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
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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The author was supported by Korea Institute for Advanced Study (KIAS) grant (MG070701) funded by the Korea government (MSIP).
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Shin, J. Three-dimensional Ricci-degenerate Riemannian manifolds satisfying geometric equations. manuscripta math. 169, 401–423 (2022). https://doi.org/10.1007/s00229-021-01342-2
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DOI: https://doi.org/10.1007/s00229-021-01342-2