Élie Cartan

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Élie Cartan
Elie Cartan.jpg
Professor Élie Joseph Cartan
Born (1869-04-09)9 April 1869
Dolomieu, Isère, France
Died 6 May 1951(1951-05-06) (aged 82)
Paris, France
Nationality French
Fields Mathematics
Institutions University of Paris
École Normale Supérieure
Alma mater University of Paris
Doctoral advisor Gaston Darboux
Sophus Lie
Doctoral students Charles Ehresmann
Mohsen Hashtroodi
Radu Rosca
Kentaro Yano
Known for Lie groups, differential geometry
Notable awards Leconte Prize (1930)
Lobachevsky Prize (1937)
Fellow of the Royal Society[1]

Élie Joseph Cartan (French: [kaʁtɑ̃]; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups and their geometric applications. He also made significant contributions to mathematical physics, differential geometry, and group theory.[2][3] He was the father of another influential mathematician, Henri Cartan, and the composer Jean Cartan.


Élie Cartan was born in the village of Dolomieu, Isère, 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.


By his own rather understated account (found in his Notice sur les travaux scientifiques), the main theme of Cartan's works (numbering 186 and published throughout the period 1893–1947) was the theory of Lie groups. He began by reworking the foundational material on the complex simple Lie algebras, clarifying the previous work by Friedrich Engel and Wilhelm Killing. This lead him to a complete classification, i.e.,the identification of the four main families and the five exceptional cases. He also introduced the concept of algebraic group, as well as many of the basic constructions of representation theory. However,from our perspective his "main theme" was far more reaching: the invention of the calculus of exterior differential systems and their differential invariants.

These contributions began with his definition of anti-symmetric differential p-forms, which greatly simplified Lie group classification via the Maurer–Cartan equations. This is a large intricate system of partial differential equations when written in local coordinate calculus, as in Lie's approach. This lead Cartan to a highly efficient description for local integrability conditions for any system of analytic partial differential equations. Historically, 1-forms had been introduced to clarify Hamiltonian method for integrating overdetermined systems of PDE for one function of several variables. This approach slowly evolved into integrating Pfaffian systems (i.e. first-order partial differential equations ). But,by introducing additional variables corresponding to higher derivatives, and describing their integrability conditions via additional differential forms, Cartan produced a natural linear algebraic "prolongation" algorithm which led him to the Cartan-Kahler theorem establishing the local existence, or non existence, of solutions to general analytic systems of PDE. Central to this "prolongation" is his discovery of the exterior derivative of a p-form, an entirely geometric and coordinate-independent operation. With this striking insight Cartan was able to condense 150 years of idiosyncratic several variable techniques. It should be taken as a pivotal moment in mathematical history.

With these devices under his control — Lie groups and differential forms — he went on to produce a very large body of work, based on novel computational techniques involving moving frames: He is the first to consider a principal frame bundle as the fundamental object of geometry, thereby significantly generalizing F. Klein's "Erlangen program"; Cartan was the first to prove that the Einstein PDE for a vacuum spacetime metric has a large set of solutions (i.e.,it is an "involutive" system per the prolongation test), a nontrivial observation which indicates that general relativity is a robust physical theory (as documented by extensive personal communications between Cartan and Einstein); The iteration of the exterior derivative is the zero operator (i.e.,defines an exact sequence) thus, Cartan pushed opened the door to algebraic topology, a subject central to the mathematics of the 20th century; The Smale-Gromov h-principal begins with Cartan's local integrability conditions and seeks global integrability conditions.

It should be noted that, rather than engaging in abstraction, Cartan's approach was to: concretely define an object via differential forms on the total space of a principal frame bundle; quickly compute relationships among their invariants via his calculus of differential forms; then state the resulting theorem. (These explicit computations are often so brief and efficient, that much reflection is required in order to understand Cartan's underlying "neo-geometrical" train of thought.) Also,since the relevant frame bundles can be realized as subsets of a high dimensional affine space, his approach cleverly avoided many of technical issues surrounding the definition of abstract manifolds, a subject which had yet to be fully delineated in the early 1900s. After his death Cartan's calculus was rediscovered and reinvigorated by: S.Chern, R.Bryant, R.Gardner,and P.Griffiths. The subject is now referred to as Exterior differential systems on jet bundles of mappings between manifolds. Anyone dissatisfied with their undergraduate course in several variable calculus will find an introductory text on differential forms reinvigorating, and the perfect memorial to the crystalline mind of E. Cartan.

In the Travaux, he breaks down his work into 15 areas. Using modern terminology, they are these:

  1. Lie theory
  2. Representations of Lie groups
  3. Hypercomplex numbers, division algebras
  4. Systems of PDEs, Cartan–Kähler theorem
  5. Theory of equivalence
  6. Integrable systems, theory of prolongation and systems in involution
  7. Infinite-dimensional groups and pseudogroups
  8. Differential geometry and moving frames
  9. Generalised spaces with structure groups and connections, Cartan connection, holonomy, Weyl tensor
  10. Geometry and topology of Lie groups
  11. Riemannian geometry
  12. Symmetric spaces
  13. Topology of compact groups and their homogeneous spaces
  14. Integral invariants and classical mechanics
  15. Relativity, spinors

See also[edit]


  • Cartan, Élie (1894), Sur la structure des groupes de transformations finis et continus, Thesis, Nony 
  • Leçons sur les invariants intégraux, Hermann, Paris, 1922
  • La Géométrie des espaces de Riemann, 1925
  • Leçons sur la géométrie des espaces de Riemann, Gauthiers-Villars, 1928
  • La théorie des groups finis et continus et l'analysis situs, Gauthiers-Villars, 1930
  • Leçons sur la géométrie projective complexe, Gauthiers-Villars, 1931
  • La parallelisme absolu et la théorie unitaire du champ, Hermann, 1932
  • La Théorie des groupes continus et des espaces généralisés, 1935
  • Leçons sur la théorie des espaces à connexion projective, Gauthiers-Villars, 1937[4]
  • La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile, Gauthiers-Villars, 1937[5]
  • Cartan, Élie (1981) [1938], The theory of spinors, New York: Dover Publications, ISBN 978-0-486-64070-9, MR 631850 
  • Les systèmes différentiels extérieurs et leurs applications géométriques, Hermann, 1945[6]
  • Oeuvres complètes, 3 parts in 6 vols., Paris 1952 to 1955, reprinted by CNRS 1984:
    • Part 1: Groupes de Lie (in 2 vols.), 1952
    • Part 2, Vol. 1: Algèbre, formes différentielles, systèmes différentiels, 1953
    • Part 2, Vol. 2: Groupes finis, Systèmes différentiels, théories d'équivalence, 1953
    • Part 3, Vol. 1: Divers, géométrie différentielle, 1955
    • Part 3, Vol. 2: Géométrie différentielle, 1955


External links[edit]

English translations of some of his books and articles: