Some Classification Studies Of Almost Contact Riemannian Geometry In Low Dimension

As an odd-dimensional analogy with the Hermitian manifolds,the almost contact Riemannian manifolds have been introduced in the fifties in the last century and have received many attentions for the last seven decades from a great number of geometers from all over the world,including the great geometer S.S.Chern.At this time of day,the geometry of almost contact Riemanian manifolds has became one of the most fundamental research fields in Riemannian geometry and in this field fruitful results have been established.It has important effects and applications not only on some branches in modern differential geometry and topology,but also on theoretical physics and mechanics.In terms of some classical methods in Riemannian geometry,together with some techniques and results in Lie group,Lie algebras and geometry of submanifolds,the aim of this thesis is to investigate some classification problems of almost contact Riemannian structures—a branch in geometry of almost contact Riemannian manifolds.Around the almost contact Riemannian structures,in this thesis some important classification theorems are given.These results generalize some well known results in almost contact Riemannian geometry and some of them answer an open question.Main results in every chapter in this thesis are listed as follows.In chapter two,locally Φ-symmetric and conformally flat almost cosymplectic manifold of dimension three are investigated.First,it is obtained that if an almost cosymplectic 3-h-amanifold is locally Φ-symmetric,then the scalar curvature is invariant along the contact distribution if and only if the manifold is locally isometric to Riemannian product R×N2(c)(where N2(c)denotes a surface of constant Gauss curvature c)or a unimodular Lie group endowed with a left invariant almost cosymplectic structure.More specifically,M is locally isometric to the universal covering of rigid motion group of Euclidian 2-plane E(2),Geisenberg group H3,or rigid motion group of the Minkowski 2-plane E(1,1).Second,it is proved that if an almost cosymplectic 3-h-a-manifold has the property that the scalar curvature is invariant along the Reeb vector field,then the manifold is conformally flat if and only if it is locally isometric to the Riemannian product R×N2(c).In particular,some examples are constructed in order to illustrate the necessary of those conditions employed in the above theorems.In chapter three,classification problems of trans-Sasakian manifolds are investigated.First,a complete classification of locally symmetric trans-Sasakian manifolds of dimension three are provided.By means of this,a classification of homogeneous trans-Sasakian manifolds of dimension three is completed.Namely,a trans-Sasakian 3-manifold is homogeneous if and only if it is isometric to the Riemannian product R×N2(c),the universal covering of rigid motion group of Euclidian 2-plane E(2),Abelian group R3,Heisenberg group H3,3-sphere group S U(2),or three kinds of nonunimodular groups.At last,an open question proposed by S.Deshmukh is investigated and a negative answer is given.It is proved that if a trans-Sasakian manifold of dimension three satisfies dβ∧η=0,then either α=0 or β=0;however,the vanishing of α or βdoes not implies that the other one is a constant even if the manifold is assumed to be compact.All these results give some new characterizations for which a three-dimensional trans-Sasakian manifold is proper.In chapter four,it is investigated the classification problem of locally symmetric almost cosymplectic manifold of dimension five.The main reuslt is that a five-dimensional CRintegrable almost cosymplectic manifold is locally symmetric if and only if it is locally isometric to the Riemannian product of R and a locally symmetric K?hler manifold of dimension four.