Luís Antoni Santaló Sors (October 9, 1911 – November 22, 2001) was a Spanish
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
.
He graduated from the
University of Madrid and he studied at the
University of Hamburg
The University of Hamburg (, also referred to as UHH) is a public university, public research university in Hamburg, Germany. It was founded on 28 March 1919 by combining the previous General Lecture System ('':de:Allgemeines Vorlesungswesen, ...
, where he received his Ph.D. in 1936. His advisor was
Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry.
Education and career
Blaschke was the son of mathematician Josef Blaschke, who taugh ...
. Because of the
Spanish Civil War
The Spanish Civil War () was a military conflict fought from 1936 to 1939 between the Republican faction (Spanish Civil War), Republicans and the Nationalist faction (Spanish Civil War), Nationalists. Republicans were loyal to the Left-wing p ...
, he moved to Argentina as a
professor
Professor (commonly abbreviated as Prof.) is an Academy, academic rank at university, universities and other tertiary education, post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin ...
in the
National University of the Littoral
The National University of the Littoral (, UNL) is a public university in Argentina. It is based in Santa Fe, Argentina, Santa Fe, the capital of Santa Fe Province. It has colleges and other academic facilities in Esperanza, Santa Fe, Esperanza ...
,
National University of La Plata
The National University of La Plata (, UNLP) is a national public research university located in the city of La Plata, capital of Buenos Aires Province, Argentina. It has over 90,000 regular students, 10,000 teaching staff, 17 departments and 10 ...
and
University of Buenos Aires
The University of Buenos Aires (, UBA) is a public university, public research university in Buenos Aires, Argentina. It is the second-oldest university in the country, and the largest university of the country by enrollment. Established in 1821 ...
.
His work with Blaschke on
convex set
In geometry, a set of points is convex if it contains every line segment between two points in the set.
For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
s is now cited in its connection with
Mahler volume. Blaschke and Santaló also collaborated on
integral geometry. Santaló wrote
textbook
A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions, but also of learners ( ...
s in Spanish on
non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...
,
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...
, and
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
s.
Works
Luis Santaló published in both English and Spanish:
''Introduction to Integral Geometry'' (1953)
Chapter I. Metric integral geometry of the plane including densities and the
isoperimetric inequality
In mathematics, 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'' ...
. 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
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
s on the plane: central-affine, unimodular affine, projective groups.
''Geometrias no Euclidianas'' (1961)
I. The Elements of Euclid
II. Non-Euclidean geometries
III., IV.
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...
and
conic
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, thou ...
s
V, VI, VII.
Hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
: 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
''Geometria proyectiva'' (1966)
A curious feature of this book on projective geometry is the opening on
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
including
laws of composition,
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
,
ring theory,
fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...
,
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
s,
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s and
linear mapping
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vec ...
. These seven introductory sections on
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
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
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
, (11)
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
, (12)
cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points , , , on a line, their cross ratio is defin ...
, (13)
harmonic quadruples, (18)
pole and polar
In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section.
Polar reciprocation in a given circle is the transformation of each point in the plane into i ...
, (21)
Klein model of
non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...
, (22–4)
quadrics
In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids.
More generally, a quadric hyper ...
. Serious and coordinated study of this text is invited by 240
exercise
Exercise or workout is physical activity that enhances or maintains fitness and overall health. It is performed for various reasons, including weight loss or maintenance, to aid growth and improve strength, develop muscles and the cardio ...
s 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” (p. 345) which involves inverting the
Radon transform.
''Vectores y tensores con sus aplicaciones'' (1977)
Includes standard vector algebra,
vector analysis
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...
, introduction to
tensor field
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...
s and
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s,
geodesic
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
curves,
curvature tensor and
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
to
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumpti ...
. Exercises distributed at an average rate of ten per section enhance the 36 instructional sections. Solutions are found on pages 343–64.
See also
*
Santaló's formula
References
*
External links
*
Rincon Matemático (Spanish)Fons Lluís Santaló, de la Universitat de Girona(Catalan)
DUGi Fons Especials. Fons Lluís Santaló(Catalan)
{{DEFAULTSORT:Santalo, Luis
1911 births
2001 deaths
20th-century Argentine mathematicians
20th-century Spanish mathematicians
Differential geometers
Scientists from Catalonia
Textbook writers
Academic staff of the University of Buenos Aires
University of Hamburg alumni