Gradient Ricci solitons with half harmonic Weyl curvature and two Ricci eigenvalues
Commun. Korean Math. Soc. 2022 Vol. 37, No. 2, 585-594
https://doi.org/10.4134/CKMS.c200423
Published online March 29, 2022
Printed April 30, 2022
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: {\rm (i)} an Einstein manifold. {\rm (ii)} a domain in the Riemannian product $ (\mathbb{R}^2, g_0) \times (N, \tilde{g})$, where $g_0$ is the flat metric on $\mathbb{R}^2$ and $(N, \tilde{g})$ is a two dimensional Riemannian manifold of constant curvature $\lambda \neq 0$. {\rm (iii)} a domain in $\mathbb{R} \times W$ with the warped product metric $ ds^2 + h(s)^2 \tilde{g},$ where $\tilde{g}$ is a constant curved metric on a three dimensional manifold $W$.
Keywords : Gradient Ricci soliton, half harmonic Weyl curvature
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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