Four-dimensional static and related critical spaces with harmonic curvature

Jongsu Kim and Jinwoo Shin

Vol. 295 (2018), No. 2, 429–462

DOI: 10.2140/pjm.2018.295.429

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

$$\nabla df=f\left(Rc-\frac{R}{n-1}g\right)+xRc+y\left(R\right)g,$$

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) ${\mathbb{S}}^2\left(\frac{R}{6}\right)\times {\mathbb{S}}^2\left(\frac{R}{3}\right)$ with $R>0$, (ii) ${ℍ}^2\left(\frac{R}{6}\right)\times {ℍ}^2\left(\frac{R}{3}\right)$ 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.

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