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Three-dimensional Ricci-degenerate Riemannian manifolds satisfying geometric equations

Abstract

In this paper, we study a three-dimensional Ricci-degenerate Riemannian manifold \((M^3,g)\) that admits a smooth nontrivial solution f to the equation

$$\begin{aligned} \nabla df=\psi Rc+\phi g, \end{aligned}$$
(1)

where \(\psi ,\phi \) are given smooth functions of f, Rc is the Ricci tensor of g. Spaces of this type include various interesting classes, namely gradient Ricci solitons, m-quasi Einstein metrics, (vacuum) static spaces, V-static spaces, and critical point metrics. The m-quasi Einstein metrics and vacuum static spaces were previously studied in Jordan (Gen Relativ Gravit 41(9):2191–2280, 2009) and Kim and Shin (Math Nachr 292(8): 1727–1750, 2019), respectively. In this paper, we refine them and develop a general approach for the solutions of (1). We specify the shape of the metric g satisfying (1) when \(\nabla f\) is not a Ricci-eigen vector. Then we focus on the remaining three classes, namely gradient Ricci solitons, V-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 g and the potential function f.

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Acknowledgements

The author was supported by Korea Institute for Advanced Study (KIAS) grant (MG070701) funded by the Korea government (MSIP).

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Correspondence to Jinwoo Shin.

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Shin, J. Three-dimensional Ricci-degenerate Riemannian manifolds satisfying geometric equations. manuscripta math. (2021). https://doi.org/10.1007/s00229-021-01342-2

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Keywords

  • Gradient Ricci soliton
  • Static space
  • Critical point metric
  • Codazzi tensor

Mathematics Subject Classification

  • Primary 53C21
  • Secondary 53C25