# Luigi Bianchi

### Quick Info

Born
18 January 1856
Parma (now Italy)
Died
6 June 1928
Pisa, Italy

Summary
Luigi Bianchi made important contributions to differential geometry.

### Biography

Luigi Bianchi's father was Francesco Saverio Bianchi who had been born on 24 November 1827 at Piacenza. Saverio Bianchi had studied law at the University of Parma, graduating in July 1848. His eldest son, Ferdinando, had been born at Parma on 6 August 1854. In January 1856 his second son, Luigi (the subject of this biography) was born at Parma and later in the same year Saverio was appointed Professor of Civil Law at the University of Parma. Ferdinando and Luigi grew up in one of the leading families of the city of Parma for Saverio not only held important positions in the University (for example Dean of the Faculty of Law from 1868 to 1873) but he also held many public positions. He was elected city councilor and then mayor of Parma in 1869 and was for many years chairman of the Provincial Council and president of the Civic Hospice.

Luigi Bianchi was educated at a school in Parma as was his elder brother Ferdinando who followed his father into the legal profession studying civil law. Luigi, however, was attracted to mathematics and he took the competitive entrance examinations for the Scuola Normale Superiore of Pisa in November 1873. In fact not only did Luigi leave Parma in 1873, but so did his father who was named to the chair of Civil Law at the University of Siena in this year. At the University of Pisa, Luigi Bianchi studied under Enrico Betti and Ulisse Dini and graduated with the highest distinction from Pisa on 30 November 1877. He completed a doctoral dissertation on applicable surfaces. He then continued his research at a number of universities throughout Europe, first at Pisa where he remained until 1879, then at Munich and finally at Göttingen where he studied with Felix Klein [17]:-
In 1879 appeared a paper on the centro-surface of a helicoid in the Giornale di Matematiche; and from that time onwards until shortly before his death a series of papers of first-rate merit appeared each year from his prolific pen in spite of the many distractions caused by domestic troubles, the duties of his professorship, and his labours as an editor.
After his return to Italy in 1881, Bianchi was appointed to a professorship at the Scuola Normale Superiore at Pisa. He taught differential geometry at the University of Pisa where he was promoted a number of times, first to extraordinary professor in differential geometry, then after a competitive examination to extraordinary professor in projective geometry in 1886. Also in 1886 he married; the Bianchi's had five sons. In the same year he was also appointed as extraordinary professor of analytic geometry, becoming a full professor of analytic geometry in 1890. We note that his father Saverio Bianchi left the academic world in 1880 when he moved to the judiciary as an advisor of the Court of Cassation of Turin, a position he held until 1882 after which he became an advisor of the Court of Cassation in Rome. When Saverio Bianchi resigned his chair at Siena in 1880, Luigi's brother Ferdinando was appointed as his successor.

Luigi Bianchi made important contributions to differential geometry. He discovered all the geometries of Riemann that allow a continuous group of transformations. His work on non-euclidean geometries was used by Einstein in his general theory of relativity. Dirk Struik gives the following overview of Bianchi's mathematical contributions [18]:-
Bianchi's productive career begins with the applicability of surfaces (1878) and in rapid succession follow papes on surfaces of constant curvature, orthogonal surfaces, Weingarten surfaces, minimal surfaces, and congruences. Bianchi also turns to non-Euclidean geometry, taking up the ideas of Beltrami; the surfaces of curvature zero in such geometry draw his special attention. Following Bäcklund he invents transformations to pass from one set of surfaces of special character to another. In the work of ... Bianchi partial differential equations play an important role. ... The principal fame of Darboux and Bianchi lies in their beautiful textbooks, in which they combine their on results with those of their predecessors. Bianchi's 'Lezioni di geometria differenziale' (1893), a new edition of autographed lectures published in 1886, is a systematic treatise of the theory of curves and surfaces, with special attention paid to more-dimensional geometry. It became known in wider circles through the German translation of M Lukat (1899).
Bianchi's mathematical contributions are also described by Hilton in [17]. He writes:-
The greater part of [his early] work is on the properties of surfaces. His methods were based on the theory of the two fundamental differential quadratic form of Gauss. ... he never published anything which did not contain a real contribution to knowledge. Other subjects which attracted him at time were Lie's theory of continuous groups and the theory of groups of substitutions of the type x' = (ax + b)/(cx + d). But even while he was writing on these topics, papers on surfaces were still appearing, and differential geometry absorbed nearly all his attention for the last twelve years or so of his life.
Perhaps Bianchi's most famous paper was the 92-page On the three-dimensional spaces which admit a continuous group of motions published in 1897. This importance of this paper can be judged from the fact that an English translation was published as recently as 2002. The original paper was reviewed several times. Here is the review written by Gino Loria:-
An $n$-dimensional space $\mathbb{R}^{n}$ is completely defined by the square of its line element $ds^{2} = ...$ . In particular, with the help of this line element, one can solve all applicability problems of $\mathbb{R}^{n}$ onto itself, i.e., the problem of motions in $\mathbb{R}^{n}$. The author restricts himself to the spaces possessing continuous motions. Such motions form a group depending on the finite number r of parameters. This group contains a continuous group $G_{r}$ defined by r infinitesimal transformations $X_{1}f, ... , X_{r} f$. The problem of finding spaces possessing a continuous group of motions is identical to the problem of finding all possible $ds^{2}$ admitting a group $G_{r} = [X_{1} f, ... , X_{r} f]$ of transformations. Although the formulas for solving this problem were essentially given already by W Killing (1892), and Lie, it was still not completely solved. As preparation for the general solution, the author applies the Lie-Killing methods for finding all three-dimensional spaces in which the motions of figures with given degrees of freedom are possible - this is enough to outline the goal and the general train of thought of Bianchi's work. The importance of his results is known to every reader who is familiar with the awards of the Royal Jablonowski Society for 1901; their citation states that the strength of the methods and the elegance of the solutions need not be pointed out when we are talking about a paper whose author is Bianchi.
Bianchi wrote a number of influential treatises which [17]:-
... show him to be not only a scientific genius but also a master of clear and attractive style.
In particular he wrote: Lezioni di geometria differenziale (1886); Lezioni sulla teoria dei gruppi di sostituzioni e delle equazioni algebriche secondo Galois (1900); Lezioni sulla teoria aritmetica delle forme quadratiche binarie e ternarie (1912); Lezionidi geometria analitica (1915); Lezioni sulla teoria delle funzioni di variabile complessa e delle funzioni ellittiche (1916); Lezioni sulla teoria dei gruppi continui finiti di trasformazioni (1918); and Lezioni sulla teoria dei numeri algebrici e principii di geometria analitica (1923). Hilton writes [17]:-
For so prolific a writer the standard was extraordinarily high; he never published anything which did not contain a real contribution to knowledge.
We should also mention his editorial duties as an editor of Annali di Matematica pura ed applicata. During his time as editor other mathematicians who shared the editorial duties with him include Luigi Cremona, Ulisse Dini, Corrado Segre, Salvatore Pincherle, Tullio Levi-Civita and Francesco Severi.

