Kahlerian Structures and \mathcal{D} Homothetic Bi Warping
To study manifolds with negative curvature, Bishop and O'Neill introduced the notion of warped product as a generalization of Riemannian product [1] In 1985, using the warped product, Oubina showed that there is a one to one correspondence between Sasakian and Kahlerian structures [14] 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 [6] 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 [14] and Blair [6] 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.