Bulletin of the
Korean Mathematical Society
BKMS

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

Article

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Bull. Korean Math. Soc. 2017; 54(3): 975-992

Online first article January 10, 2017      Printed May 31, 2017

https://doi.org/10.4134/BKMS.b160414

Copyright © The Korean Mathematical Society.

Real hypersurfaces with $*$-Ricci tensors in complex two-plane Grassmannians

Xiaomin Chen

China University of Petroleum-Beijing

Abstract

In this article, we consider a real hypersurface of complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2}),m\geq3$, admitting commuting $*$-Ricci and pseudo anti-commuting $*$-Ricci tensor, respectively. As the applications, we prove that there do not exist $*$-Einstein metrics on Hopf hypersurfaces as well as $*$-Ricci solitons whose potential vector field is the Reeb vector field on any real hypersurfaces.

Keywords: commuting $*$-Ricci tensor, pseudo anti-commuting $*$-Ricci tensor, Hopf hypersurfaces, complex two-plane Grassmannians, $*$-Ricci soliton

MSC numbers: Primary 53C40, 53C15