Republication of: The physical reality of some normal spaces of Bianchi


This is an English translation of the first of two papers by Tullio Levi-Civita, first published in 1917 and 1919. The papers are remarkable as being among the earliest in which exact solutions of Einstein’s equations were derived. Of the two solutions presented, the first (republished as this Golden Oldie) is better known today as the Bertotti-Robinson solution, and the second one (republished as an accompanying Golden Oldie) is the gravitational field of an infinite cylinder. The papers have been selected by the Editors of General Relativity and Gravitation for republication in the Golden Oldies series of the journal. This republication is accompanied by an editorial note written by Malcolm MacCallum, and by a brief biography of the author, compiled from internet sources by Andrzej Krasiński.

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Correspondence to T. Levi-Civita.

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An editorial note to this paper and a biography can be found in this issue preceeding this Golden Oldie and online via doi:10.1007/s10714-011-1187-5. The companion paper to this one can be found in this issue after this Golden Oldie and online via doi:10.1007/s10714-011-1189-3.

Original paper: Realtá fisica di alcuni spazi normali del Bianchi, Rendiconti della Reale Accademia dei Lincei 26, 519–531(1917). Reprinted with the kind permission of the Accademia Nazionale dei Lincei in Rome. Translated by Robert Jantzen, Department of Mathematical Sciences, Villanova University, Villanova, PA 19085–1699, USA.

T. Levi-Civita (Deceased December 29, 1941)

Editorial responsibility: A. Krasiński, e-mail:

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Levi-Civita, T. Republication of: The physical reality of some normal spaces of Bianchi. Gen Relativ Gravit 43, 2307–2320 (2011).

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  • Normal Space
  • Covariant Derivative
  • Constant Curvature
  • Principal Curvature
  • Physical Reality