Élie Cartan

Élie Cartan
Professor Élie Joseph Cartan
Born	(1869-04-09)9 April 1869 Dolomieu, Isère, France
Died	Paris, France
Nationality	France
Fields	Mathematics and physics
Institutions	University of Paris École Normale Supérieure
Alma mater	University of Paris
Thesis	Sur la structure des groupes de transformations finis et continus (1894)
Doctoral advisor	Gaston Darboux Sophus Lie
Doctoral students	Charles Ehresmann Mohsen Hashtroodi Radu Rosca Kentaro Yano
Known for	Lie groups Differential geometry Special and general relativity Quantum mechanics : spinor, rotating vectors
Notable awards	Leconte Prize (1930) Lobachevsky Prize (1937) Fellow of the Royal Society

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. He was the father of another influential mathematician, Henri Cartan, and the composer Jean Cartan.

Life

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. Also in 1938 he participated in the International Committee composed to organise the International Congresses for the Unity of Science. He died in Paris after a long illness.

In 1937 he became foreign member of the Royal Netherlands Academy of Arts and Sciences.

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 groups. 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 equations 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 frames, 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:

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

Publications

• 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 groupes 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 méthode de repère mobile, la théorie des groupes continus, et les espaces généralisés, 1935
• Leçons sur la théorie des espaces à connexion projective, Gauthiers-Villars, 1937
• La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile, Gauthiers-Villars, 1937
• Les systèmes différentiels extérieurs et leurs applications géométriques, Hermann, 1945
• 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

See also

• List of things named after Élie Cartan
• Integrability conditions for differential systems
• CAT(k) space

External links

• M.A. Akivis & B.A. Rosenfeld (1993) Élie Cartan (1869–1951), translated from Russian original by V.V. Goldberg, American Mathematical Society ISBN 0-8218-4587-X .
  • Shiing-Shen Chern (1994) Book review: Elie Cartan by Akivis & Rosenfeld Bulletin of the American Mathematical Society 30(1)
• Shiing-Shen Chern and Claude Chevalley (1951) Élie Cartan and his mathematical work, Bulletin of the American Mathematical Society 58: 217-250.

English translations of some of his books and articles:

• "On certain differential expressions and the Pfaff problem"
• "On the integration of systems of total differential equations"
• Lessons on integral invariants.
• "The structure of infinite groups"
• "Spaces with conformal connections"
• "On manifolds with projective connections"
• "The unitary theory of Einstein-Mayer"
• "E. Cartan, Exterior Differential Systems and its Applications, (Translated in to English by M. Nadjafikhah)"