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13-542: It was founded on December 7, 1922, by Luigi Bianchi , Vito Volterra , and most notably, Salvatore Pincherle , who became the Union's first President. Salvatore Pincherle , professor at the University of Bologna , sent on 31 March 1922 a letter to all Italian mathematicians in which he planned the establishment of a national mathematical society. The creation was inspired by similar initiatives in other countries, such as

26-547: A leading differential geometer who is today best remembered for his seminal contributions to topology , and Ulisse Dini , a leading expert on function theory . Bianchi was also greatly influenced by the geometrical ideas of Bernhard Riemann and by the work on transformation groups of Sophus Lie and Felix Klein . Bianchi became a professor at the Scuola Normale Superiore in Pisa in 1896, where he spent

39-533: A subgroup of P S L 2 ( C ) {\displaystyle PSL_{2}(\mathbb {C} )} , a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space H 3 {\displaystyle \mathbb {H} ^{3}} . The quotient space M d = P S L 2 ( O d ) ∖ H 3 {\displaystyle M_{d}=PSL_{2}({\mathcal {O}}_{d})\backslash \mathbb {H} ^{3}}

52-734: Is a non-compact, hyperbolic 3-fold with finite volume, which is also called Bianchi orbifold . An exact formula for the volume, in terms of the Dedekind zeta function of the base field Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} , was computed by Humbert as follows. Let D {\displaystyle D} be the discriminant of Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} , and Γ = S L 2 ( O d ) {\displaystyle \Gamma =SL_{2}({\mathcal {O}}_{d})} ,

65-620: Is essentially the same thing as classifying, up to isomorphism , the three-dimensional real Lie algebras . This complements the earlier work of Lie himself , who had earlier classified the complex Lie algebras. Through the influence of Luther P. Eisenhart and Abraham Haskel Taub , Bianchi's classification later came to play an important role in the development of the theory of general relativity . Bianchi's list of nine isometry classes, which can be regarded as Lie algebras, Lie groups, or as three-dimensional homogeneous (possibly nonisotropic) Riemannian manifolds, are now often called collectively

78-419: Is the ring of integers of the imaginary quadratic field Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} . The groups were first studied by Bianchi  ( 1892 ) as a natural class of discrete subgroups of P S L 2 ( C ) {\displaystyle PSL_{2}(\mathbb {C} )} , now termed Kleinian groups . As

91-564: The Bianchi groups . In 1902, Bianchi rediscovered what are now called the Bianchi identities for the Riemann tensor , which play an even more important role in general relativity . (They are essential for understanding the Einstein field equation .) According to Tullio Levi-Civita , these identities had first been discovered by Ricci in about 1889, but Ricci apparently forgot all about

104-791: The Société mathématique de France (1872), the Deutsche Mathematiker-Vereinigung (1891), the American Mathematical Society (1891) and, above all, the International Mathematical Union (1920). The most important Italian mathematicians of the time - among all Luigi Bianchi and Vito Volterra - encouraged Pincherle's initiative also by personally sending articles for the future Bulletin; overall, about 180 mathematicians replied to Pincherle's letter. On December 7 of

117-405: The discontinuous action on H {\displaystyle {\mathcal {H}}} , then The set of cusps of M d {\displaystyle M_{d}} is in bijection with the class group of Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} . It is well known that every non-cocompact arithmetic Kleinian group

130-507: The matter, which led to Bianchi's rediscovery. However, the contracted Bianchi identities , which are sufficient for the proof that the divergence of the Einstein tensor always vanishes, had been published by Aurel Voss in 1880. Bianchi group In mathematics , a Bianchi group is a group of the form where d is a positive square-free integer . Here, PSL denotes the projective special linear group and O d {\displaystyle {\mathcal {O}}_{d}}

143-407: The remainder of his career. At Pisa, his colleagues included the talented Ricci . In 1890, Bianchi and Dini supervised the dissertation of the noted analyst and geometer Guido Fubini . In 1898, Bianchi worked out the Bianchi classification of nine possible isometry classes of three-dimensional Lie groups of isometries of a (sufficiently symmetric) Riemannian manifold . As Bianchi knew, this

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156-718: The same year the first meeting was held. In 1928 the Italian Mathematical Union hosted the International Congress of Mathematicians in Bologna . The Union's journal is the Bollettino dell'Unione Matematica Italiana , which contains two sections: one for research papers, and one for expository articles. The Italian Mathematical Union awards the following prizes: Luigi Bianchi Luigi Bianchi (18 January 1856 – 6 June 1928)

169-514: Was an Italian mathematician . He was born in Parma , Emilia-Romagna , and died in Pisa . He was a leading member of the vigorous geometric school which flourished in Italy during the later years of the 19th century and the early years of the twentieth century. Like his friend and colleague Gregorio Ricci-Curbastro , Bianchi studied at the Scuola Normale Superiore in Pisa under Enrico Betti ,

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