On Generalized M-quasi-Einstein Manifold

The study of Einstein manifolds and their various generalizations is always an attractive topic in modern Riemannian geometry.In particular,there has been increasing interest on weighted quasi Einstein manifolds.In this dissertation,we study generalized m-quasi-Einstein manifolds with typical geometrical conditions.The main results are as follows.(1)We establish a complete classification of generalized m-quasi-Einstein manifolds with parallel Ricci tensor.Afterward,under a more general curvature condition,we study generalized m-quasi-Einstein manifold with constant Ricci curvature,and we get that these manifolds must be Einstein under suitable assumptions.Consequently,every homogeneous proper generalized m-quasi-Einstein manifold with m = 1 must be Einstein.(2)We obtain the rigidity of generalized m-quasi-Einstein manifold with constant scalar curvature.In addition,for the m-quasi-Einstein manifold with constant scalar curvature,by using theory of isoparametric function,we can determine its scalar curvature in explicit form and show that these values are attainable by presenting examples.(3)We study generalized m-quasi-Einstein manifold with typical geometrical structure.Firstly,we discuss the existence and uniqueness of the generalized mquasi-Einstein structure on 3-dimensional homogeneous manifolds,and we obtain that space forms are the only 3-dimensional homogeneous manifolds that can carry proper generalized m-quasi-Einstein structure.Secondly,we get some properties ongeneralized m-quasi-Einstein manifold with warped product structure.