On the geometry of vector bundles with flat connections
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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.
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.
참고문헌 (7)
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