Cones and Cartan geometry

TL;DR

This paper explores the structure of Cartan geometries of a specific type, showing their relation to affine frame bundles, and classifies their holonomy groups, revealing geometric decompositions of Riemannian manifolds.

Contribution

It establishes an isomorphism between extended principal bundles of certain Cartan geometries and affine frame bundles, and classifies their holonomy groups in relation to Riemannian geometry.

Findings

Extended principal bundle is isomorphic to affine frame bundle with classical affine connection.

Holonomy groups of associated Cartan geometries are classified.

Riemannian manifolds with compact holonomy are locally products of cones.

Abstract

We show that the extended principal bundle of a Cartan geometry of type $(A(m,\mathbb{R}),GL(m,\mathbb{R}))$, endowed with its extended connection $\hat\omega$, is isomorphic to the principal $A(m,\mathbb{R})$-bundle of affine frames endowed with the affine connection as defined in classical Kobayashi-Nomizu volume I. Then we classify the local holonomy groups of the Cartan geometry canonically associated to a Riemannian manifold. It follows that if the holonomy group of the Cartan geometry canonically associated to a Riemannian manifold is compact then the Riemannian manifold is locally a product of cones.