Élie Cartan
Encyclopedia
Élie Joseph Cartan was an influential French
mathematician
, who did fundamental work in the theory of Lie group
s and their geometric applications. He also made significant contributions to mathematical physics
, differential geometry, and group theory
.
He was the father of another influential mathematician, Henri Cartan
.
, the son of a blacksmith
. He attended the Lycée Janson de Sailly
before studying at the École Normale Supérieure
in Paris in 1888 and obtaining his doctorate in 1894. He subsequently held lecturing positions in Montpellier
and Lyon
, becoming a professor in Nancy in 1903. He took a lecturing position at the Sorbonne
in Paris in 1909, becoming professor there in 1912 until his retirement in 1940. He died in Paris after a long illness.
s. He began by working over the foundational material on the complex simple Lie algebras, tidying up the previous work by Friedrich Engel
and Wilhelm Killing
. This proved definitive, as far as the classification went, with the identification of the four main families and the five exceptional cases. He also introduced the algebraic group
concept, which was not to be developed seriously before 1950.
He defined the general notion of anti-symmetric differential form
, in the style now used; his approach to Lie groups through the Maurer–Cartan equations
required 2-forms for their statement. At that time what were called Pfaffian systems (i.e. first-order differential equation
s given as 1-forms) were in general use; by the introduction of fresh variables for derivatives, and extra forms, they allowed for the formulation of quite general PDE
systems. Cartan added the exterior derivative
, as an entirely geometric and coordinate-independent operation. It naturally leads to the need to discuss p-forms, of general degree p. Cartan writes of the influence on him of Charles Riquier’s general PDE theory.
With these basics — Lie groups and differential forms — he went on to produce a very large body of work, and also some general techniques such as moving frame
s, that were gradually incorporated into the mathematical mainstream.
In the Travaux, he breaks down his work into 15 areas. Using modern terminology, they are these:
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
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....
, who did fundamental work in the theory of 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 and their geometric applications. He also made significant contributions to mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, differential geometry, and 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...
.
He was the father of another influential mathematician, Henri Cartan
Henri Cartan
Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:...
.
Life
Élie Cartan was born in the village of Dolomieu, IsèreDolomieu, Isère
Dolomieu is a commune in the Isère department in south-eastern France.-Notable resident:Mathematician Élie Joseph Cartan was born here in 1869.Also geologist Déodat Gratet de Dolomieu was born here in 1750.-See also:*Communes of the Isère department...
, the son of a blacksmith
Blacksmith
A blacksmith is a person who creates objects from wrought iron or steel by forging the metal; that is, by using tools to hammer, bend, and cut...
. He attended the Lycée Janson de Sailly
Lycée Janson de Sailly
Lycée Janson de Sailly is a lycée located in the XVIe arrondissement of Paris, France. It is generally considered as one of the most prestigious lycées in Paris...
before studying at the École Normale Supérieure
École Normale Supérieure
The École normale supérieure is one of the most prestigious French grandes écoles...
in Paris in 1888 and obtaining his doctorate in 1894. He subsequently held lecturing positions in Montpellier
Montpellier
-Neighbourhoods:Since 2001, Montpellier has been divided into seven official neighbourhoods, themselves divided into sub-neighbourhoods. Each of them possesses a neighbourhood council....
and Lyon
Lyon
Lyon , is a city in east-central France in the Rhône-Alpes region, situated between Paris and Marseille. Lyon is located at from Paris, from Marseille, from Geneva, from Turin, and from Barcelona. The residents of the city are called Lyonnais....
, becoming a professor in Nancy in 1903. He took a lecturing position at the Sorbonne
Sorbonne
The Sorbonne is an edifice of the Latin Quarter, in Paris, France, which has been the historical house of the former University of Paris...
in Paris in 1909, becoming professor there in 1912 until his retirement in 1940. He died in Paris after a long illness.
Work
By his own account, in his Notice sur les travaux scientifiques, the main theme of his works (numbering 186 and published throughout the period 1893–1947) was the theory of Lie groupLie 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. He began by working over the foundational material on the complex simple Lie algebras, tidying up the previous work by Friedrich Engel
Friedrich Engel (mathematician)
Friedrich Engel was a German mathematician.Engel was born in Lugau, Saxony, as the son of a Lutheran pastor. He attended the Universities of both Leipzig and Berlin, before receiving his doctorate from Leipzig in 1883.Engel studied under Felix Klein at Leipzig, and collaborated with Sophus Lie for...
and Wilhelm Killing
Wilhelm Killing
Wilhelm Karl Joseph Killing was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry....
