The Study Of Some Problems On Almost Kenmotsu Manifolds

In1969, S. Tanno[103]proved the well known classification theorem: the connectedalmost contact metric manifolds whose automorphism groups have maximal dimensionsare classified into three cases by the signatures of their characteristic sectional curvatures(>0,=0or <0). Later, in1972, K. Kenmotsu[62]introduced a new type of normal almostcontact metric manifolds (with respective to Sasakian and cosymplectic manifolds) beingthe characterization of the third class of Tanno’s classification. In1981, D. Janssens andL. Vanhecke[59]regarded such manifolds as Kenmotsu manifolds and introduced the notionof almost Kenmotsu manifolds by generalizing the previous one. Since then, Kenmotsumanifolds were studied and generalized by many geometers. In terms of some knownresults in Kenmotsu geometry, the method of tensorial analysis, the theory of movingframes and exterior diferentiation, this dissertation aims to study canonical metrics,local structure theorems, Ricci solitons and Yamabe solitons of some types of almostKenmotsu manifolds. The structure of this dissertation is organized as follows:In Chapter1, after introducing the brief history of almost contact metric manifolds,we present many recent results in almost Kenmotsu geometry. In addition, our mainresearch results in this framework are also listed. Chapter2provides some necessarypreliminaries to read this dissertation. We first recall some important facts and propertiesof complex manifolds, emphasis on almost contact metric manifolds together with itsspecial case, namely, almost Kenmotsu manifolds.Chapter3contains many new results regarding the canonical metrics and local struc-ture theorems of (k, μ)-almost Kenmotsu manifolds (M2n+1, φ, η, ξ, g). We prove in thefirst section that, if M2n+1admits a second order parallel symmetric tensor α, then ei-ther M2n+1is locally isometric to the productHn+1(4)×Rnor α is a constant multipleof g. Next, we consider ξ-Riemann, ξ-Ricci or ξ-Weyl semisymmetric M2n+1and provethat M2n+1is locally isometric to either the hyperbolic spaceH2n+1(1) or the productspaceHn+1(4)×Rn. The harmonicities of the Riemannian curvature tensor and theWeyl conformal curvature tensor of M2n+1are also obtained respectively. In the last sec-tion, we study φ-recurrent and φ-symmetric almost Kenmotsu manifolds satisfying thegeneralized nullity conditions, generalizing some known results.In Chapter4, we focus on the study of CR-integrable almost Kenmotsu manifolds(M2n+1, φ, η, ξ, g). We prove that if M2n+1is locally symmetric, then it is locally iso-metric to either the hyperbolic spaceH2n+1(1) or the productHn+1(4)×Rn. The above result gives an afrmative answer of the well-known open problem proposed byG. Dileo and A. M. Pastore[39]in some ways. It is also proved that if M2n+1is Weylconformally flat and the scalar curvature is a constant, then M2n+1is locally isometricto the hyperbolic spaceH2n+1(1). Making use of the condition of strong η-parallelism,we obtain a Schur-type theorem for CR-integrable almost Kenmotsu manifolds in section3. Consequently, CR-integrable almost Kenmotsu space forms,(k, μ)-almost Kenmotsuspace forms and the expressions of their curvatures are obtained. The last section con-tains some classification theorems of three dimensional almost Kenmotsu manifolds withconditions of local symmetry and Qφ=φQ respectively.Ricci solitons and Yabame solitons on some types of almost Kenmotsu manifolds arestudied in Chapter5. If the Riemannian metric g of a (k, μ)-almost Kenmotsu manifold(M2n+1, g) with h=0is a gradient Ricci soliton, we prove that M2n+1is locally isometricto the productHn+1(4)×Rn. We also prove that if the metric g of a3-dimensional η-Einstein almost Kenmotsu manifold (M3, g) is a Ricci soliton, then M3is locally isometricto the hyperbolic spaceH3(1) and in this case the soliton is expanding, extending theearlier results (see A. Ghosh[44]). At last, we prove that the metric g of a3-dimensionalKenmotsu manifold is a Yamabe soliton if and only if it is an expanding Ricci soliton.