On Quasi-einstein Manifolds And Its Generalized Forms

Einstein manifolds are important research objects in Riemannian geometry and play an important role in general relativity.As a generalization of Einstein manifolds,quasi-Einstein manifolds are one of the main research objects in Riemannian geometry and mathematical physics.B tensor is a(0,2)type tensor formed by the combination of Ricci tensor and metric tensor with scale expansion and contraction,respectively.In this paper,we introduce special manifolds via the study of B tensors,and prove that these manifolds are quasi-Einstein manifolds,and then we further generalize quasi-Einstein manifolds.The details are as follows.1.Based on the study of B tensor,we further study weak B symmetric manifold.We give a sufficient conditions for weak B symmetric manifolds to be second-order Einstein manifolds and prove that weak B symmetric manifolds are quasi-Einstein manifolds.Then,we introduce almost pseudo B symmetric manifolds.We give the geometric properties of this manifold and prove that it is a quasi-Einstein manifold under certain conditions.We also prove that a conformally flat almost pseudo B symmetric manifolds are quasi-Einstein manifolds.Finally,we discuss curvature limit structures with B tensor in N(k)quasi-Einstein manifold.2.We introduce a new curvature tensor T,whose trace is B tensor.Further more,we give the basic symmetry properties of T tensor and introduce pseudo T symmetric manifolds.On the premise of the associated vector field is parallel,we prove that the manifold is a quasiEinstein manifold.Next,we introduce weak T symmetric manifolds and give the geometric properties of decomposable weak T symmetric manifolds.3.We consider that the generators of mixed quasi-Einstein manifolds are special vector fields,such as Killing vector field,concircular vector field,and recurrent vector field.We discuss different curvature limiting structures in such manifolds,such as Ricci semi-symmetry,concircular Ricci pseudo symmetry,Wi Ricci pseudo symmetry.Finally,we study mixed quasi-Einstein manifolds with Ricci soliton.4.We consider quasi-Einstein manifolds with three generators and introduce extended quasi-Einstein manifolds.We give the existence theorem and basic geometric properties of extended quasi-Einstein manifolds.We consider the case where the generators of extended quasi-Einstein manifolds are special vector fields,and obtain many interesting properties.Finally,we study the geometric properties of extended quasi-Einstein manifolds with generalized Ricci soliton and Riemannian soliton,respectively.