Professor Élie Joseph Cartan
9 April 1869|
Dolomieu, Isère, France
|Died||6 May 1951
|Institutions||University of Paris
École Normale Supérieure
|Alma mater||University of Paris|
|Doctoral advisor||Gaston Darboux
|Doctoral students||Charles Ehresmann
|Known for||Lie groups, differential geometry|
|Notable awards||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.
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 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:
- Lie groups
- Representations of Lie groups
- Hypercomplex numbers, division algebras
- Systems of PDEs, Cartan–Kähler theorem
- Theory of equivalence
- Integrable systems, theory of prolongation and systems in involution
- Infinite-dimensional groups and pseudogroups
- Differential geometry and moving frames
- Generalised spaces with structure groups and connections, Cartan connection, holonomy, Weyl tensor
- Geometry and topology of Lie groups
- Riemannian geometry
- Symmetric spaces
- Topology of compact groups and their homogeneous spaces
- Integral invariants and classical mechanics
- Relativity, spinors
- List of things named after Élie Cartan
- Integrability conditions for differential systems
- CAT(k) space
- 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
- La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile, Gauthiers-Villars, 1937
- Cartan, Élie (1981) , 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
- 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
- Whitehead, J. H. C. (1952). "Elie Joseph Cartan. 1869-1951". Obituary Notices of Fellows of the Royal Society 8 (21): 71–26. doi:10.1098/rsbm.1952.0005. JSTOR 768800.
- O'Connor, John J.; Robertson, Edmund F., "Élie Cartan", MacTutor History of Mathematics archive, University of St Andrews.
- Élie Cartan at the Mathematics Genealogy Project
- Vanderslice, J. L. (1938). "Review: Leçons sur la théorie des espaces à connexion projective". Bull. Amer. Math. Soc. 44 (1, Part 1): 11–13. doi:10.1090/s0002-9904-1938-06648-7.
- Weyl, Hermann (1938). "Cartan on Groups and Differential Geometry". Bull. Amer. Math. Soc. 44 (9, part 1): 598–601.
- Thomas, J. M. (1947). "Review: Les systèmes différentiels extérieurs et leurs applications géométriques". Bull. Amer. Math. Soc. 53 (3): 261–266. doi:10.1090/s0002-9904-1947-08750-4.
- 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 and Claude Chevalley, Élie Cartan and his mathematical work, Bull. Amer. Math. Soc. 58 (1952), 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)"