Holonomy reductions of Cartan geometries and curved orbit decompositions

TL;DR

This paper introduces a method to decompose Cartan geometries into submanifolds based on holonomy reductions, linking geometric structures to solutions of invariant differential equations in various geometries.

Contribution

It develops a general holonomy reduction framework for Cartan geometries and applies it to analyze invariant differential equations in projective, conformal, and CR geometries.

Findings

Decomposition of manifolds into initial submanifolds via holonomy orbits.

Canonical induced Cartan geometries on each submanifold.

Application to existence criteria for Einstein and Kähler-Einstein metrics.

Abstract

We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initial submanifolds. Each such submanifold corresponds to an orbit of the holonomy group on the modelling homogeneous space and carries a canonical induced Cartan geometry. The result can therefore be understood as a `curved orbit decomposition'. The theory is then applied to the study of several invariant overdetermined differential equations in projective, conformal and CR-geometry. This makes use of an equivalent description of solutions to these equations as parallel sections of a tractor bundle. In projective geometry we study a third order differential equation that governs the existence of a compatible Einstein metric. In CR-geometry we discuss an invariant equation that governs the existence of a compatible K\"{a}hler-Einstein metric.