Luis Santaló
Encyclopedia
Luís Antoni Santaló Sors (October 9, 1911 – November 22, 2001) was a Spanish
mathematician
.
He graduated from the University of Madrid
and he studied at the University of Hamburg
, where he received his Ph.D. in 1936. His advisor was Wilhelm Blaschke
. Because of the Spanish Civil War
, he moved to Argentina where he became a very famous mathematician.
He studied integral geometry
and many other topics of mathematics and science.
He worked as a teacher in the National University of the Littoral
, National University of La Plata and University of Buenos Aires
.
. Ch. II. Integral geometry on surfaces including Blaschke’s
formula and the isoperimetric inequality on surfaces of constant curvature. Ch. III. General integral geometry: Lie group
s on the plane: central-affine, unimodular affine, projective groups.
II. Non-Euclidean geometries
III., IV. Projective geometry
and conics
V,VI,VII. Hyperbolic geometry
: graphic properties, angles and distances, areas and curves.
(This text develops the Klein model
, the earliest instance of a model.)
VIII. Other models of non-Euclidean geometry
including laws of composition
, group theory
, ring theory
, fields
, finite field
s, vector space
s and linear mapping. These seven introductory section on algebraic structure
s provide an enhanced vocabulary for the treatment of 15 classical topics of projective geometry. Furthermore sections (14) projectivities with non-commutative fields, (22) quadrics over non-commutative fields, and (26) finite geometries
embellish the classical study. The usual topics are covered such as (4) Fundamental theorem of projective geometry, (11) projective plane
, (12) cross-ratio
, (13) harmonic quadruples
, (18) pole and polar
, (21) Klein model
of non-Euclidean geometry
, (22–4) quadrics
. Serious and coordinated study of this text is invited by 240 exercises at the end of 25 sections, with solutions on pages 347–65.
For instance, in Chapter 19, he notes “Trends in Integral Geometry” and includes “The integral geometry of Gelfand
” (p. 345) which involves inverting the Radon transform
.
s and Riemannian manifold
s, geodesic
curves, curvature tensor
and general relativity
to Schwarzschild metric
. Exercises distributed at an average rate of ten per section enhance the 36 instructional sections. Solutions are found on pages 343–64.
Spain
Spain , officially the Kingdom of Spain languages]] under the European Charter for Regional or Minority Languages. In each of these, Spain's official name is as follows:;;;;;;), is a country and member state of the European Union located in southwestern Europe on the Iberian Peninsula...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
.
He graduated from the University of Madrid
University of Madrid
The Complutense University of Madrid is a public university in Madrid, Spain, and one of the oldest universities in the world.The University of Madrid may also refer to:* The Autonomous University of Madrid, a public university founded in 1968...
and he studied at the University of Hamburg
University of Hamburg
The University of Hamburg is a university in Hamburg, Germany. It was founded on 28 March 1919 by Wilhelm Stern and others. It grew out of the previous Allgemeines Vorlesungswesen and the Kolonialinstitut as well as the Akademisches Gymnasium. There are around 38,000 students as of the start of...
, where he received his Ph.D. in 1936. His advisor was Wilhelm Blaschke
Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke was an Austro-Hungarian differential and integral geometer.His students included Shiing-Shen Chern, Luis Santaló, and Emanuel Sperner....
. Because of the Spanish Civil War
Spanish Civil War
The Spanish Civil WarAlso known as The Crusade among Nationalists, the Fourth Carlist War among Carlists, and The Rebellion or Uprising among Republicans. was a major conflict fought in Spain from 17 July 1936 to 1 April 1939...
, he moved to Argentina where he became a very famous mathematician.
He studied integral geometry
Integral geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant transformations from the space of functions on one geometrical space to the...
and many other topics of mathematics and science.
He worked as a teacher in the National University of the Littoral
National University of the Littoral
The National University of the Littoral is a university in Argentina. It is based in Santa Fe, the capital of the province of the same name, and it has colleges and other academic facilities in Esperanza, Reconquista and Gálvez, also in Santa Fe Province.-History:The original institution was...
, National University of La Plata and University of Buenos Aires
University of Buenos Aires
The University of Buenos Aires is the largest university in Argentina and the largest university by enrollment in Latin America. Founded on August 12, 1821 in the city of Buenos Aires, it consists of 13 faculties, 6 hospitals, 10 museums and is linked to 4 high schools: Colegio Nacional de Buenos...
