Journal of the
Korean Mathematical Society

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



J. Korean Math. Soc. 2023; 60(6): 1303-1336

Online first article October 23, 2023      Printed November 1, 2023

Copyright © The Korean Mathematical Society.

On the $\eta$-parallelism in almost Kenmotsu $3$-manifolds

Jun-ichi Inoguchi, Ji-Eun Lee

Hokkaido University; Chonnam National University


In this paper, we study the $\eta$-parallelism of the Ricci operator of almost Kenmotsu $3$-manifolds. First, we prove that an almost Kenmotsu $3$-manifold $M$ satisfying $\nabla_{\xi}h=-2\alpha h \varphi$ for some constant $\alpha$ has dominantly $\eta$-parallel Ricci operator if and only if it is locally symmetric. Next, we show that if $M$ is an $H$-almost Kenmotsu $3$-manifold satisfying $\nabla_{\xi}h=-2\alpha h \varphi$ for a constant $\alpha$, then $M$ is a Kenmotsu $3$-manifold or it is locally isomorphic to certain non-unimodular Lie group equipped with a left invariant almost Kenmotsu structure. The dominantly $\eta$-parallelism of the Ricci operator is equivalent to the local symmetry on homogeneous almost Kenmotsu $3$-manifolds.

Keywords: Almost Kenmotsu manifold, local symmetry, $\eta$-parallelism

MSC numbers: Primary 53D15, 53C25, 53C30

Supported by: The first named author is partially supported by Kakenhi JP19K03461, JP23K03081.

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