Non-regular $|2|$-graded geometries II: classifying geometries, and generic six-in-nine distributions

TL;DR

This paper classifies $|2|$-graded parabolic geometries, detailing their properties, exploring specific cases, and partially solving the equivalence problem for generic six distributions on nine-dimensional manifolds.

Contribution

It provides a comprehensive classification of $|2|$-graded geometries and advances the understanding of their structure and equivalence problems.

Findings

Classified key properties of $|2|$-graded parabolic geometries.

Explored specific geometries in detail.

Partially solved the equivalence problem for certain distributions.

Abstract

Complementing the previous paper in the series, this paper classifies $|2|$-graded parabolic geometries, listing their important properties: the group $G_0$, the graded tangent bundle $gr(T)$ and its algebra\"ic bracket, the relevant cohomology spaces and the standard Tractor bundle $\mc{T}$. Several of these geometries are then explored in more detail, and the paper ends with a case study that that partially solves the equivalence problem for generic six distributions on nine dimensional manifolds.