Symmetric-like Riemannian Spaces And Parakahler Manifolds

The main objects of study in this thesis are some nilpotent and solvable Lie groups and some manifolds that are closely related to Lie groups. This dissertation consists of six chapters. It includes two parts. The first three chapters is part I: Our main goal is to study the geometric properties of some nilpotent and solvable Lie groups.Nilpotent and solvable Lie groups with left-invariant Riemannian metrics play a remarkable role in Riemannian geometry. At many occasions they arise quite naturally. For example, they appear in the Iwasawa decomposition of the isometry group of a non-compact Riemannian symmetric space. Also, every connected homogeneous Riemannian manifold of non-positive sectional curvature can be represented as a connected solvable Lie group with a left-invariant Riemannian metric.Among the nilpotent Lie groups the two-step ones are of particular significance. The latter ones are the nonabelian Lie groups that come as close as possible to being abelian, but they admit interesting phenomena that do not arise in abelian groups. Two-step nilpotent Lie groups endowed with a left-invariant Riemannian metric, often called two-step homogeneous nilmanifolds, have attracted considerable attention in the last twenty years, specially in Riemannian geometry, harmonic analysis and spectral geometry.Nilpotent and solvable Lie groups with left-invariant Riemannian metrics are closely related to symmetric-like Riemannian spaces(for example: naturally reductive Riemannian homogeneous spaces, Riemannian g.o.spaces, weakly symmetric spaces, commutative spaces, D扐tri spaces, etc.).The arrangement of part I of this dissertation is as the following. In the first chapter, we recall the definitions of generalized Heisenberg groups, naturally reductive Riemannian homogeneous spaces, Riemannian g.o. spaces, weakly symmetric spaces, commutative spaces and D扐tri spaces, and some fundamental results.In the second chapter, we study some geometric properties on a subclass of twostep nilpotent Lie groups(we call them c5-H-type groups). In the first section, we discuss the curvatures , geodesics, Integrability of some subbundles and geometric structures in case of integrable, isometry groups on the 5-H-type groups. In the second section, our main goal will be to study, on the 5-H-type groups, some special classes of symmetric-like Riemannian spaces, i.e., Riemannian spaces which may be regarded as generalizations of Riemannian(locally) symmetric spaces. We give a complete classification of the c5-H-type groups which belong to each one of four classes of symmetric-like Riemannian spaces, thus obtaining new examples of naturally reductive Riemannian homogeneous spaces(resp. Riemannian g.o. spaces, weakly symmetric spaces, commutative spaces). In the third section, we prove that some semi-direct extension of a H-type group (N, (.,.)) is isometric to 6-H-type group (N, (., .)~).iiiln the third chapter, We study some geometric properties and spectral propertiesof the Jacobi operator on a solvable extension G of Heisenberg grouPs H. In the firstsection, we give the definition of G and some ftlndamental geometric properties on G. Inthe second section, we di8cuss the Integrability of certain subbundles and the geometricstructure of the induced fOliations in case of integrability. In the third section, we 8tudythe spectral properties of the Jacobi operator of G.The last three chapters i8 part II: Our main gOa1 is to study dipolarizations andparakdhler manifolds. Let M be a symplectic manifO1d, if M admits two 8moothtransversal Lagrangian fOliations, then M is called a parakdh1er manifo1d. The de finition of parakdhler manifOld was first given by P.Libermann. The ear1y researche8on this kind of manifolds were closely re1ated to Phy8ics and Mechanic8(But since1991, S. Kaneyuki published his result on the algebraic condition fOr the existence ofinvariant parakdhler structures on a coset space, Lie theory has p1ayed the most important role in the study of this kind of man...