Élie Cartan
Élie Cartan | |
---|---|
Gaston Darboux Sophus Lie | |
Doctoral students | Charles Ehresmann Mohsen Hashtroodi Kentaro Yano |
Other notable students | Shiing-Shen Chern |
Élie Joseph Cartan
His son Henri Cartan was an influential mathematician working in algebraic topology.
Life
Élie Cartan was born 9 April 1869 in the village of Dolomieu, Isère to Joseph Cartan (1837–1917) and Anne Cottaz (1841–1927). Joseph Cartan was the village blacksmith; Élie Cartan recalled that his childhood had passed under "blows of the anvil, which started every morning from dawn", and that "his mother, during those rare minutes when she was free from taking care of the children and the house, was working with a spinning-wheel". Élie had an elder sister Jeanne-Marie (1867–1931) who became a dressmaker; a younger brother Léon (1872–1956) who became a blacksmith working in his father's smithy; and a younger sister
Élie Cartan entered an elementary school in Dolomieu and was the best student in the school. One of his teachers, M. Dupuis, recalled "Élie Cartan was a shy student, but an unusual light of great intellect was shining in his eyes, and this was combined with an excellent memory".
Cartan enrolled in the
After graduation from the École Normale Superieure in 1891, Cartan was drafted into the French army, where he served one year and attained the rank of sergeant. For the next two years (1892–1894) Cartan returned to ENS and, following the advice of his classmate Arthur Tresse (1868–1958) who studied under Sophus Lie in the years 1888–1889, worked on the subject of classification of simple Lie groups, which was started by Wilhelm Killing. In 1892 Lie came to Paris, at the invitation of Darboux and Tannery, and met Cartan for the first time.
Cartan defended his dissertation, The structure of finite continuous groups of transformations in 1894 in the Faculty of Sciences in the Sorbonne. Between 1894 and 1896 Cartan was a lecturer at the University of Montpellier; during the years 1896 through 1903, he was a lecturer in the Faculty of Sciences at the University of Lyon.
In 1903, while in Lyon, Cartan married Marie-Louise Bianconi (1880–1950); in the same year, Cartan became a professor in the Faculty of Sciences at the
As a student of Cartan, the geometer
He died in 1951 in Paris after a long illness.
In 1976, a lunar crater was named after him. Before, it was designated Apollonius D.
Work
In the Travaux, Cartan breaks down his work into 15 areas. Using modern terminology, they are:
- Lie theory
- 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
Cartan's mathematical work can be described as the development of analysis on differentiable manifolds, which many now consider the central and most vital part of modern mathematics and which he was foremost in shaping and advancing. This field centers on Lie groups, partial differential systems, and differential geometry; these, chiefly through Cartan's contributions, are now closely interwoven and constitute a unified and powerful tool.
Lie groups
Cartan was practically alone in the field of Lie groups for the thirty years after his dissertation. Lie had considered these groups chiefly as systems of analytic transformations of an
After 1925 Cartan grew more and more interested in topological questions. Spurred by Weyl's brilliant results on compact groups, he developed new methods for the study of global properties of Lie groups; in particular he showed that topologically a connected Lie group is a product of a Euclidean space and a compact group, and for compact Lie groups he discovered that the possible fundamental groups of the underlying manifold can be read from the structure of the Lie algebra of the group. Finally, he outlined a method of determining the Betti numbers of compact Lie groups, again reducing the problem to an algebraic question on their Lie algebras, which has since been completely solved.
Lie pseudogroups
After solving the problem of the structure of Lie groups which Cartan (following Lie) called "finite continuous groups" (or "finite transformation groups"), Cartan posed the similar problem for "infinite continuous groups", which are now called Lie pseudogroups, an infinite-dimensional analogue of Lie groups (there are other infinite generalizations of Lie groups). The Lie pseudogroup considered by Cartan is a set of transformations between subsets of a space that contains the identical transformation and possesses the property that the result of composition of two transformations in this set (whenever this is possible) belongs to the same set. Since the composition of two transformations is not always possible, the set of transformations is not a group (but a groupoid in modern terminology), thus the name pseudogroup. Cartan considered only those transformations of manifolds for which there is no subdivision of manifolds into the classes transposed by the transformations under consideration. Such pseudogroups of transformations are called primitive. Cartan showed that every infinite-dimensional primitive pseudogroup of complex analytic transformations belongs to one of the six classes: 1) the pseudogroup of all analytic transformations of n complex variables; 2) the pseudogroup of all analytic transformations of n complex variables with a constant Jacobian (i.e., transformations that multiply all volumes by the same complex number); 3) the pseudogroup of all analytic transformations of n complex variables whose Jacobian is equal to one (i.e., transformations that preserve volumes); 4) the pseudogroup of all analytic transformations of 2n > 4 complex variables that preserve a certain double integral (the symplectic pseudogroup); 5) the pseudogroup of all analytic transformations of 2n > 4 complex variables that multiply the above-mentioned double integral by a complex function; 6) the pseudogroup of all analytic transformations of 2n + 1 complex variables that multiply a certain form by a complex function (the contact pseudogroup). There are similar classes of pseudogroups for primitive pseudogroups of real transformations defined by analytic functions of real variables.
