The Projective Parabolic Geometry of Riemannian, K\"ahler and Quaternion-K\"ahler Metrics

TL;DR

This paper introduces a unified framework called projective parabolic geometries that generalizes classical geometries like projective, c-projective, and quaternionic geometries, providing new classifications and insights into compatible metrics.

Contribution

It develops a comprehensive theory of projective parabolic geometries, extending classical results and classifying geometries modeled on special symmetric R-spaces, including the Cayley plane and conformal geometries.

Findings

Classification of projective parabolic geometries with irreducible models

Control over Lie brackets via Jordan algebra structures

Characterization of compatible metrics through the first BGG operator

Abstract

We present a uniform framework generalising and extending the classical theories of projective differential geometry, c-projective geometry, and almost quaternionic geometry. Such geometries, which we call \emph{projective parabolic geometries}, are abelian parabolic geometries whose flat model is an R-space $G\cdot\mathfrak{p}$ in the infinitesimal isotropy representation $\mathbb{W}$ of a larger self-dual symmetric R-space $H\cdot\mathfrak{q}$. We also give a classification of projective parabolic geometries with $H\cdot\mathfrak{q}$ irreducible which, in addition to the aforementioned classical geometries, includes a geometry modelled on the Cayley plane $\mathbb{OP}^2$ and conformal geometries of various signatures. The larger R-space $H\cdot\mathfrak{q}$ severely restricts the Lie-algebraic structure of a projective parabolic geometry. In particular, by exploiting a Jordan algebra structure on $\mathbb{W}$, we obtain a $\mathbb{Z}^2$-grading on the Lie algebra of $H$ in which we have tight control over Lie brackets between various summands. This allows us to generalise known results from the classical theories. For example, which riemannian metrics are compatible with the underlying geometry is controlled by the first BGG operator associated to $\mathbb{W}$. In the final chapter, we describe projective parabolic geometries admitting a $2$-dimensional family of compatible metrics. This is the usual setting for the classical projective structures; we find that many results which hold in these settings carry over with little to no changes in the general case.