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 FranceLezioni sulla teoria dei numeri algebriciLezioni 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