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

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

df = f( Rc R n 1g) + xRc+y(R)g,

where Rc is the Ricci tensor of g, x is a constant and y(R) 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) S2(R 6 ) × S2(R 3 ) with R > 0, (ii) 2(R 6 ) × 2(R 3 ) with R < 0, where S2(k) and 2(k) 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.

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