Eugenio Beltrami

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 non-Euclidean geometry by modeling it on a surface of constant curvature, the pseudosphere, and in the interior of an n-dimensional unit sphere, the so-called 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 Ricci-Curbastro and Tullio Levi-Civita. Beltrami was born in 1835 in Cremona (Lombardy), then a part of the Austrian Empire, and now part of Italy. Both parents were artists Giovanni Beltrami and the Venetian Elisa Barozzi. 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. In 1868 Beltrami published two memoirs (written in Italian; French translations by J. Hoüel appeared in 1869) dealing with consistency and interpretations of non-Euclidean geometry of János Bolyai and Nikolai Lobachevsky. In his "Essay on an interpretation of non-Euclidean geometry", Beltrami proposed that this geometry could be realized on a surface of constant negative curvature, a pseudosphere.

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Related lectures (5)

Non-Euclidean Geometries

Explores non-Euclidean geometries, including hyperbolic geometry and the tractricoid model, challenging Euclidean principles and introducing projective geometry.

Introduction to Euclidean Elements

Introduces the 5th postulate in Euclidean geometry and explores non-Euclidean geometries and their implications.

Surfaces and Curvature in Geometry

Explores surfaces with variable curvature and the profound impact of curvature sign on geometric properties.

Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface.

Eugenio Beltrami

Poincaré disk model

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle. The group of orientation preserving isometries of the disk model is given by the projective special unitary group PSU(1,1), the quotient of the special unitary group SU(1,1) by its center {I, −I}.