  We introduce the concept of a base conformal warped product of two
pseudo-Riemannian manifolds. We also define a subclass of this structure called
as a special base conformal warped product. After, we explicitly mention many
of the relevant fields where metrics of these forms and also considerations
about their curvature related properties play important rolls. Among others, we
cite general relativity, extra-dimension, string and super-gravity theories as
physical subjects and also the study of the spectrum of Laplace-Beltrami
operators on p-forms in global analysis. Then, we give expressions for the
Ricci tensor and scalar curvature of a base conformal warped product in terms
of Ricci tensors and scalar curvatures of its base and fiber, respectively.
Furthermore, we introduce specific identities verified by particular families
of, either scalar or tensorial, nonlinear differential operators on
pseudo-Riemannian manifolds. The latter allow us to obtain new interesting
expressions for the Ricci tensor and scalar curvature of a special base
conformal warped product and it turns out that not only the expressions but
also the analytical approach used are interesting from the physical,
geometrical and analytical point of view. Finally, we analyze, investigate and
characterize possible solutions for the conformal and warping factors of a
special base conformal warped product, which guarantee that the corresponding
product is Einstein. Besides, we apply these results to a generalization of the
Schwarzschild metric.
