## Reviews of Élie Cartan's books

**1. Leçons sur la Géométrie des Espaces de Riemann (1928), by Elie Cartan.**

1.1. Review by: M S Knebelman.

Amer. Math. Monthly36(10) (1929), 528-530.This book begins with very simple and familiar ideas of vectors in Euclidean space in rectangular Cartesian coordinates and gradually arrives at the notion of a tensor and the algebraic and differential operations with tensors. In fact the study of the differential geometry of a Riemann space does not really begin until the fourth chapter. The method of study consists of associating with each point of the Riemann space an osculating Euclidean space; it then follows that the properties (at the point) of the Euclidean space which depend on the fundamental tensor and its first partial derivatives are valid for the Riemann space. ... Although a great many theorems of classical differential geometry have been generalized or extended to any Riemann space, the author usually con- fines his attention to two and three dimensional Riemann spaces with positive definite metrics. ... On the whole, one leaves the book with a feeling of having read about something very concrete, the language throughout being vividly geometrical; yet there is no lack of analytical rigour. ... A great many interesting and important problems are not mentioned at all in this book but, as the author states in the preface, they will probably form the subject matter of another volume which will no doubt be welcomed by every student of geometry.

**2. Leçons sur la Géométrie des Espaces de Riemann (Second Edition) (1946), by Elie Cartan.**

3.2.1. Review by: S Bochner.

Mathematical Reviews, MR0020842(8,206g).The book is pervaded with tensor analysis from first to last, and yet tensor formalism for its own sake is rather underplayed. In this sense, this is not a book from which to learn the skill of tensor formalism; just as a book in complex variables, if leaning in a geometric direction, need not be the appropriate source from which to learn the technique of power series manipulation. The most engaging feature of the book, fully retained in the new edition, is the insistence on problems in the large, or of global incipience at least. There are ample discussions of covering spaces and fundamental groups for complete Riemannian spaces of constant curvature, positive, zero or negative. ... Entirely new is first of all a chapter on symmetric spaces. ... New also is a full-length chapter on Lie groups of motions, replete with contents. ... Finally there is a new chapter which methodically unifies the problem of deciding the (local) equivalence of two metrics with Killing's approach to determining the largest group of motions for any one metric

**La méthode du repère mobile, la théorie des groupes continus et les espaces généralisés (1935), by Elie Cartan.**

3.1. Review by: J A Todd.

The Mathematical Gazette19(233) (1935), 154.This tract gives an account of a generalisation of the idea of moving axes, familiar in elementary differential geometry, to a form which can be adapted to study the intrinsic geometry of spaces of various kinds. The subject is intimately connected with the Lie theory of continuous groups, since each such group may be regarded as a set of transformations in a space, the space being the group itself. The principle of the method is to associate a configuration with each point of the space, with the property that there is just one displacement in the space which changes the figure associated with a point A into that associated with any other point B. The equations deter- mining the local variations of these figures can be determined, and are intimately related to the structure-equations of the group. These equations, in fact, contain in essence the whole of the differential geometry of the space.

**4.**

**La Topologie des Groupes de Lie (1936), by Elie Cartan.**

4.1. Review by: J H C Whitehead.

The Mathematical Gazette23(255) (1939), 318.This book contains a survey of what is known of the subject, most of which knowledge is due to Prof Cartan himself. After two chapters on the generalities of the theory there is a chapter on closed groups and one on closed simple groups. ... There follows a chapter on open groups, containing the theorem that the first Betti number of an open simple group is 0 or 1. In the next chapter (Chapter VI) there is a very pretty proof of the third fundamental theorem of Lie, asserting the existence of a group with given constants of structure. Of course this was proved by Lie himself for what is now known as a "gruppenkeim" (in general a "gruppenkeim" is not a group, since a product xy will only exist if x and y lie near enough to the identity). But it was left for Cartan himself, in 1930, to prove the existence of an actual group with given constants of structure. The book concludes with a statement of all known theorems on the Betti numbers of closed simple groups - among others the results of L Pontrjagin and R Brauer, who have calculated them for the four main types of simple group.

**5. Leçons sur la Théorie des Espaces à Connexion Projective (1937), by Élie Cartan.**

6.5.1. Review by: J L Vanderslice.

Bull. Amer. Math. Soc.44(1) (1938), 11-13.In a recent monograph, 'La Méthode de Repère mobile, ...' E Cartan presented a splendid outline of his general method of approach toward all branches of differential geometry. In a word, it consists of a far-reaching generalization of the familiar moving trihedral with assistance from the theory of groups. The analytical formulation employs the exterior differential calculus, a discipline extensively used by Cartan since the turn of the century. And there is frequent recourse to the theory of Pfaffian systems. Undoubtedly his unusual analytical machinery is, to many, a source of difficulty. Most differential geometers use, instead, Ricci's tensor calculus and theorems on total differential equations stemming from Christoffel. But every disciple of Ricci knows the profit which lies in the study of Cartan. In the above monograph the author devoted one paragraph to showing how projective differential geometry fits into his general scheme. In the work here under review this paragraph is presented to us in more satisfactory form as a book of three hundred odd pages. The book has two principal divisions, the first devoted to classical projective differential geometry, the second, to the geometry of projectively connected spaces. ... Of all Cartan's books this is one of the most clearly written but, like the others, it can be fully appreciated only by those having some previous acquaintance with its field. The author works through his subject informally, enriching it with his unified point of view and his unsurpassable geometric insight, finding new approaches leading to a better understanding, and giving here and there a new result for good measure. This is probably the first major publication in book form to have classical and generalized projective differential geometry fraternizing within its covers. And the author was a pioneer in both fields.

