Communications of the
Korean Mathematical Society
CKMS

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

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Commun. Korean Math. Soc. 2020; 35(2): 613-623

Online first article November 12, 2019      Printed April 30, 2020

https://doi.org/10.4134/CKMS.c190089

Copyright © The Korean Mathematical Society.

Ricci $\rho$-Solitons on 3-dimensional $\eta$-Einstein almost Kenmotsu manifolds

Shahroud Azami, Ghodratallah Fasihi-Ramandi

Imam Khomeini International University; Imam Khomeini International University

Abstract

The notion of quasi-Einstein metric in theoretical physics and in relation with string theory is equivalent to the notion of Ricci soliton in differential geometry. Quasi-Einstein metrics or Ricci solitons serve also as solution to Ricci flow equation, which is an evolution equation for Riemannian metrics on a Riemannian manifold. Quasi-Einstein metrics are subject of great interest in both mathematics and theoretical physics. In this paper the notion of Ricci $\rho$-soliton as a generalization of Ricci soliton is defined. We are motivated by the Ricci-Bourguignon flow to define this concept. We show that if a 3-dimensional almost Kenmotsu Einstein manifold $M$ is a $\rho$-soliton, then $M$ is a Kenmotsu manifold of constant sectional curvature $-1$ and the $\rho$-soliton is expanding with $\lambda=2$.

Keywords: Almost Kenmotsu manifold, $\rho$-Ricci soliton, $\eta$-Einstein, generalized k-nullity distribution

MSC numbers: Primary 53Axx, 53Bxx

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