Parallel Connections Over Symmetric Spaces

TL;DR

This paper characterizes vector bundles with parallel curvature over symmetric spaces, reducing their classification to representations of the structure group of a canonical principal bundle.

Contribution

It establishes that such vector bundles are associated bundles of a canonical principal bundle, simplifying their classification over symmetric spaces.

Findings

Vector bundles with parallel curvature are associated to canonical principal bundles.

Classification reduces to finding representations of the structure group.

Applicable to Riemannian and unitary vector bundles over symmetric spaces.

Abstract

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding Riemannian (or unitary) vector bundles with parallel curvature then reduces to finding representations of the structure group of the canonical principal bundle.