VI.69 Élie Joseph Cartan
b. Dolomieu, France, 1869; d. Paris, 1951
Lie algebras; differential geometry; differential equations
Cartan was one of the leading mathematicians of his generation, particularly influential for his work on geometry and the theory of LIE ALGESRAS [III.48 §§2, 3]. In the bleak years after World War I he was one of the most prominent mathematicians in France. He eventually became a notable influence on the BOURBAKI [VI.96] group, of which his son Henri, another distinguished mathematician, was one of the seven founder members. Cartan held lecturing positions in Montpellier and Lyon before becoming a professor in Nancy in 1903. He went on to gain a lecturing position at the Sorbonne in 1909, becoming a professor in 1912 and remaining there until his retirement.
In his doctoral thesis of 1894 Cartan classified the simple Lie algebras over the field of complex numbers, refining and correcting earlier work of Wilhelm Killing and emphasizing the deep general abstract structures inherent in the theory. In later years he returned to these ideas and drew out their implications for the study of the corresponding LIE GROUPS [III.48 §1]—these groups have a major bearing on symmetry considerations in physics.
Cartan spent much of his life working on geometry. In the 1870s and the 1890s KLEIN [VI.57] had analyzed geometry and shown how the major branches (Euclidean, non-Euclidean, projective, and affine) could be unified and treated as special cases of projective geometry. Cartan became interested in the extent to which the group-theoretic ideas that had animated Klein could be adapted to the setting of differential geometry, and especially to spaces of variable CURVATURE [III.78]—the mathematical setting for EINSTEIN’S GENERAL THEORY OF RELATIVITY [IV.13]. In that subject the observations of different observers are related by coordinate transformations, and changes in the gravitational field are expressed through changes in the metric, and hence curvature, of the underlying spacetime manifold. In the 1920s Cartan broadened the setting to what are today called FIBER BUNDLES [IV.6 § 5], and showed that Klein’s approach could be carried through by concentrating on the possible types of coordinate transformation and the Lie groups to which they can belong.
There are many problems in which one has a multitude of possible observations at each point of a space: for example, the weather at each point of Earth’s surface. In Cartan’s formulation, Earth’s surface is taken as the base MANIFOLD [I.3 §6.9] and the possible observations at each point form another manifold, called the fiber at the point. The pair consisting of all fibers and all points of the base manifold is, roughly, a fiber bundle; the precise concept has proved to be fundamental across the whole field of modern differential geometry. It was to prove a natural setting for the study of what are called connections on a manifold, which deal with the way objects, such as vectors, are transformed as they move along curves in the manifold. Cartan’s fundamental idea was to capture the symmetry of a geometrical problem by allowing fibers to have a common symmetry group, although aspects of the geometry of the base manifold, such as its curvature, were allowed to vary from point to point in such a way that the base manifold admits no symmetries at all.
Cartan also applied his geometric approach to the study of differential equations, which had earlier been a motivating concern for LIE [VI.53] in the creation of the theory of Lie algebras. He did important work on systems of equations, and this led him to emphasize the role of what are called exterior forms. Familiar examples include the 1-FORM [III.16] that represents the element of length along a curve, the 2-form that represents the element of area of a surface, and so on. The main thing one does to a 1-form is integrate it; integrating the 1-form that describes arc-length gives length along a curve. Cartan studied systems of equations involving arbitrary 1-forms and was led to discover ways in which the algebra of 1-forms, and more generally the algebra of k-forms for arbitrary k, captures features of the geometry of the manifold on which they are defined. This led him to reformulate a method of studying the geometry of curves and surfaces that had been pursued by Gaston Darboux, the leading French geometer of the previous generation, and to proclaim his method of “moving frames” that again related to the study of fiber bundles and symmetries in differential geometry. This work, together with his work on fiber bundles, remains a major source of ideas for the study of differentiable manifolds to this day.
Further Reading
Chern, S.-S., and C. Chevalley. 1984. Élie Cartan and his mathematical work. In Oeuvres Complétes de Élie Cartan, volume III.2 (1877-1910). Paris: CNRS.
Hawkins, T. 2000. Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics, 1869-1926. New York: Springer.