Élie Joseph Cartan was a French differential geometer. His results include the classification of complex semisimple Lie algebras (“Cartan classification”), extension of these results to a class of symmetric spaces, the proof of the Lie–Cartan theorem (after Serre sometimes called “Lie's third theorem”) on integration of Lie algebras to Lie groups (Lie proved just the integration to local Lie groups), the method of moving frames, the introduction of Cartan’s connection, numerous results in Riemannian geometry, results related to the formal integrability of PDEs (Cartan involutive equations, Pfaffian system), etc.
Father of Henri Cartan.
Introducing what came to be known as Maurer-Cartan forms:
Introducing the re-formulation of Riemannian geometry that came to be known as Cartan geometry via Cartan structural equations for curvature and torsion of Cartan moving frames and Cartan connections:
Élie Cartan, Sur une généralisation de la notion de courbure de Riemann et les espaces á torsion, C. R. Acad. Sci. 174 (1922) 593-595 .
Élie Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), Annales scientifiques de l’École Normale Supérieure, Sér. 3, 40 (1923) 325-412 [doi:ASENS_1923_3_40__325_0]
Élie Cartan, La géométrie des espaces de Riemann, Mémorial des sciences mathématiques 9 (1925) [numdam:MSM_1925__9__1_0]
Élie Cartan (translated by Vladislav Goldberg from Cartan’s lectures at the Sorbonne in 1926–27): Riemannian Geometry in an Orthogonal Frame, World Scientific (2001) [doi:10.1142/4808, pdf]
Élie Cartan (translated by Robert Hermann from Cartan’s lectures in 1951): Geometry of Riemannian Spaces, Lie Groups: History, Frontiers and Applications XIII, Math Sci Press (1983) [ark:/13960/s28rzmj9xrv]
as reviewed in
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