Eugenio Beltrami (November 16, 1835,
CremonaCremona is a city in northern Italy, situated in Lombardy, on the left bank of the Po River in the middle of the Pianura Padana...
– February 18, 1900,
RomeRome is the capital of Italy and the country's largest and most populated municipality , with over 2.7 million residents in , while the population of the urban area is estimated by Eurostat to be 3.46 million. The metropolitan area of Rome is estimated by OECD to have a population of 3.7 million...
) was an
ItalianItaly , officially the Italian Republic , is a country located on the Italian Peninsula in Southern Europe and on the two largest islands in the Mediterranean Sea, Sicily and Sardinia. Italy shares its northern, Alpine boundary with France, Switzerland, Austria and Slovenia...
mathematician notable for his work in differential geometry and
mathematical physicsMathematical physics is the scientific discipline concerned with the interface of mathematics and physics. The Journal of Mathematical Physics defines it as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the...
. Beltrami's work was noted for his modern approach and the clarity of exposition. He was the first to prove consistency of
non-Euclidean geometryA non-Euclidean geometry is characterized by a non-vanishing Riemann curvature tensor. Examples of non-Euclidean geometries include the hyperbolic and elliptic geometry, which are contrasted with a Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the...
by modeling it on a surface of
constant curvatureIn mathematics, constant curvature in differential geometry is a concept most commonly applied to surfaces. For those the scalar curvature is a single number determining the local geometry, and its constancy has the obvious meaning that it is the same at all points...
, the
pseudosphereIn geometry, a pseudosphere of radius R is a surface of curvature −1/R2 , by analogy with the sphere of radius R, which is a surface of curvature 1/R2...
, and in the interior of an
n-dimensional unit ball, the so-called Beltrami–Klein model. He also pioneered the
singular value decompositionIn linear algebra, the singular value decomposition is an important factorization of a rectangular real or complex matrix, with many applications in signal processing and statistics...
for matrices, which has been subsequently rediscovered several times. Beltrami's use of differential calculus in problems of mathematical physics indirectly influenced development of tensor calculus by
Gregorio Ricci-CurbastroGregorio Ricci-Curbastro was an Italian mathematician. He was born at Lugo di Romagna. He is most famous as the inventor of the tensor calculus but published important work in many fields....
and
Tullio Levi-CivitaTullio Levi-Civita was an Italian mathematician, most famous for his work on absolute differential calculus and its applications to the theory of relativity but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus...
.
Short biography
Beltrami was born in
CremonaCremona is a city in northern Italy, situated in Lombardy, on the left bank of the Po River in the middle of the Pianura Padana...
in
LombardyLombardy is one of the 20 regions of Italy. The capital is Milan. One-sixth of Italy's population lives in Lombardy and about one fifth of Italy's GDP is produced in this region...
, then a part of the
Austrian EmpireThe Austrian Empire was a modern era successor empire founded on a remnant of the Holy Roman Empire centered on what is today's Austria that officially lasted from 1804 to 1867...
, and now part of Italy. He began studying mathematics at
University of PaviaThe University of Pavia is a university located in Pavia, Lombardy, Italy. It was founded in 1361 and is organized in 9 Faculties.-History:...
in 1853, but was expelled
from
Ghislieri CollegeThe Ghislieri College , founded in 1567 by Pope Pius V, is one of the most ancient colleges in Pavia and co-founder of the IUSS, located in Pavia as well....
in 1856 due to his political views. During this time he was taught and influenced by
Francesco BrioschiFrancesco Brioschi was an Italian mathematician.Brioschi was born in Milan in 1824. From 1850 he taught analytical mechanics in the University of Pavia. After the Italian unification in 1861, he was elected depute in the Parliament of Italy and then appointed twice secretary of the Education...
.
He had to discontinue his studies because of financial hardship and spent next several years as a secretary working for Lombardy–Venice railroad company. He was appointed to the
University of BolognaThe University of Bologna is the oldest continually operating degree-granting university in Europe, the word 'university' being first used by this institution at its foundation. The true date of its founding is uncertain, but believed by most accounts to have been 1088...
as a professor in 1862, the year he published his first paper. Throughout his life, Beltrami held various positions at universities in
PisaPisa is a city in Tuscany, central Italy, on the right bank of the mouth of the Arno River on the Ligurian Sea. It is the capital city of the Province of Pisa...
,
RomeRome is the capital of Italy and the country's largest and most populated municipality , with over 2.7 million residents in , while the population of the urban area is estimated by Eurostat to be 3.46 million. The metropolitan area of Rome is estimated by OECD to have a population of 3.7 million...
and Pavia. From 1891 until the end of his life Beltrami lived in Rome. He became the president of the
Accademia dei LinceiThe Accademia dei Lincei, , is an Italian science academy, located at the Palazzo Corsini on the Via della Lungara in Rome, Italy....
in 1898 and a senator of the Kingdom of Italy in 1899.
