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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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참고문헌 (7)
참고문헌 신청
M. T. K. Abbassi / 2009 / g-natural metrics: new horizons in the geometry of tangent bundles of Riemannian manifolds / Note Mat 28:6~35
M. T. K. Abbassi / 2012 / Metriques Naturelles Riemanniennes sur le Fibre tangent a une variete Rie-mannienne / Editions Universitaires Europeennes
Mohamed Tahar Kadaoui Abbassi / 2005 / On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds / Differential Geometry and its Applications 22(1):19~47
R. Albuquerque / 2017 / On vector bundle manifolds with spherically symmetric metrics / Annals of Global Analysis and Geometry 51(2):129~154
I. Belegradek / 2002 / Metrics of positive Ricci curvature on vector bundles over nilmanifolds / Geometric And Functional Analysis 12(1):56~72
M. T. K. Abbassi / 2012 / Metriques Naturelles Riemanniennes sur le Fibre tangent a une variete Rie-mannienne / Editions Universitaires Europeennes
Mohamed Tahar Kadaoui Abbassi / 2005 / On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds / Differential Geometry and its Applications 22(1):19~47
R. Albuquerque / 2017 / On vector bundle manifolds with spherically symmetric metrics / Annals of Global Analysis and Geometry 51(2):129~154
I. Belegradek / 2002 / Metrics of positive Ricci curvature on vector bundles over nilmanifolds / Geometric And Functional Analysis 12(1):56~72
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