Eugenio Beltrami
Eugenio Beltrami  

Eugenio Beltrami


Born 
Cremona, Lombardy, Kingdom of Lombardy–Venetia, Austrian Empire 
16 November 1835
Died  18 February 1900 Rome, Kingdom of Italy 
(aged 64)
Residence  Italy 
Nationality  Italian 
Alma mater  Ghislieri College, Pavia (no degree) 
Known for 
Beltrami equation Beltrami identity Beltrami's theorem Laplace–Beltrami operator Beltrami vector field Beltrami–Klein model 
Scientific career  
Fields  Mathematician 
Institutions 
University of Bologna University of Pisa University of Rome University of Pavia 
Academic advisors  Francesco Brioschi 
Doctoral students  Giovanni Frattini 
Eugenio Beltrami (16 November 1835 – 18 February 1900) was an Italian mathematician notable for his work concerning differential geometry and mathematical physics. His work was noted especially for clarity of exposition. He was the first to prove consistency of nonEuclidean geometry by modeling it on a surface of constant curvature, the pseudosphere, and in the interior of an ndimensional unit sphere, the socalled Beltrami–Klein model. He also developed singular value decomposition for matrices, which has been subsequently rediscovered several times. Beltrami's use of differential calculus for problems of mathematical physics indirectly influenced development of tensor calculus by Gregorio RicciCurbastro and Tullio LeviCivita.
Life
Beltrami was born in Cremona in Lombardy, then a part of the Austrian Empire, and now part of Italy. He began studying mathematics at University of Pavia in 1853, but was expelled from Ghislieri College in 1856 due to his political opinions—he was sympathetic with the Risorgimento. During this time he was taught and influenced by Francesco Brioschi. He had to discontinue his studies because of financial hardship and spent the next several years as a secretary working for the Lombardy–Venice railroad company. He was appointed to the University of Bologna as a professor in 1862, the year he published his first research paper. Throughout his life, Beltrami had various professorial jobs at the universities of Pisa, Rome and Pavia. From 1891 until the end of his life Beltrami lived in Rome. He became the president of the Accademia dei Lincei in 1898 and a senator of the Kingdom of Italy in 1899.
Contributions to nonEuclidean geometry
In 1868 Beltrami published two memoirs (written in Italian; French translations by J. Hoüel appeared in 1869) dealing with consistency and interpretations of nonEuclidean geometry of Bolyai and Lobachevsky. In his "Essay on an interpretation of nonEuclidean geometry", Beltrami proposed that this geometry could be realized on a surface of constant negative curvature, a pseudosphere. For Beltrami's concept, lines of the geometry are represented by geodesics on the pseudosphere and theorems of nonEuclidean geometry can be proved within ordinary threedimensional Euclidean space, and not derived in an axiomatic fashion, as Lobachevsky and Bolyai had done previously. In 1840, Minding already considered geodesic triangles on the pseudosphere and remarked that the corresponding "trigonometric formulas" are obtained from the corresponding formulas of spherical trigonometry by replacing the usual trigonometric functions with hyperbolic functions; this was further developed by Codazzi in 1857, but apparently neither of them noticed the association with Lobachevsky's work. In this way, Beltrami attempted to demonstrate that twodimensional nonEuclidean geometry is as valid as the Euclidean geometry of the space, and in particular, that Euclid's parallel postulate could not be derived from the other axioms of Euclidean geometry. It is often stated that this proof was incomplete due to the singularities of the pseudosphere, which means that geodesics could not be extended indefinitely. However, John Stillwell remarks that Beltrami must have been well aware of this difficulty, which is also manifested by the fact that the pseudosphere is topologically a cylinder, and not a plane, and he spent a part of his memoir designing a way around it. By a suitable choice of coordinates, Beltrami showed how the metric on the pseudosphere can be transferred to the unit disk and that the singularity of the pseudosphere corresponds to a horocycle on the nonEuclidean plane. On the other hand, in the introduction to his memoir, Beltrami states that it would be impossible to justify "the rest of Lobachevsky's theory", i.e. the nonEuclidean geometry of space, by this method.
In the second memoir published during the same year (1868), "Fundamental theory of spaces of constant curvature", Beltrami continued this logic and gave an abstract proof of equiconsistency of hyperbolic and Euclidean geometry for any dimension. He accomplished this by introducing several models of nonEuclidean geometry that are now known as the Beltrami–Klein model, the Poincaré disk model, and the Poincaré halfplane model, together with transformations that relate them. For the halfplane model, Beltrami cited a note by Liouville in the treatise of Monge on differential geometry. Beltrami also showed that ndimensional Euclidean geometry is realized on a horosphere of the (n + 1)dimensional hyperbolic space, so the logical relation between consistency of the Euclidean and the nonEuclidean geometries is symmetric. Beltrami acknowledged the influence of Riemann's groundbreaking Habilitation lecture "On the hypotheses on which geometry is based" (1854; published posthumously in 1868).
Although today Beltrami's "Essay" is recognized as very important for the development of nonEuclidean geometry, the reception at the time was less enthusiastic. Cremona objected to perceived circular reasoning, which even forced Beltrami to delay the publication of the "Essay" by one year. Subsequently, Felix Klein failed to acknowledge Beltrami's priority in construction of the projective disk model of the nonEuclidean geometry. This reaction can be attributed in part to the novelty of Beltrami's reasoning, which was similar to the ideas of Riemann concerning abstract manifolds. J. Hoüel published Beltrami's proof in his French translation of works of Lobachevsky and Bolyai.
Works
 Beltrami, Eugenio (1868). "Saggio di interpretazione della geometria noneuclidea". Giornale di Mathematiche. VI: 285–315.
 Beltrami, Eugenio (1868). "Teoria fondamentale degli spazii di curvatura costante". Annali. di Mat., ser II. 2: 232–255. doi:10.1007/BF02419615.
 Opere matematiche di Eugenio Beltrami pubblicate per cura della Facoltà di scienze della r. Università di Roma (volumes 1–2) (U. Hoepli, Milano, 1902–1920)^{[1]}
 Same edition, vols. 1–4
References
 ^ Study, E. (1909). "Review: Opere matematiche di Eugenio Beltrami". Bull. Amer. Math. Soc. 16 (3): 147–149. doi:10.1090/s000299041909018828.
 Stillwell, John (1996). Sources of hyperbolic geometry. History of Mathematics. 10. Providence, R.I.: American Mathematical Society. ISBN 9780821805299. MR 1402697
 Jeremy Gray, Poincaré and Klein — Groups and Geometries. In 1830–1930: a Century of Geometry (ed L.Boi, D.Flament and J.M.Salanskis), Springer, 1992, 35–44
External links
 O'Connor, John J.; Robertson, Edmund F., "Eugenio Beltrami", MacTutor History of Mathematics archive, University of St Andrews.
 Eugenio Beltrami at the Mathematics Genealogy Project
 Eugenio Beltrami  Œuvres complètes GallicaMath