Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2019; 56(5): 1219-1233

Online first article August 6, 2019      Printed September 30, 2019

https://doi.org/10.4134/BKMS.b180983

Copyright © The Korean Mathematical Society.

On the geometry of vector bundles with flat connections

Mohamed Tahar Kadaoui Abbassi, Ibrahim Lakrini

University Sidi Mohamed Ben Abdallah, University Sidi Mohamed Ben Abdallah

Abstract

Let $E \rightarrow M$ be an arbitrary vector bundle of rank $k$ over a Riemannian manifold $M$ equipped with a fiber metric and a compatible connection $D^{E}$. 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 $D^E$ is flat. We study also the Einstein property on $E$ proving, among other results, that if $k \geq 2$ 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.

Keywords: vector bundle, spherically symmetric metric, curvatures, Einstein manifold, local symmetry

MSC numbers: 53C07, 53C24, 53C25