Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2022; 37(2): 585-594

Online first article March 29, 2022      Printed April 30, 2022

https://doi.org/10.4134/CKMS.c200423

Copyright © The Korean Mathematical Society.

Gradient Ricci solitons with half harmonic Weyl curvature and two Ricci eigenvalues

Yutae Kang, Jongsu Kim

Sogang University; Sogang University

Abstract

In this article we classify four dimensional gradient Ricci solitons $(M, g, f)$ with half harmonic Weyl curvature and at most two distinct Ricci-eigenvalues at each point. Indeed, we showed that, in a neighborhood $V$ of each point in some open dense subset of $M$, $(V, g)$ is isometric to one of the following: {m (i)} an Einstein manifold. {m (ii)} a domain in the Riemannian product $ (mathbb{R}^2, g_0) imes (N, ilde{g})$, where $g_0$ is the flat metric on $mathbb{R}^2$ and $(N, ilde{g})$ is a two dimensional Riemannian manifold of constant curvature $lambda
eq 0$. {m (iii)} a domain in $mathbb{R} imes W$ with the warped product metric $ ds^2 + h(s)^2 ilde{g},$ where $ ilde{g}$ is a constant curved metric on a three dimensional manifold $W$.

Keywords: Gradient Ricci soliton, half harmonic Weyl curvature

MSC numbers: 53C21, 53C25

Supported by: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2020R1A2B5B01001862).

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