Djelloul DJEBBOURI

Dualistic structures on generalized warped products

In this work, wr gerneralize rhe dualistic structures on warped product manifold to thr dualistic structure on doubly warped prodrct and generalized warprd product manifolds. We construct a symmetric tensor field G_f_1f_2 on product manifold and we give conditions under wich G_f_1f_2 becomes a metric tensor. These a tensor field will be called the generalized warped product and then we develope an expression of curvature for the connexion of the generalized warped product in relation to those corresponding analogue of its base and fiber and warping functions. By construction a frame field in M_1*M_2 with respect to the riemannian metric G , then we calculate the Laplacizn-Beltrami operator of a function on generalized warped product wich may be expressed in terme of the local restrictions of the functions to the base and fiber. Finally, we show also that the dualistic structures on the base M_1 and fiber M_2 manifold induce on the generalized warped product M_1*M_2 a dualistic structure.