Vol. 295, No. 2, 2018

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Four-dimensional static and related critical spaces with harmonic curvature

Jongsu Kim and Jinwoo Shin

Vol. 295 (2018), No. 2, 429–462
Abstract

We study any four-dimensional Riemannian manifold $\left(M,g\right)$ with harmonic curvature which admits a smooth nonzero solution $f$ to the equation

$\nabla \phantom{\rule{0.3em}{0ex}}df=f\left(Rc-\frac{R}{n-1}g\right)+xRc+y\left(R\right)g,$

where $Rc$ is the Ricci tensor of $g$, $x$ is a constant and $y\left(R\right)$ a function of the scalar curvature $R$. We show that a neighborhood of any point in some open dense subset of $M$ is locally isometric to one of the following five types: (i) ${\mathbb{S}}^{2}\left(\frac{R}{6}\right)×{\mathbb{S}}^{2}\left(\frac{R}{3}\right)$ with $R>0$, (ii) ${ℍ}^{2}\left(\frac{R}{6}\right)×{ℍ}^{2}\left(\frac{R}{3}\right)$ with $R<0$, where ${\mathbb{S}}^{2}\left(k\right)$ and ${ℍ}^{2}\left(k\right)$ are the two-dimensional Riemannian manifolds with constant sectional curvatures $k>0$ and $k<0$, respectively, (iii) the static spaces we describe in Example 3, (iv) conformally flat static spaces described by Kobayashi (1982), and (v) a Ricci flat metric.

We then get a number of corollaries, including the classification of the following four-dimensional spaces with harmonic curvature: static spaces, Miao–Tam critical metrics and $V$-static spaces.

For the proof we use some Codazzi-tensor properties of the Ricci tensor and analyze the equation displayed above depending on the various cases of multiplicity of the Ricci-eigenvalues.

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Keywords
static space, harmonic curvature, Codazzi tensor, critical point metric, Miao-Tam critical metric, $V$-static space
Mathematical Subject Classification 2010
Primary: 53C21, 53C25
Milestones
Received: 18 December 2016
Revised: 2 November 2017
Accepted: 9 January 2018
Published: 11 April 2018
Authors
 Jongsu Kim Department of Mathematics Sogang University Seoul South Korea Jinwoo Shin Department of Mathematics Sogang University Seoul South Korea