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 ...

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 ...

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

, (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 ...

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 ...

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.

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