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Luigi Bianchi (18 January 1856 – 6 June 1928) was an
Italian Italian(s) may refer to: * Anything of, from, or related to the people of Italy over the centuries ** Italians, an ethnic group or simply a citizen of the Italian Republic or Italian Kingdom ** Italian language, a Romance language *** Regional It ...
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.


Biography

Like his friend and colleague
Gregorio Ricci-Curbastro Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician. He is most famous as the discoverer of tensor calculus. With his former student Tullio Levi-Civita, he wrote his most famous single publication, a pioneering work on the ...
, Bianchi studied at the Scuola Normale Superiore in Pisa under
Enrico Betti Enrico Betti Glaoui (21 October 1823 – 11 August 1892) was an Italian mathematician, now remembered mostly for his 1871 paper on topology that led to the later naming after him of the Betti numbers. He worked also on the theory of equations, gi ...
, 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 Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
and by the work on transformation groups of Sophus Lie and
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
. Bianchi became a professor at the Scuola Normale Superiore in Pisa in 1896, where he spent 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 In mathematics, the Bianchi classification provides a list of all real 3-dimensional Lie algebras (up to isomorphism). The classification contains 11 classes, 9 of which contain a single Lie algebra and two of which contain a continuum-sized fam ...
of nine possible isometry classes of three-dimensional
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
s of isometries of a (sufficiently symmetric) Riemannian manifold. As Bianchi knew, this 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 the Bianchi groups. In 1902, Bianchi rediscovered what are now called the Bianchi identities for the
Riemann tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
, which play an even more important role in general relativity. (They are essential for understanding the
Einstein field equation In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the for ...
.) According to
Tullio Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus ( tensor calculus) and its applications to the theory of relativity, but who also made signi ...
, these identities had first been discovered by Ricci in about 1889, but Ricci apparently forgot all about the matter, which led to Bianchi's rediscovery. Where in the footnote one can read: These identities were staded without proof by PADOVA, on the strength of a verbal communication of RICCI (cf. 'Sulle deformazioni infinitesime', in ''Rend. della R. Acc. dei Lincei'', (4), Vol. V (first half-year, 1889, p. 176). They were then forgotten even by Ricci himself. BIANCHI rediscovered them and published a proof obtained by direct calculation in 1902 (''Ibid''., (5), Vol. XI (first half-year, 1902, pp. 3-7). However, the
contracted Bianchi identities In general relativity and tensor calculus, the contracted Bianchi identities are: : \nabla_\rho _\mu = \nabla_ R where _\mu is the Ricci tensor, R the scalar curvature, and \nabla_\rho indicates covariant differentiation. These identities are na ...
, which are sufficient for the proof that the divergence of the
Einstein tensor In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein fie ...
always vanishes, had been published by
Aurel Voss Aurel Voss (7 December 1845 – 19 April 1931) was a German mathematician, best known today for his contributions to geometry and mechanics. He served as president of the German Mathematical Society for the 1898 term. He was a professor at the Uni ...
in 1880.


Publications


Articles

*


Books

* * *''Lezioni sulla teoria dei gruppi di sostituzioni e delle equazioni algebriche secondo Galois'', Pisa 1899 *''Lezioni sulla teoria delle funzioni di variabile complessa e delle funzioni ellittiche'' 1916 * *''Lezioni sulla teoria dei numeri algebrici e principi d'aritmetica analitica'', 1921


See also

*
Contracted Bianchi identities In general relativity and tensor calculus, the contracted Bianchi identities are: : \nabla_\rho _\mu = \nabla_ R where _\mu is the Ricci tensor, R the scalar curvature, and \nabla_\rho indicates covariant differentiation. These identities are na ...


References


Sources

* * * *


Further reading

*


External links


Robert T. Jantzen
(Villanova University) offers translations of some of Bianchi's papers, plus a biography of Bianchi.

PDF copy a
Gallica, Bibliothèque Nationale de France

Lezioni sulla teoria dei numeri algebrici

Lezioni sulla teoria delle funzioni di variabile complessa e delle funzioni ellittiche
(Images at Cornell)
Lezioni sulla teoria dei gruppi di sostituzioni e delle equazioni algebriche secondo Galois
(Images at Cornell)
Vorlesungen über differentialgeometrie
(PDF/DjVu at archive.org) {{DEFAULTSORT:Bianchi, Luigi 1856 births 1928 deaths People from Parma Differential geometers 19th-century Italian mathematicians 20th-century Italian mathematicians Members of the Senate of the Kingdom of Italy 20th-century Italian politicians Members of the Göttingen Academy of Sciences and Humanities