Intrinsic holonomy and curved cosets of Cartan geometries

TL;DR

This paper introduces an intrinsic approach to curved cosets in Cartan geometries, defining an intrinsic holonomy group that aligns with the classical one, and applies this to generalize the de Rham decomposition theorem.

Contribution

It provides a new intrinsic notion of curved cosets and holonomy groups for Cartan geometries, simplifying existing constructions and enabling new geometric decompositions.

Findings

Intrinsic holonomy group coincides with classical holonomy.

Curved cosets preserve properties of homogeneous counterparts.

Generalization of de Rham decomposition theorem.

Abstract

We provide an intrinsic notion of curved cosets for arbitrary Cartan geometries, simplifying the existing construction of curved orbits for a given holonomy reduction. To do this, we define an intrinsic holonomy group, which is shown to coincide precisely with the standard definition of the holonomy group for Cartan geometries in terms of an associated principal connection. These curved cosets retain many characteristics of their homogeneous counterparts, and they behave well under the action of automorphisms. We conclude the paper by using the machinery developed to generalize the de Rham decomposition theorem for Riemannian manifolds and give a potentially useful characterization of inessential automorphism groups for parabolic geometries.