Non-diagonal metric on a product riemanniann manifold

Abstract: In this paper, We construct the symmetric tensor field $G_{f_1f_2}$ and $h_{f_1f_2}$ on a product manifold and we give conditions under which $G_{f_1f_2}$ becomes a metric tensor, theses tensors fields will be called the generalized warped product, and then we develop an expression of curvature for the connection of the generalized warped product in relation to those corresponding analogues of its base and fiber and warping functions. By constructing a frame field in $M_1\times_{f_1f_2}M_2$ with respect to the Riemannian metric $G_{f_1f_2}$ and $h_{f_1f_2}$, then we calculate the Laplacian$-$Beltrami operator of a function on a generalized warped product which may be expressed in terms of the local restrictions of the functions to the base and fiber. Finally, we conclude some interesting relationships between the geometry of the couples $(M_1,g_1)$ and $(M_2,g_2)$ and that of $(M_1\times M_2,h_{f_1f_2})$.