.
Introduction to Integral Geometry (1953)
Chapter I. Metric integral geometry of the plane including densities and the isoperimetric inequalityIsoperimetry
The isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. Isoperimetric literally means "having the same perimeter"...
. Ch. II. Integral geometry on surfaces including Blaschke’s
Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke was an Austro-Hungarian differential and integral geometer.His students included Shiing-Shen Chern, Luis Santaló, and Emanuel Sperner....
formula and the isoperimetric inequality on surfaces of constant curvature. Ch. III. General integral geometry: Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s on the plane: central-affine, unimodular affine, projective groups.
Geometrias no Euclidianas (1961)
I. The Elements of EuclidII. Non-Euclidean geometries
III., IV. Projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
and conics
V,VI,VII. Hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
: graphic properties, angles and distances, areas and curves.
(This text develops the Klein model
Klein model
In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of n-dimensional hyperbolic geometry in which points are represented by the points in the interior of the n-dimensional unit ball and lines are represented by the...
, the earliest instance of a model.)
VIII. Other models of non-Euclidean geometry
Geometria proyectiva (1966)
A curious feature of this book on projective geometry is the opening on abstract algebraAbstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
including laws of composition
Binary function
In mathematics, a binary function, or function of two variables, is a function which takes two inputs.Precisely stated, a function f is binary if there exists sets X, Y, Z such that\,f \colon X \times Y \rightarrow Z...
, group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s, vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s and linear mapping. These seven introductory section on algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
s provide an enhanced vocabulary for the treatment of 15 classical topics of projective geometry. Furthermore sections (14) projectivities with non-commutative fields, (22) quadrics over non-commutative fields, and (26) finite geometries
Finite geometry
A finite geometry is any geometric system that has only a finite number of points.Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact as many points as there are real numbers...
embellish the classical study. The usual topics are covered such as (4) Fundamental theorem of projective geometry, (11) projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
, (12) cross-ratio
Cross-ratio
In geometry, the cross-ratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line...
, (13) harmonic quadruples
Projective harmonic conjugates
In projective geometry, the harmonic conjugate point of a triple of points on the real projective line is defined by the following construction due to Karl von Staudt:...
, (18) pole and polar
Pole and polar
In geometry, the terms pole and polar are used to describe a point and a line that have a unique reciprocal relationship with respect to a given conic section...
, (21) Klein model
Klein model
In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of n-dimensional hyperbolic geometry in which points are represented by the points in the interior of the n-dimensional unit ball and lines are represented by the...
of non-Euclidean geometry
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
, (22–4) quadrics
Quadrics
Quadrics was a supercomputer company formed in 1996 as a joint venture between Alenia Spazio and the technical team from Meiko Scientific. They produced hardware and software for clustering commodity computer systems into massively parallel systems. Their highpoint was in June 2003 when six out of...
. Serious and coordinated study of this text is invited by 240 exercises at the end of 25 sections, with solutions on pages 347–65.
Integral Geometry and Geometric Probability (1976)
Amplifies and extends the 1953 text.For instance, in Chapter 19, he notes “Trends in Integral Geometry” and includes “The integral geometry of Gelfand
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand was a Soviet mathematician who made major contributions to many branches of mathematics, including group theory, representation theory and functional analysis...
” (p. 345) which involves inverting the Radon transform
Radon transform
thumb|right|Radon transform of the [[indicator function]] of two squares shown in the image below. Lighter regions indicate larger function values. Black indicates zero.thumb|right|Original function is equal to one on the white region and zero on the dark region....
.
Vectores y tensores con sus aplicaciones (1977)
Includes standard vector algebra, vector analysis, introduction to tensor fieldTensor field
In mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...
s and Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
s, geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
curves, curvature tensor
Curvature tensor
In differential geometry, the term curvature tensor may refer to:* the Riemann curvature tensor of a Riemannian manifold — see also Curvature of Riemannian manifolds;* the curvature of an affine connection or covariant derivative ;...
and general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
to Schwarzschild metric
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...
. Exercises distributed at an average rate of ten per section enhance the 36 instructional sections. Solutions are found on pages 343–64.