**La Théorie des Groupes Finis et Continus et la Géométrie Différentielle traitées par la Méthode du Repère Mobile (1937), by Élie Cartan.**

7.6.1. Review by: Hermann Weyl.

Bull. Amer. Math. Soc.44(9) (1938), 598-601.This book, which originated from a course of lectures given in 1931-1932 at the Sorbonne, covers in a somewhat more explicit form essentially the same material as the 'Actualités Scientifiques et Industrielles' (1935) ... The book under review pursues a three-fold purpose: it contains (1) an exposition of the general theory of finite continuous Lie groups in a terminology adapted to its differential geometric applications; (2) a general description of the method of 'repères mobiles'; and (3) its application to a number of important examples. ... All of the author's books, the present one not excepted, are highly stimulating, full of original viewpoints, and profuse in interesting geometric details. Cartan is undoubtedly the greatest living master of differential geometry. ... We should not let pass unmentioned Jean Leray's merit in molding the lecture notes he took into something which is a true book and yet catches some of the vividness of the original lectures. Nevertheless, I must admit that I found the book, like most of Cartan's papers, hard reading. Does the reason lie only in the great French geometric tradition on which Cartan draws, and the style and contents of which he takes more or less for granted as a common ground for all geometers, while we, born and educated in other countries, do not share it?

**Leçons sur la théorie des spineurs (2 vols.) (1938), by Élie Cartan.**

7.1. Review by: H S Ruse.

The Mathematical Gazette23(255) (1939), 320-323.In the preface to the two volumes under review M Cartan points out that, in their most general mathematical form, spinors were discovered by him in 1913 in his work on linear representations of simple groups, and he emphasises their connection, shown in Vol. II, with Clifford-Lipschitz hypercomplex numbers. In the text he develops the theory from the beginning, the first volume being devoted to spinors in three-dimensional space and the second to spinors in n-dimensional space, with special reference to the four-dimensional space-time of Special Relativity. The point of view is fundamentally geometrical, the theory being presented as one of linear representations of the rotation-groups of determinants + 1 and -1. ... M Cartan's book will be indispensable to mathematicians interested in the geometrical and physical aspects of group theory, giving, as it does, a complete and authoritative survey of the algebraic theory of spinors treated from a geometrical point of view. It is not perhaps to be recommended as an introductory book, for it would be difficult to follow without some previous knowledge either of group theory in general or of spinor theory in particular, but it is and will remain a standard work. Whether it will prove to contain the last word on the subject of spinors remains to be seen

**8. Les systèmes différentiels extérieurs et leurs applications géométriques (1945), by Élie Cartan.**

8.1. Review by: J M Thomas.

Bull. Amer. Math. Soc.53(3) (1945), 261-266.This book gives a revised account of lectures delivered in 1936-1937 at the Sorbonne. The content is based almost exclusively on the author's own outstanding contributions to the subject at the beginning of the century. In developing many of these old results, however, the author's present viewpoint is new as well as stimulating. The whole treatment, even at the few points where it touches upon contemporary work by other writers, bears the stamp of the author's individuality. The fundamental calculus employed is Grassmann algebra. Cartan's manner of regarding this discipline might be described as follows. The indeterminates are differentials and the polynomials are forms.

8.2. Review by: J A Schouten.

Mathematical Reviews, MR0016174(7,520d).The first part of this book contains the theory of integration of total differential equations connected with a general system of exterior differential forms (covariant alternating quantities). ... The second part of the book contains applications to several problems of differential geometry.