. This proved definitive, as far as the classification went, with the identification of the four main families and the five exceptional cases. He also introduced the algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
concept, which was not to be developed seriously before 1950.
He defined the general notion of anti-symmetric differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
, in the style now used; his approach to Lie groups through the Maurer–Cartan equations
Maurer-Cartan form
In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G...
required 2-forms for their statement. At that time what were called Pfaffian systems (i.e. first-order differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s given as 1-forms) were in general use; by the introduction of fresh variables for derivatives, and extra forms, they allowed for the formulation of quite general PDE
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
systems. Cartan added the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
, as an entirely geometric and coordinate-independent operation. It naturally leads to the need to discuss p-forms, of general degree p. Cartan writes of the influence on him of Charles Riquier’s general PDE theory.
With these basics — Lie groups and differential forms — he went on to produce a very large body of work, and also some general techniques such as moving frame
Moving frame
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.-Introduction:...
s, that were gradually incorporated into the mathematical mainstream.
In the Travaux, he breaks down his work into 15 areas. Using modern terminology, they are these:
- Lie groups
- Representations of Lie groups
- Hypercomplex numberHypercomplex numberIn mathematics, a hypercomplex number is a traditional term for an element of an algebra over a field where the field is the real numbers or the complex numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established...
s, division algebraDivision algebraIn the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...
s - Systems of PDEs, Cartan–Kähler theoremCartan–Kähler theoremIn mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals I. It is named for Élie Cartan and Erich Kähler....
- Theory of equivalence
- Integrable systems, theory of prolongation and systems in involution
- Infinite-dimensional groups and pseudogroupPseudogroupIn mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra...
s - Differential geometry and moving frameMoving frameIn mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.-Introduction:...
s - Generalised spaces with structure groups and connectionConnection (mathematics)In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
s, Cartan connectionCartan connectionIn the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the...
, holonomyHolonomyIn differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections,...
, Weyl tensorWeyl tensorIn differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic... - Geometry and topology of Lie groups
- Riemannian geometryRiemannian geometryRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
- Symmetric spaceSymmetric spaceA symmetric space is, in differential geometry and representation theory, a smooth manifold whose group of symmetries contains an "inversion symmetry" about every point...
s - Topology of compact groupCompact groupIn mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
s and their homogeneous spaceHomogeneous spaceIn mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
s - Integral invariants and classical mechanicsClassical mechanicsIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
- RelativityGeneral relativityGeneral 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...
, spinorSpinorIn mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
s
See also
- Cartan connectionCartan connectionIn the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the...
, Cartan connection applicationsCartan connection applicationsThe vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifolds. It applies to metrics of any signature. This section is an approach to tetrads, but written in general terms... - Cartan matrixCartan matrixIn mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. In fact, Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan.- Lie algebras :A generalized...
- Cartan's theoremCartan's theoremIn mathematics, three results in Lie group theory are called Cartan's theorem, named after Élie Cartan:See also Cartan's theorems A and B, results of Henri Cartan, and Cartan's lemma for various other results attributed to Élie and Henri Cartan....
- Cartan subalgebra
- Cartan's equivalence methodCartan's equivalence methodIn mathematics, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up to a diffeomorphism...
- Einstein–Cartan theory
- Integrability conditions for differential systemsIntegrability conditions for differential systemsIn mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the way a differential form restricts to a...
- CAT(k) spaceCAT(k) spaceIn mathematics, a CAT space is a specific type of metric space. Intuitively, triangles in a CAT space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature k. In a CAT space, the curvature is bounded from above by k...
- Cartan–Dieudonné theorem
- Cartan–Hadamard theoremCartan–Hadamard theoremThe Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point...
External links
- Shiing-Shen ChernShiing-Shen ChernShiing-Shen Chern was a Chinese American mathematician, one of the leaders in differential geometry of the twentieth century.-Early years in China:...
and Claude ChevalleyClaude ChevalleyClaude Chevalley was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, and the theory of algebraic groups...
, Élie Cartan and his mathematical work, Bull. Amer. Math. Soc. 58 (1952), 217-250.