Inclusions between parabolic geometries

TL;DR

This paper classifies special inclusions in parabolic geometries, especially those leading to spinorial structures, using cohomological methods and classical classifications, and discusses their geometric and connection properties.

Contribution

It provides a complete classification of inclusions between parabolic geometries that produce new structures, including a detailed analysis of a novel series involving spinorial geometries.

Findings

Classified all cases of inclusions leading to new structures using Onishchik's lists.

Identified a unique series of inclusions with spinorial structures on manifolds.

Discussed the normality of resulting spinorial connections and identified subclasses with normal connections.

Abstract

Some of the well known Fefferman like constructions of parabolic geometries end up with a new structure on the same manifold. In this paper, we classify all such cases with the help of the classical Onishchik's lists \cite{onish1} and we treat in detail the only new series of inclusions providing the spinorial structures on the manifolds with generic free distributions. Our technique relies on the cohomological understanding of the canonical normal Cartan connections for parabolic geometries and the classical computations with exterior forms. Apart of the complete discussion of the distributions from the geometrical point of view and the new functorial construction of the inclusion into the spinorial geometry, we also discuss the normality problem of the resulting spinorial connections. In particular, there is a non--trivial subclass of distributions providing normal spinorial connections directly by the construction.