Produit de deux variétés munies de quelques structures.

Abstract

The product of Riemannian manifolds is one way to exhibit new Riemannian manifolds. To study manifolds with negative curvature, Bishop and O’Neill introduced the notion of warped product as a generalization of Riemannian product. By means of a natural change of the product metric, one can widely construct remarkable structures from the structures of the two factors. Our goal is to construct some structures on the product of two Riemannian manifolds by providing both factors with some essential structures. The metric called D-homothetic bi-warping that we introduced on the product of a Riemannian manifold with an almost contact metric manifold as a generalization of warped product and D-homothetic warping allows us to construct: - A family of Kählerian structures starting from a Sasakian manifold. - A 1-parameter family of conformal Kähler structures with a cosymplectic or Kenmotsu structure. - A 1-parameter family of Kenmotsu structures from a single Sasakian manifold. - A quaternionic structure using a Sasakian 3-structure. - New generalized Kähler manifolds starting from both classical almost contact metric and almost Kählerian manifolds. On the other hand, we construct an almost contact metric 3-structure and an almost quaternionic metric structure starting from an almost contact manifold almost hermitian structure. Next, we construct an almost quaternionic metric structures on the product of two almost contact manifold almost hermitian structure.