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

We consider general relativity with a cosmological constant minimally coupled to an electromagnetic field and assume that a four-dimensional space-time manifold is the warped product of two surfaces with Lorentzian and Euclidean signature metrics. Einstein's equations imply that at least one of the surfaces must be of constant curvature. It means that the symmetry of the metric arises as the consequence of the equations of motion ("spontaneous symmetry emergence"). We give a classification of global solutions in two cases: (i) both surfaces are of a constant curvature and (ii) the Riemannian surface is of a constant curvature. The latter case includes spherically symmetric solutions [a sphere S² with a S O (3 )-symmetry group], planar solutions [two-dimensional Euclidean space R² with an I O (2 )-symmetry group], and hyperbolic solutions [a two-sheeted hyperboloid H² with a S O (1 ,2 )-symmetry]. Totally, we get 37 topologically different solutions. There is a new one among them, which describes the changing topology of space in time already at the classical level.