  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 $(M_1\times
M_2,h_{f_1f_2})$.
