Geodesic
Global Solutions in Gravity. Lorentzian Signature
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Problems of the modern mathematical physics, Tome 228 (2000), pp. 168-195

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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. 
@article{TM_2000_228_a13,
     author = {M. O. Katanaev},
     title = {Global {Solutions} in {Gravity.} {Lorentzian} {Signature}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {168--195},
     publisher = {mathdoc},
     volume = {228},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2000_228_a13/}
}
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M. O. Katanaev. Global Solutions in Gravity. Lorentzian Signature. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Problems of the modern mathematical physics, Tome 228 (2000), pp. 168-195. http://geodesic.mathdoc.fr/item/TM_2000_228_a13/