On Ricci Solitons as Quasi-Einstein metrics

Abstract

This thesis is the key to good understanding of differential geometry with para- Kenmotsu and Lorentzian Para- Sasakian structure and it is organized as follows. In chapter one, the preliminaries and definitions are introduced,where, Manifolds, differentiable structures, Riemannian Manifolds and Ricci flows are defined. In chapter two the relevant literature is reviewed and Propositions and theorems proved in area are included. In chapter three, Ricci solitons on para- Kenmotsu Manifolds satisfying (ξ,.)s.W8 = 0 and (ξ,.)W8.S = 0 are discussed and we have proved that the Para- Kenmotsu manifolds satisfying (ξ,.)W8.S = 0. are quasi- Einstein Manifolds and those satisfying (ξ,.)S.W8 = 0, are Einstein Manifolds.Also it has been proved that the para- Kenmotsu manifolds with cyclic Ricci tensor and η− Ricci soliton structure are quasi-Einstein manifolds . In chapter four, Ricci solitons on Lorentzian Para- Sasakian manifolds satisfying (ξ,.)s.W8 = 0 and (ξ,.)W8.S = 0 are treated and it has been proved that Lorentzian ParaSasakian manifolds satisfying (ξ,.)s.W8 = 0 and having η− Ricci soliton structure are quasi-Einstein manifolds and those satisfying (ξ,.)W8.S = 0 are Einstein manifolds. In chapter five, we discuss Ricci solitons on Lorentzian Para- Sasakian manifolds satisfying (ξ,.)s.W2 = 0 and (ξ,.)W2.S = 0 and it was found that, Lorentzian Para- Sasakian manifolds satisfying (ξ,.)s.W2 = 0 and having η− Ricci soliton structure are Einstein or quasi-Einstein manifolds according to the value of µ and λ. In Chapter six, results are discussed and the connection between Ricci solitons and Einstein metrics on ParaKenmotsu and Lorentzian Para Sasakian Manifolds has been established.