Extrinsic geometry and linear differential equations of $\mathfrak{sl}_3$-type

TL;DR

This paper explores the extrinsic geometry of frag varieties and classifies certain linear PDE systems linked to the adjoint representation of sl(3), revealing seven new second-order PDE systems with rich symmetry properties.

Contribution

It provides a complete local classification of homogeneous structures related to sl(3) and introduces seven new second-order PDE systems with specific symmetry and solution space characteristics.

Findings

Identified seven new second-order PDE systems on 3D contact manifolds.

Each PDE system has an 8-dimensional solution space.

Discovered PDE systems include contact Cayley's surface and an sl(2)-symmetric system.

Abstract

As an application of the general theory on extrinsic geometry, we investigate extrinsic geometry in frag varieties and systems of linear PDE's for a class of special interest associated with the adjoint representation of $\mathfrak{sl}(3)$. We carry out a complete local classification of the homogeneous structures in this class. As a result, we find 7 kinds of new systems of linear PDE's of second order on a 3-dimensional contact manifold each of which has a solution space of dimension 8. Among them there are included a system of PDE's called contact Cayley's surface and one which has $\mathfrak{sl}(2)$ symmetry.