The Cayley cubic and differential equations

Abstract

We define Cayley structures as a field of Cayley's ruled cubic surfaces over a four dimensional manifold and motivate their study by showing their similarity to indefinite conformal structures and their link to differential equations. In particular, for Cayley structures an extension of certain notions defined for indefinite conformal structures in dimension four are introduced, e.g., half-flatness, existence of a null foliation, ultra-half-flatness, an associated pair of second order ODEs, and a dispersionless Lax pair. After solving the equivalence problem we obtain the fundamental invariants, find the local generality of several classes of Cayley structures and give examples.

arXiv comment40 pages; Minor changes in the introduction and bibliography; Final version

References in corpus (13)

• Differential equations and conformal structures
• On the definition and examples of cones and Finsler spacetimes
• Paraconformal geometry of $n$th order ODEs, and exotic holonomy in dimension four
• On the Einstein-Weyl and conformal self-duality equations
• The gap phenomenon in parabolic geometries
• Einstein--Weyl geometry, dispersionless Hirota equation and Veronese webs
• Anti-self-dual conformal structures with null Killing vectors from projective structures
• Twistor geometry of a pair of second order ODEs
• Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski-West Construction
• Cayley Hypersurfaces
• Differential Geometric Aspects of Causal Structures
• Reduction of beta-integrable 2-Segre structures
• $\mathrm{GL}(2)$-structures in dimension four, $H$-flatness and integrability

Cited by in corpus (2)

• GL(2)-geometry and complex structures
• The Schwarzian derivative and Euler--Lagrange equations