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, structure, space, models, and change. History On ...

. He graduated from the University of Madrid and he studied at the

University of Hamburg The University of Hamburg (german: link=no, Universität Hamburg, also referred to as UHH) is a public research university in Hamburg, Germany. It was founded on 28 March 1919 by combining the previous General Lecture System ('' Allgemeines Vor ...

, 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 taught ...

. Because of the

Spanish Civil War The Spanish Civil War ( es, Guerra Civil Española)) or The Revolution ( es, La Revolución, link=no) among Nationalists, the Fourth Carlist War ( es, Cuarta Guerra Carlista, link=no) among Carlists, and The Rebellion ( es, La Rebelión, lin ...

, he moved to Argentina as a

professor Professor (commonly abbreviated as Prof.) is an Academy, academic rank at university, universities and other post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin as a "person who pr ...

in the

National University of the Littoral The National University of Litoral ( es, Universidad Nacional del Litoral, UNL) is a public university in Argentina. It is based in Santa Fe, the capital of Santa Fe Province. It has colleges and other academic facilities in Esperanza, Reconqu ...

National University of La Plata The La Plata National University ( es, Universidad Nacional de La Plata, UNLP) is one of the most important Argentine national universities and the biggest one situated in the city of La Plata, capital of Buenos Aires Province. It has over 9 ...

and

University of Buenos Aires The University of Buenos Aires ( es, Universidad de Buenos Aires, UBA) is a public university, public research university in Buenos Aires, Argentina. Established in 1821, it is the premier institution of higher learning in the country and one o ...

. His work with Blaschke on

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...

s is now cited in its connection with

Mahler volume In convex geometry, the Mahler volume of a centrally symmetric convex body is a dimensionless quantity that is associated with the body and is invariant under linear transformations. It is named after German-English mathematician Kurt Mahler. It is ...

. Blaschke and Santaló also collaborated on

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 (or equivariant) transformation ...

. 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. Schoolbooks are textboo ...

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 geo ...

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, pro ...

, and

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...

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 perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...

. 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 that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

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, pro ...

and

conic In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special ...

s V,VI,VII.

Hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

: graphic properties, angles and distances, areas and curves. (This text develops the

Klein model Klein may refer to: People *Klein (surname) *Klein (musician) Places *Klein (crater), a lunar feature *Klein, Montana, United States *Klein, Texas, United States *Klein (Ohm), a river of Hesse, Germany, tributary of the Ohm *Klein River, a river ...

, 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. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

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 In algebra, ring theory is the study of rings— algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their re ...

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 that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

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 mapping V \to W between two vector spaces that pre ...

. These seven introductory sections on

algebraic structure In mathematics, an algebraic structure 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 multiplication), and a finite set of ...

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. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...

, (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 ''A'', ''B'', ''C'' and ''D'' on a line, the ...

, (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 it ...

, (21)

, (22–4)

quadrics In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...

. Serious and coordinated study of this text is invited by 240

exercise Exercise is a body activity that enhances or maintains physical fitness and overall health and wellness. It is performed for various reasons, to aid growth and improve strength, develop muscles and the cardiovascular system, hone athletic ...

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 ''Gelfand'' is a surname meaning "elephant" in the Yiddish language and may refer to: * People: ** Alan Gelfand, the inventor of the ollie, a skateboarding move ** Alan E. Gelfand, a statistician ** Boris Gelfand, a chess grandmaster ** Israel Gel ...

” (p. 345) which involves inverting the

Radon transform In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the l ...

''Vectores y tensores con sus aplicaciones'' (1977)

Includes standard vector algebra,

vector analysis Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...

, introduction to

tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...

s and

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

geodesic In geometry, a geodesic () is a curve representing in some sense the 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 connection. ...

curves, curvature tensor and

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

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 University of Buenos Aires faculty