Non-regular $|2|$-graded geometries I: general theory

TL;DR

This paper explores non-regular |2|-graded geometries, demonstrating they share key properties with regular geometries, including unique normal Cartan connections and harmonic curvature, and investigates their determination by underlying distributions.

Contribution

It extends the theory of |2|-graded geometries to non-regular cases, establishing existence of Cartan connections and analyzing conditions for their determination by distributions.

Findings

Non-regular |2|-graded geometries share properties with regular ones.

Existence of unique normal Cartan connections for these geometries.

Conditions under which geometries are determined by distributions.

Abstract

This paper analyses non-regular $|2|$-graded geometries, and show that they share many of the properties of regular geometries -- the existence of a unique normal Cartan connection encoding the structure, the harmonic curvature as obstruction to flatness of the geometry, the existence of the first two BGG splitting operators and of (in most cases) invariant prolongations for the standard Tractor bundle $\mc{T}$. Finally, it investigates whether these geometries are determined entirely by the distribution $H = T_{-1}$ and concludes that this is generically the case, up to a finite choice, whenever $H^1(\mf{g}^1,\mf{g})$ vanishes in non-negative homogeneity.