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

Commun. Korean Math. Soc. Published online November 12, 2019

Shahroud Azami and Ghodratallah Fasihi Ramandi
Department of Mathematics, Faculty of Sciences, Imam Khomeini International University, Qazvin, Iran

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 be a $\rho$-soliton, then M is a Kenmotsu manifold of constant sectional curvature $-1$ and the $\rho$-soliton is expanding, with $\lambda=2$.