Paper · Jan 1, 2018
Three-dimensional Ricci-degenerate Riemannian manifolds satisfying geometric equations
As on arXiv
Abstract
In this paper, we study a three-dimensional Ricci-degenerate Riemannian manifold that admits a smooth nonzero solution to the equation \beginalign \nabla df=ψRc+φg, \endalign where are given smooth functions of , is the Ricci tensor of . Spaces of this type include various interesting classes, namely gradient Ricci solitons, -quasi Einstein metrics, (vacuum) static spaces, -static spaces, and critical point metrics. The -quasi Einstein metrics and vacuum static spaces were previously studied in, respectively. In this paper, we refine them and develop a general approach for the solutions of (); we specify the shape of the metric satisfying () when is not a Ricci-eigen vector. Then we focus on the remaining three classes, namely gradient Ricci solitons, -static spaces, and critical point metrics. Furthermore, we present classifications of local three-dimensional Ricci-degenerate spaces of these three classes by explicitly describing the metric and the potential function .
- Submitted
- Jan 1, 2018
- Last revised
- Mar 9, 2018 (v2)
- Subjects
- math.DG
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Background reading
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In the guidebook
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The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.
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