# Journal of theKorean Mathematical SocietyJKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

## Article

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J. Korean Math. Soc. 2020; 57(6): 1435-1449

Published online November 1, 2020 https://doi.org/10.4134/JKMS.j190685

Copyright © The Korean Mathematical Society.

## Static and related critical spaces with harmonic curvature and three Ricci eigenvalues

Jongsu Kim

Sogang University

### Abstract

In this article we make a local classification of $n$-dimensional Riemannian manifolds $(M,g)$ with harmonic curvature and less than four Ricci eigenvalues which admit a smooth non constant solution $f$ to the following equation \begin{align} \label{0002bxu} \nabla df = f(r -\frac{R}{n-1} g) + x \cdot r+ y(R) g, \end{align} where $\nabla$ is the Levi-Civita connection of $g$, $r$ is the Ricci tensor of $g$, $x$ is a constant and $y(R)$ a function of the scalar curvature $R$. Indeed, we showed that, in a neighborhood $V$ of each point in some open dense subset of $M$, either {\rm (i)} or {\rm (ii)} below holds; {\rm (i)} $(V, g, f+x)$ is a static space and isometric to a domain in the Riemannian product of an Einstein manifold $N$ and a static space $(W, g_W, f+x)$, where $g_W$ is a warped product metric of an interval and an Einstein manifold. {\rm (ii)} $(V, g)$ is isometric to a domain in the warped product of an interval and an Einstein manifold. For the proof we use eigenvalue analysis based on the Codazzi tensor properties of the Ricci tensor.

Keywords: Static space, critical metric, harmonic curvature

MSC numbers: 53C21, 53C25

Supported by: This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2017R1A2B4004460)

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