**9. The Theory of Spinors (1966), by Élie Cartan.**

10.9.1. Review by: Robert Hermann.

Amer. Math. Monthly90(10) (1983), 719-720.Cartan is certainly one of the greatest and most original minds of mathematics, whose work on Lie groups, differential geometry, and the geometric theory of differential equations is at the foundation of much of what we do today. In my view, his place in mathematics is similar to that of the great turn-of-the-century masters in other areas of intellectual life. Just as Freud was influenced by the mechanistic world view of 19th century science, but used this background to create something new and revolutionary which has profoundly influenced 20th century thought, so Cartan built, on a foundation of the mathematics which was fashionable in the 1890's in Paris, Berlin and Göttingen, a mathematical edifice whose implications we are still investigating. His work was highly intuitive and geometric, but was also based on a formidable combination of original methods of calculation and analysis, ranging in mathematical expertise from algebra to topology. For example, he completed the work of Killing and Lie on the classification of simple Lie algebras. As Hawkins so convincingly demonstrates [1], this required the mastery of the most advanced algebraic technique of the 1890's, a task at which Killing himself (who learned his algebra from Weierstrass!) had despaired. In the 1920's, when he was already in his 50's, he proved that the second homotopy group of a Lie group was zero, which was one of the first great general theorems about topology. As one can see in his Collected Works, he was a master of brutal calculations, and all of his work was based on an intimate knowledge of computational details and examples. In short, he was comparable to such great figures of mathematics as Gauss, Riemann, and Poincar6. Thus it is unfortunate that his original work is so inaccessible to the wide spectrum of mathematicians and scientists who now make use of it. (Of course, it is accessible through the expositions by many others in the last twenty years.) This short book is a translation of one which was written in French in the 1930's, and is perhaps the most readable of his works. It was first published in 1966, when the work of Killing and Cartan on the classification of simple Lie groups was beginning to be applied in elementary particle physics. In terms of contemporary Lie group theory, it deals with the B and D series of simple Lie algebras and the Lie groups which go along with them, i.e., the orthogonal matrix groups over the real and complex numbers and their simply connected covering groups. ... I hope I am not insulting the memory of my greatest hero to say that this book is a fraud! I don't believe that Cartan thought about the subject in the form in which it is presented here. Clearly, he is presenting a "vulgarization" of the general theory of semi-simple Lie algebras and groups, which he developed almost single-handedly (with the help of Hermann Weyl!) in the period 1893-1930. Cartan was very much a fan of physics, and he clearly is trying to teach the physicists of his day some of his profound knowledge in a form which they might find more palatable. The recently published correspondence between Einstein and Cartan is very illuminating about the habits of mind of these two great men, and even somewhat sad. They were like ships passing in the night: Cartan enthusiastically tried to communicate some of his great geometric ideas to Einstein, who was rather closed-minded and even condescending. Finally, I would like to use this occasion to convey my thanks to Dover Publications. Their efforts to keep in print, at modest prices, the classics of science have been a great help in my own work. Many times I have picked up a Dover book at my local bookstore and discovered a jewel of which I was previously unaware. Even more important, it is one of the last links in today's world with certain precious scholarly traditions.

9.2. Description by the publisher of the 1981 reprint by Courier Dover Publications.The French mathematician Élie Cartan (1869-1951) was one of the founders of the modern theory of Lie groups, a subject of central importance in mathematics and also one with many applications. In this volume, he describes the orthogonal groups, either with real or complex parameters including reflections, and also the related groups with indefinite metrics. He develops the theory of spinors (he discovered the general mathematical form of spinors in 1913) systematically by giving a purely geometrical definition of these mathematical entities; this geometrical origin makes it very easy to introduce spinors into Riemannian geometry, and particularly to apply the idea of parallel transport to these geometrical entities. The book is divided into two parts. The first is devoted to generalities on the group of rotations in n-dimensional space and on the linear representations of groups, and to the theory of spinors in three-dimensional space. Finally, the linear representations of the group of rotations in that space (of particular importance to quantum mechanics) are also examined. The second part is devoted to the theory of spinors in spaces of any number of dimensions, and particularly in the space of special relativity (Minkowski space). While the basic orientation of the book as a whole is mathematical, physicists will be especially interested in the final chapters treating the applications of spinors in the rotation and Lorentz groups. In this connection, Cartan shows how to derive the "Dirac" equation for any group, and extends the equation to general relativity. One of the greatest mathematicians of the 20th century, Cartan made notable contributions in mathematical physics, differential geometry, and group theory. Although a profound theorist, he was able to explain difficult concepts with clarity and simplicity. In this detailed, explicit treatise, mathematicians specializing in quantum mechanics will find his lucid approach a great value.

**Geometry of Riemannian spaces (1983), by Élie Cartan.**

10.1. Review by: H Rund.

Mathematical Reviews, MR0709421(85m:53001).The first edition of this profound treatise was based on lectures given by the author in the 1925-26 academic year. A somewhat modernized and extensively augmented edition appeared in 1946; a reprint appeared in 1951. In view of the great influence which this work has exerted on the subsequent development of differential geometry, it is perhaps appropriate that a review of its translation, even at this late stage, should present an outline of its contents. ... Extensive commentaries by R Hermann on various sections of the main text should enable the reader to grasp some of its material from the perspective of modern techniques and terminology. An elucidation of the concepts that occur in these commentaries is presented in a series of three appendices by Hermann, entitled "The formalism of connection theory", "Cartan's method of the moving frame as a generalization of Klein's 'Erlanger Programm' ", "Excursions into the theory of Cartan connections".

**11. Riemannian geometry in an orthogonal frame (2001), by Élie Cartan.**

11.1. Review by: Mathematical Reviews.

Mathematical Reviews, MR1877071(2002h:53001).From lectures delivered by Élie Cartan at the Sorbonne in 1926-27. With a preface to the Russian edition by S P Finikov. Translated from the 1960 Russian edition by Vladislav V Goldberg and with a foreword by S S Chern.

JOC/EFR November 2014

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