Ricci And *-Ricci Operators On Two Classes Of Almost Contact Metric Manifolds

The notion of the Ricci tensor is one of the most basic geometric concepts on the Rie-mannian manifolds.Some of its properties and behaviors profoundly reflect the curvature,local structure and topological properties of a Riemannian manifold.In this dissertation,a local classification of a special type of almost contact metric manifolds by means of the Ricci cyclic parallelism is given,and such a result generalizes some well-known results obtained by some others.In 1959,the notion of the*-Ricci tensor was first proposed on the almost Hermitian manifolds,which played an important role in solving some impor-tant problems in almost complex geometry.As an analogy of the*-Ricci tensors on an Hermitian manifold,the notion of the*-Ricci tensor on an almost contact metric manifold was proposed in 2002,and soon after received a lot of attention from scholars.Replacing the Ricci operator by the*-Ricci operator,some authors proposed the notion of*-Ricci soliton,which quickly became a research hotspot in almost contact geometry.In this dis-sertation,the existence and local classification problems on two classes of almost contact Riemannian manifolds admitting the*-Ricci soliton are investigated.The first chapter includes the background knowledge and structure of this disserta-tion.The second chapter contains some preliminary knowledge related to almost complex manifolds and almost contact Riemannian manifolds.Furthermore,some basic concepts and geometric properties of almost cosymplectic manifolds and almost Kenmotsu mani-folds are introducedIn the third chapter,a 3-dimensional almost Kenmotsu manifold M satisfying ??h=?h+2a?h(where 2h is the Lie derivative of ? along the Reeb flow and a is a smooth function invariant along the contact distribution)is studied.It is proved that the Ricci tensor of M is cyclic-parallel if and only if it is locally isometric to the hyperbolic space H3(-1)or a non-unimodular Lie group endowed with a left invariant almost Kenmotsu structure.In the fourth chapter,it is proved that if there exists a*-Ricci soliton on a(?,?)'-almost Kenmotsu manifold M of dimension 2n+1,then either M is locally isometric to the product Hn+1(-4)×Rn or the potential vector field of the soliton is a strict infinitesimal contact transformationIn the fifth chapter,it is proved that there are no*-Ricci solitons on a(?,?)-almost cosymplectic manifold.