Élie Cartan

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 analytic manifold, depending analytically on a finite number of parameters. A very fruitful approach to the study of these groups was opened in 1888 when Wilhelm Killing systematically started to study the group in itself, independent of its possible actions on other manifolds. At that time (and until 1920) only local properties were considered, so the main object of study for Killing was the Lie algebra of the group, which exactly reflects the local properties in purely algebraic terms. Killing's great achievement was the determination of all simple complex Lie algebras; his proofs, however, were often defective, and Cartan's thesis was devoted mainly to giving a rigorous foundation to the local theory and to proving the existence of the exceptional Lie algebras belonging to each of the types of simple complex Lie algebras that Killing had shown to be possible. Later Cartan completed the local theory by explicitly solving two fundamental problems, for which he had to develop entirely new methods: the classification of simple real Lie algebras and the determination of all irreducible linear representations of simple Lie algebras, by means of the notion of weight of a representation, which he introduced for that purpose. It was in the process of determining the linear representations of the orthogonal groups that Cartan discovered in 1913 the spinors, which later played such an important role in quantum mechanics.

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.

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 ${\displaystyle 2n>4}$ complex variables that preserve a certain double integral (the symplectic pseudogroup); 5) the pseudogroup of all analytic transformations of ${\displaystyle 2n>4}$ complex variables that multiply the above-mentioned double integral by a complex function; 6) the pseudogroup of all analytic transformations of ${\displaystyle 2n+1}$ 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.

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 exterior differential forms, which he helped to create and develop in the ten years following his thesis and then proceeded to apply to a variety of problems in differential geometry, Lie groups, analytical dynamics, and general relativity. He discussed a large number of examples, treating them in an extremely elliptic style that was made possible only by his uncanny algebraic and geometric insight.

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 Riemann and Darboux was being lost in dreary computations and minor results, much as had happened to elementary geometry and invariant theory a generation earlier. His guiding principle was a considerable extension of the method of "moving frames" of Darboux and Ribaucour, to which he gave a tremendous flexibility and power, far beyond anything that had been done in classical differential geometry. In modern terms, the method consists in associating to a fiber bundle E the principal fiber bundle having the same base and having at each point of the base a fiber equal to the group that acts on the fiber of E at the same point. If E is the tangent bundle over the base (which since Lie was essentially known as the manifold of "contact elements"), the corresponding group is the general linear group (or the orthogonal group in classical Euclidean or Riemannian geometry). Cartan's ability to handle many other types of fibers and groups allows one to credit him with the first general idea of a fiber bundle, although he never defined it explicitly. This concept has become one of the most important in all fields of modern mathematics, chiefly in global differential geometry and in algebraic and differential topology. Cartan used it to formulate his definition of a connection, which is now used universally and has superseded previous attempts by several geometers, made after 1917, to find a type of "geometry" more general than the Riemannian model and perhaps better adapted to a description of the universe along the lines of general relativity.

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.