# The Cayley cubic and differential equations

## TL;DR

This paper introduces Cayley structures as a geometric framework on four-dimensional manifolds, linking them to differential equations and extending concepts from conformal geometry, with detailed invariants and classifications.

## Contribution

It defines Cayley structures, explores their properties, and extends notions from conformal geometry to this new setting, including invariants and classification results.

## Key findings

- Introduction of Cayley structures on 4-manifolds
- Extension of conformal structure notions like half-flatness
- Classification and examples of Cayley structures

## 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.