Doubly warped product manifolds: investigations through the concircular curvature tensor and relativistic applications

Abstract

This paper presents a description of a doubly warped product manifold under several conditions imposed on the concircular curvature tensor. We prove that the factor manifolds of a concircularly flat (symmetric) doubly warped product manifold possess constant sectional curvature, and are themselves concircularly flat. In the flatness scenario, a doubly warped product manifold reduces to a singly warped product manifold. We establish that the factor manifolds of a doubly warped product manifold with a harmonic concircular curvature tensor are Einstein manifolds; moreover, a harmonic concircular curvature tensor is shown to force the scalar curvature to be constant, which is a stronger conclusion than the one available for the projective curvature tensor. We further clarify the position of concircular flatness among the conformal and projective flatness conditions and illustrate the theory with an explicit example. In the application section, we prove that a concircularly flat (symmetric) Generalized Robertson–Walker (GRW) space-time is an Einstein perfect fluid which is static and obeys the equation of state p = −ρ, so that it behaves as a dark-energy (cosmological constant) era and satisfies the weak and dominant energy conditions.