Global properties of warped solutions in general relativity with an electromagnetic field and a cosmological constant. II.

Abstract

We consider general relativity with a cosmological constant minimally coupled to the electromagnetic field and assume that the four-dimensional spacetime manifold is a warped product of two surfaces with Lorentzian and Euclidean signature metrics. Field equations imply that at least one of the surfaces must be of constant curvature leading to the symmetry of the metric (``spontaneous symmetry emergence''). We classify all global solutions in the case when the Lorentzian surface is of constant curvature (case C). These solutions are invariant with respect to the Lorentz $\mathbb{S}\mathbb{O}(1,2)$ or Poincare $\mathbb{I}\mathbb{O}(1,1)$ groups acting on the Lorentzian surface.