Differential invariants of Einstein-Weyl structures in 3D

TL;DR

This paper investigates the classification of Einstein-Weyl structures in three dimensions by analyzing their invariants under local diffeomorphisms, utilizing symmetries of an integrable system to describe the moduli space of solutions.

Contribution

It introduces a method to describe the quotient of Einstein-Weyl PDEs by the symmetry pseudogroup using integrable system symmetries.

Findings

Characterization of Einstein-Weyl structures via differential invariants.

Description of the quotient equation representing equivalence classes.

Application of integrable system symmetries to geometric classification.

Abstract

Einstein-Weyl structures on a three-dimensional manifold $M$ is given by a system $E$ of PDEs on sections of a bundle over $M$. This system is invariant under the Lie pseudogroup $G$ of local diffeomorphisms on $M$. Two Einstein-Weyl structures are locally equivalent if there exists a local diffeomorphism taking one to the other. Our goal is to describe the quotient equation $E/G$ whose solutions correspond to nonequivalent Einstein-Weyl structures. The approach uses symmetries of the Manakov-Santini integrable system and the action of the corresponding Lie pseudogroup.