Differential systems
Cartan's methods in the theory of differential systems are perhaps his most profound achievement. Breaking with tradition, he sought from the start to formulate and solve the problems in a completely invariant fashion, independent of any particular choice of variables and unknown functions. He thus was able for the first time to give a precise definition of what is a "general" solution of an arbitrary differential system. His next step was to try to determine all "singular" solutions as well, by a method of "prolongation" that consists in adjoining new unknowns and new equations to the given system in such a way that any singular solution of the original system becomes a general solution of the new system. Although Cartan showed that in every example which he treated his method led to the complete determination of all singular solutions, he did not succeed in proving in general that this would always be the case for an arbitrary system; such a proof was obtained in 1955 by Masatake Kuranishi.
Cartan's chief tool was the calculus of
Differential geometry
Cartan's contributions to differential geometry are no less impressive, and it may be said that he revitalized the whole subject, for the initial work of
Cartan showed how to use his concept of connection to obtain a much more elegant and simple presentation of Riemannian geometry. His chief contribution to the latter, however, was the discovery and study of the symmetric Riemann spaces, one of the few instances in which the initiator of a mathematical theory was also the one who brought it to its completion. Symmetric Riemann spaces may be defined in various ways, the simplest of which postulates the existence around each point of the space of a "symmetry" that is involutive, leaves the point fixed, and preserves distances. The unexpected fact discovered by Cartan is that it is possible to give a complete description of these spaces by means of the classification of the simple Lie groups; it should therefore not be surprising that in various areas of mathematics, such as automorphic functions and analytic number theory (apparently far removed from differential geometry), these spaces are playing a part that is becoming increasingly important.
Alternative theory to general relativity
Cartan created a competitor theory of gravity also Einstein–Cartan theory.
Publications
Cartan's papers have been collected in his Oeuvres complètes, 6 vols. (Paris, 1952–1955). Two excellent obituary notices are S. S. Chern and C. Chevalley, in Bulletin of the American Mathematical Society, 58 (1952); and J. H. C. Whitehead, in Obituary Notices of the Royal Society (1952).
- Cartan, Élie (1894), Sur la structure des groupes de transformations finis et continus, Thesis, Nony
- Cartan, Élie (1899), "Sur certaines expressions différentielles et le problème de Pfaff", Annales Scientifiques de l'École Normale Supérieure, Série 3 (in French), 16, Paris: Gauthier-Villars: 239–332, JFM 30.0313.04
- Leçons sur les invariants intégraux, Hermann, Paris, 1922
- Cartan, Élie (1925). "La géométrie des espaces de Riemann". Paris, Gauthier-Villars (Mémorial des sciences mathématiques, fasc. 9.) (in French): IV + 60. JFM 51.0566.01.
- Cartan, Elie (1946). Leçons sur la géométrie des espaces de Riemann (2ème. ed. rev. et aug. ed.). Paris: Gauthier-Villars. p. VIII, 378. Zbl 0060.38101.
- Cartan, Élie (1931). "La théorie des groupes finis et continus et l'analysis situs". Mémorial des sciences mathématiques (42): 68. JFM 56.0370.08.
- Cartan, Elie (1950). Leçons sur la géométrie projective complexe (2d ed.). Paris : Gauthier-Villars. p. VII + 325. Zbl 0003.06801.
- La parallelisme absolu et la théorie unitaire du champ, Hermann, 1932
- Les Espaces Métriques Fondés sur la Notion d'Arie, Hermann, 1933[7]
- La méthode de repère mobile, la théorie des groupes continus, et les espaces généralisés, 1935[8]
- Leçons sur la théorie des espaces à connexion projective, Gauthiers-Villars, 1937[9]
- La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile, Gauthiers-Villars, 1937[10]
- Cartan, Élie (1981) [1938], The theory of spinors, New York:
- Les systèmes différentiels extérieurs et leurs applications géométriques, Hermann, 1945[14]
- Oeuvres complètes, 3 parts in 6 vols., Paris 1952 to 1955, reprinted by CNRS 1984:[15]
- 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
- Élie Cartan and Albert Einstein: Letters on Absolute Parallelism, 1929–1932 / original text in French & German, English trans. by Jules Leroy & Jim Ritter, ed. by Robert Debever, Princeton University Press, 1979[16]
See also
- Exterior derivative
- Integrability conditions for differential systems
- Isotropic line
- CAT(k) space
- Einstein – Cartan theory
- Hermitian symmetric space
- Moving frame
- Pseudogroup
- Pure spinor
References
- ^ 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
- ^ a b O'Connor, J J; Robertson, E F (1999). Great Mathematicians of the 20th century (PDF).
- ^ Jackson, Allyn (1998). "Interview with Shiing Shen Chern" (PDF).
- ^ "Élie J. Cartan (1869–1951)". Royal Netherlands Academy of Arts and Sciences. Retrieved 19 July 2015.
- ^ Neurath, Otto (1938). "Unified Science as Encyclopedic Integration". International Encyclopedia of Unified Science. 1 (1): 1–27.
- ISSN 0002-9904.
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- JSTOR 3606453.
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- ^ "Review of Élie Cartan, Albert Einstein: Letters on Absolute Parallelism, 1929–1932 edited by Robert Debever". Bulletin of the Atomic Scientists. 36 (3): 51. March 1980.
External links
- Media related to Élie Cartan at Wikimedia Commons
- M.A. Akivis & B.A. Rosenfeld (1993) Élie Cartan (1869–1951), translated from Russian original by V.V. Goldberg, ISBN 0-8218-4587-X.
- Chern, Shiing-Shen; .
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 into English by M. Nadjafikhah)"