Query: quasi sasakian d-homothetic deformation

title: Curvature Properties of Quasi Para Sasakian Manifolds
abstract: The present paper is devoted to quasi Para Sasakian manifolds. Basic properties of such manifolds are obtained and general curvature identities are investigated. Next it is proved that if $M$ is quasi Para Sasakian manifold of constant curvature $K$ Then $K$ \leq 0$ and (i)$if $K=0$ the manifold is paracosymplectic, (ii) if $K<0$ the quasi para Sasakian structure of $M$ is obtained by a homothetic deformation of a para Sasakian structure.

title: D homothetic deformation of an h Einstein Para Sasakian manifold
abstract: The main purpose of this paper is to study the theory of Dhomothetic deformation of an e Einstein Para Sasakian manifold.  A deformation technique is employed to solve the resulting problem. We provide its application in the general theory of relativity.  Section 2 deals with recurrent and symmetric Para Sasakian Manifolds. In section 3 we have defined and studied projectively symmetric Para Sasakianmanifold. The notion of ph holomorphic sectional curvature in an e Einstein Para Sasakian manifold has been delineated in the section 4. In this very section, we shall study the relevant commutation formulae which give rise to the required curvature tensor. The obtained results were compared with the previous works (1 3) in this field andwere found to be in good agreement. The study concludes the following results: 1. If n dimensional Para Sasakian manifold is to be recurrent projective symmetric then the recurrent projective curvature tensor vanishes identically. 2. If n dimensional Para Sasakian manifold is to be projective semi symmetric then the semi recurrent projective curvature tensor vanishes identically. 3. If n dimensional Para Sasakian manifold is to be projective scalar symmetric then the scalar projective curvature vanishes identically. 4. If D homothetic deformation of e Einstein Para Sasakian manifold is constant ph holomorphic sectional curvature thenwe shall obtain the value of scalar

title: D Homothetic deformation of LP Sasakian manifolds
abstract: The object of the present paper is to study a transformation called D homothetic deformation of LP Sasakian manifolds. Among others it is shown that in an LP Sasakian manifold, the Ricci operator Q commutes with the structure tensor ph. We also discuss about the invariance of e Einstein manifolds, ph sectional curvature, the locally ph Ricci symmetry and e parallelity of the Ricci tensor under the D homothetic deformation. Finally, we give an example of such a manifold

title: ON D HOMOTHETIC DEFORMATION OF TRANS SASAKIAN STRUCTURE
abstract: Abs t rac t The object of the present paper is to study a transformation called Dhomothetic deformation of trans Sasakian structure. Among others it is shown that in a trans Sasakian manifold, the Ricci operator Q does not commute with the structure tensor and the operator Q(j> Q is conformal under a D homothetic deformation. Also the sectional curvature of a trans Sasakian manifold is conformal under such a deformation. Some non trivial examples of trans Sasakian (non Sasakian) manifolds with global vector fields are obtained.

title: On D homothetic deformation of trans Sasakian structure
abstract: The object of the present paper is to study a transformation called D homothetic deformation of trans Sasakian structure. Among others it is shown that in a trans Sasakian manifold, the Ricci operator Q does not commute with the structure tensor ph and the operator Qph phQ is conformal under a D homothetic deformation. Also the ph sectional curvature of a trans Sasakian manifold is conformal under such a deformation. Some non trivial examples of trans Sasakian (non Sasakian) manifolds with global vector fields are obtained.

title: On Quasi Sasakian Manifolds
abstract: In this paper we study three dimensional quasi Sasakian manifolds admitting the Schouten van Kampen connection. Also, we study D homothetic deformations on three dimensional quasi Sasakian manifolds admitting Schouten van connection and projectively flat three dimensional quasi Sasakian manifolds admitting scv connection.

title: D A Homothetic Deformation of K Contact Manifolds
abstract: We study Da homothetic deformations of K contact manifolds. We prove that Da homothetically deformed K contact manifold is a generalized Sasakian space form if it is conharmonically flat. Further, we find expressions for scalar curvature of Da homothetically deformed K contact manifolds.

title: Einstein and \eta $e Einstein Sasakian submanifolds in spheres
abstract: The aim of this paper is to study Sasakian immersions of compact Sasakian manifolds into the odd dimensional sphere equipped with the standard Sasakian structure. We obtain a complete classification of such manifolds in the Einstein and \eta $e Einstein cases when the codimension of the immersion is 4. Moreover, we exhibit infinite families of compact Sasakian \eta $e Einstein manifolds which cannot admit a Sasakian immersion into any odd dimensional sphere. Finally, we show that, after possibly performing a \mathcal {D}$D homothetic deformation, a homogeneous Sasakian manifold can be Sasakian immersed into some odd dimensional sphere if and only if S is regular and either S is simply connected or its fundamental group is finite cyclic.

title: Closed orbits of Reeb fields on Sasakian manifolds and elliptic curves on Vaisman manifolds
abstract: A compact complex manifold V is called Vaisman if it admits a Hermitian metric which is conformal to a Kahler one, and a non isometric conformal action by \mathbb {C} It is called quasi regular if the \mathbb {C} action has closed orbits. In this case the corresponding leaf space is a projective orbifold, called the quasi regular quotient of V. It is known that the set of all quasi regular Vaisman complex structures is dense in the appropriate deformation space. We count the number of closed elliptic curves on a Vaisman manifold, proving that their number is either infinite or equal to the sum of all Betti numbers of a Kahler orbifold obtained as a quasi regular quotient of V. We also give a new proof of a result by Rukimbira showing that the number of Reeb orbits on a Sasakian manifold M is either infinite or equal to the sum of all Betti numbers of a Kahler orbifold obtained as an $S^1$ quotient of M.

title: Kahlerian Structures and \mathcal{D} Homothetic Bi Warping
abstract: To study manifolds with negative curvature, Bishop and O'Neill introduced the notion of warped product as a generalization of Riemannian product  In 1985, using the warped product, Oubina showed that there is a one to one correspondence between Sasakian and Kahlerian structures  Recently, building on the work of Tanno [16] (the homothetic deformation on contact metric manifold) Blair [6] introduced the notion of D homothetic warping. He used it for generating further results and examples of various structures. In particular, he showed in another way that there is a one to one correspondence between Sasakian and Kahlerian structures. Here by generalizing the D homothetic warping and following what made Blair in  we exceed this correspondence and we show that every Sasakian manifold M generates a one parameter family of Kahlerian manifolds, thereby generalizing the results of Oubina  and Blair  On the other hand, we define a two parameter family of conformally Kahler manifolds structures on the product manifold IxM of an open interval and a cosymplectic or Kenmotsu manifold M (Theorem 7) which is the first main result of the present paper. This text is organized in the following way. Section 2 is devoted to the background of the structures which will be used in the sequel.