Sp(3,R) Monge geometries in dimension 8

TL;DR

This paper investigates a special class of 8-dimensional geometries linked to rank 3 distributions with a specific Lie algebra symmetry, constructing their invariants and classifying homogeneous models with degenerate harmonic curvature.

Contribution

It develops a complete set of local differential invariants for these geometries and classifies homogeneous models with maximally degenerate harmonic curvature quintic.

Findings

Constructed full system of local differential invariants.

Identified all homogeneous models with degenerate harmonic curvature.

Reduced EDS to lower dimensions for special cases.

Abstract

We study a geometry associated with rank 3 distributions in dimension 8, whose symbol algebra is constant and has a simple Lie algebra sp(3,R) as Tanaka prolongation. We restrict our considerations to only those distributions that are defined in terms of a systems of ODEs of the form $\dot{z}_{ij}=\frac{\partial^2 f(\dot{x}_1,\dot{x}_2)}{\partial \dot{x}_i\partial \dot{x}_j}$, $i\leq j=1,2$. For them we built the full system of local differential invariants, by solving an equivalence problem a'la Cartan, in the spirit of his 1910's five variable paper. The considered geometry is a parabolic geometry, and we show that its main invariant - the harmonic curvature - is a certain quintic. In the case when this quintic is maximally degenerate but nonzero, we use Cartan's reduction procedure and reduce the EDS governing the invariants to 11, 10 and 9 dimensions. As a byproduct all homogeneous models having maximally degenerate harmonic curvature quintic are found. They have symmetry algebras of dimension 11 (a unique structue), 10 (a 1-parameter family of nonequivalent structures) or 9 (precisely two nonequivalent structures).