Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON THE GEOMETRY OF VECTOR BUNDLES WITH FLAT CONNECTIONS

  • Abbassi, Mohamed Tahar Kadaoui (Laboratory of Algebra, Geometry and Arithmetic Department of Mathematics Faculty of Sciences Dhar El Mahraz University Sidi Mohamed Ben Abdallah) ;
  • Lakrini, Ibrahim (Laboratory of Algebra, Geometry and Arithmetic Department of Mathematics Faculty of Sciences Dhar El Mahraz University Sidi Mohamed Ben Abdallah)
  • Received : 2018.10.16
  • Accepted : 2019.03.08
  • Published : 2019.09.30
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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.

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References

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