Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online July 8, 2022

Hopf hypersurfaces of the homogeneous nearly K\"{a}hler $\mathbb{S}^3\times\mathbb{S}^3$ satisfying certain commuting conditions

Xiao Min Chen and Yi Fan Yang

China Uniersity of Petroleum-Beijing

Abstract

In this article, we first introduce the notion of commuting Ricci tensor and pseudo-anti commuting Ricci tensore for Hopf hypersurfaces in the homogeneous nearly K\"{a}hler $\mathbb{S}^3\times\mathbb{S}^3$ and prove that the mean curvature of hypersurface is constant under certain assumptions. Next, we prove the nonexistence of Ricci soliton on Hopf hypersurface
with potential Reeb vector field, which improves a result of Hu-Moruzb-Vranckenc-Yao in \cite{HMVY}.

Keywords: nearly K\"ahler $\mathbb{S}^3\times\mathbb{S}^3$; Hopf hypersurface; commuting Ricci tensor; pseudo-anti commuting Ricci tensor; Ricci soliton.

MSC numbers: 53B35; 53C25; 53C30