On the geometry of vector bundles with flat connections

摘要

Let E→M be an arbitrary vector bundle of rank k over a Riemannian manifold M equipped with a fiber metric and a compatible connection DE. R.~Albuquerque constructed a general class of (two-weights) spherically symmetric metrics on E. In this paper, we give a characterization of locally symmetric spherically symmetric metrics on E in the case when DE is flat. We study also the Einstein property on E proving, among other results, that if k≥2 and the base manifold is Einstein with positive constant scalar curvature, then there is a 1-parameter family of Einstein spherically symmetric metrics on E, which are not Ricci-flat.