Curvature in Special Base Conformal Warped Products

TL;DR

This paper introduces a new class of warped product manifolds called special base conformal warped products, derives their curvature properties, and explores conditions under which these manifolds are Einstein, with applications to general relativity.

Contribution

It defines special base conformal warped products, derives explicit curvature formulas, and investigates Einstein conditions, extending geometric and physical applications.

Findings

Derived explicit Ricci and scalar curvature formulas for these manifolds.

Identified conditions for the manifolds to be Einstein.

Applied results to generalize the Schwarzschild metric.

Abstract

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.