Bianchi received many honours including being made a corresponding member of the Accademia dei Lincei in 1887 and a fellow of the Academy in 1893. He became Senator of the Kingdom of Italy in 1924. He was elected an honorary member of the London Mathematical Society and he was similarly honoured by many other Societies.

### References (show)

1. E Carruccio, G Piccoli, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. L Bianchi, Opere. Vol. I : Parte prima (Edizioni Cremonese, Rome, 1952).
3. L Bianchi, Opere. Vol. I : Parte seconda (Edizioni Cremonese, Rome, 1953).
4. L Bianchi, Opere. Vol. II : Applicabilità e problemi di deformazione (Edizioni Cremonese, Rome, 1953).
5. L Bianchi, Opere. Vol. III : Sistemi tripli ortogonali (Edizioni Cremonese, Rome, 1955).
6. L Bianchi, Opere. Vol. IV : Deformazioni delle quadriche, teoria delle trasformazioni delle superficie applicabili sulle quadriche. Parte prima (Edizioni Cremonese, Rome, 1956).
7. L Bianchi, Opere. Vol. IV : Deformazioni delle quadriche, teoria delle trasformazioni delle superficie applicabili sulle quadriche. Parte seconda (Edizioni Cremonese, Rome, 1956).
8. L Bianchi, Opere. Vol. V : Trasformazioni delle superficie e delle curve (Edizioni Cremonese, Rome, 1957).
9. L Bianchi, Opere. Vol. VI : Congruenze di rette e di sfere e loro deformazioni (Edizioni Cremonese, Rome, 1957).
10. L Bianchi, Opere. Vol. VII : Problemi di rotolamento (Edizioni Cremonese, Rome, 1957).
11. L Bianchi, Opere. Vol. VIII : Classi speciali di superficie (Edizioni Cremonese, Rome, 1958).
12. L Bianchi, Opere. Vol. IX : Geometria degli spazi di Riemann and Vol. X : Ricerche varie (Edizioni Cremonese, Rome, 1958).
13. L Bianchi, Opere. Vol. XI : Corrispondenza (Edizioni Cremonese, Rome, 1959).
14. W Blaschke, Luigi Bianchi e la geometria differenziale, Ann. Scuola Norm. Super. Pisa (3) 8 (1954), 43-52.
15. G Fubini, Luigi Bianchi e la sua opera scientifica, Annali di matematica (4) 6 (1928-29), 45-83.
16. G Fubini, Commemorazione di Luigi Bianchi, Rendiconti della Accademia nazionale dei Lincei, Classe di scienze fisiche matematiche e naturali (6a) 10 (1929), xxxiv-xliv.
17. H Hilton, Luigi Bianchi, J. London Math. Soc. 4 (1) (1929), 79-80.
18. D J Struik, Outline of a History of Differential Geometry (II), Isis 20 (1) (1933), 161-191.
19. A Sym, Luigi Bianchi: a short biography, in Nonlinearity & geometry, Warsaw, 1995 (PWN, Warsaw, 1998), 9.
20. P Vincensini, Vue d'ensemble sur l'oeuvre géométrique de Luigi Bianchi, Univ. e Politec. Torino. Rend. Sem. Mat. 16 (1956/1957), 115-157.

### Honours (show)

Honours awarded to Luigi Bianchi

### Cross-references (show)

Written by J J O'Connor and E F Robertson
Last Update May 2010