Global Solutions in Gravity. Lorentzian Signature

A constructive method of conformal blocks is developed for the construction of global solutions for two-dimensional metrics having one Killing vector. The method is proved to yield a smooth universal covering space with a smooth pseudo-Riemannian metric. The Schwarzschild, Reisner–Nordström solutions, extremal black hole, dilaton black hole, and constant curvature surfaces are considered as examples.