$
• Two problems in the theory of differential equations