Ricci $ρ$-solitons on 3-dimensional $η$-Einstein almost Kenmotsu mandifolds

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 ρ-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 3dimensional almost Kenmotsu Einstein manifold M be a ρ-soliton, then M is a Kenmotsu manifold of constant sectional curvature −1 and the ρ-soliton is expanding, with λ = 2. 2010 Mathematics Subject Classification. 53Axx, 53Bxx.