Contributions to non-Euclidean geometry
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 geometryA non-Euclidean geometry is characterized by a non-vanishing Riemann curvature tensor. Examples of non-Euclidean geometries include the hyperbolic and elliptic geometry, which are contrasted with a Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the...
of Bolyai and 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
curvatureIn differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the...
, a
pseudosphereIn geometry, a pseudosphere of radius R is a surface of curvature −1/R2 , by analogy with the sphere of radius R, which is a surface of curvature 1/R2...
. In Beltrami's approach, lines of the geometry are represented by
geodesicIn mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".In the presence of a metric, geodesics are defined to be the shortest path between points on the space...
s on the pseudosphere and theorems of non-Euclidean geometry can be proved within ordinary three-dimensional
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...
, and not derived in an axiomatic fashion, as Lobachevsky and Bolyai had done previously. In 1840,
MindingFerdinand Minding was a German–Russian mathematician known for his contributions to differential geometry. He continued the work of Gauss on differential geometry of surfaces, especially its intrinsic aspects. Minding considered questions of bending of surfaces and proved the invariance of...
already considered geodesic triangles on the pseudosphere and remarked that the corresponding "trigonometric formulas" are obtained from the corresponding formulas of
spherical trigonometrySpherical trigonometry is a branch of spherical geometry, which deals with polygons on the sphere and the relationships between the sides and the angles...
by replacing the usual trigonometric functions with
hyperbolic functionIn mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh", and the hyperbolic cosine "cosh", from which are derived the hyperbolic tangent "tanh", etc., in analogy to the derived...
s; this was further developed by
CodazziDelfino Codazzi was an Italian mathematician. He made some important contributions to the differential geometry of surfaces, such as the Gauss-Codazzi-Mainardi equations....
in 1857, but apparently neither of them noticed the connection with Lobachevsky's work. In this way, Beltrami attempted to demonstrate that two-dimensional non-Euclidean geometry is as valid as the Euclidean geometry of the space, and in particular, that Euclid's
parallel postulateIn geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...
could not be derived from the other axioms of
Euclidean geometryEuclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content...
. 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 in the fact that the pseudosphere is topologically a
cylinderA cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called 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 diskIn mathematics, the open unit disk around P , is the set of points whose distance from P is less than 1:...
and that the
singularityIn mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor. Probably there will appear a number of...
of the pseudosphere corresponds to a
horocycleIn hyperbolic geometry, a horocycle is a curve whose normals all converge asymptotically. It is the two-dimensional example of a horosphere....
on the non-Euclidean 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 non-Euclidean geometry of space, by this method.
In the second memoir published in the same year (1868), "Fundamental theory of spaces of constant curvature", Beltrami went much farther and gave an abstract proof of
equiconsistencyIn mathematics, specifically in mathematical logic, formal theories are studied as mathematical objects. Since some theories are powerful enough to model different mathematical objects, it is natural to wonder about their own consistency....
of hyperbolic and Euclidean geometry in any dimension. He accomplished this by introducing several models of non-Euclidean geometry that are now known as the Beltrami–Klein model, the
Poincaré disk modelIn geometry, the Poincaré disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or unit ball, and the straight lines of the hyperbolic geometry are segments of circles contained in the disk...
, and the
Poincaré half-plane modelIn non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry....
, together with transformations that relate them. For the half-plane model, Beltrami cited a note by Liouville in the treatise of
MongeGaspard Monge, Comte de Péluse , was the a French mathematician and inventor of descriptive geometry.-Biography:...
on differential geometry. Beltrami also showed that
n-dimensional Euclidean geometry is realized on a horosphere of the (
n + 1)-dimensional
hyperbolic spaceIn mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. Hyperbolic space is the principal example of a space exhibiting hyperbolic geometry...
, so the logical relation between consistency of the Euclidean and the non-Euclidean 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 a milestone in the development of non-Euclidean geometry, the reception at the time was less enthusiastic.
CremonaLuigi Cremona was an Italian mathematician. His life was devoted to the study of geometry and reforming advanced mathematical teaching in Italy. His reputation mainly rests on his Introduzione ad una teoria geometrica delle curve piane...
objected to perceived circular reasoning, which even forced Beltrami to delay the publication of the "Essay" by one year. Subsequently,
Felix KleinFelix Christian Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
failed to acknowledge Beltrami's priority in construction of the projective disk model of the non-Euclidean geometry. This reaction can be attributed in part to the novelty of Beltrami's approach, which was close to the ideas of Riemann concerning abstract
manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
s. J. Hoüel published Beltrami's proof in his French translation of works of Lobachevsky and Bolyai.
Works
See also
- Beltrami equation
- Beltrami identity
The Beltrami identity is an identity in the calculus of variations. It says that a function u which is an extremal of the integralsatisfies the differential equation...
- Beltrami's theorem
In mathematics — specifically, in Riemannian geometry — Beltrami's theorem is a result named after the Italian mathematician Eugenio Beltrami which states that geodesic maps preserve the property of having constant curvature...
- Laplace–Beltrami operator
External links