On the geometry of vector bundles with flat connections
Bull. Korean Math. Soc. 2019 Vol. 56, No. 5, 1219-1233
Published online September 30, 2019
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
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