Differential Geometry, Lie Groups, and Symmetric Spaces. by Sigurdur HelgasonReview by: Robert HermannSIAM Review, Vol. 22, No. 4 (Oct., 1980), pp. 524-526Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2029828 .

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524 BOOK REVIEWS

Nonlinear programming deals with the problem of optimizing an objective func- tion subject to equality and inequality constraints when some of the functions in- volved are not linear. The various topics associated with the mathematical founda- tions and computational methods of nonlinear programming exceed what can be covered in a book of reasonable size. This book is a good introductory sample of this material. The language used is clear and distinct. This book is well organized and relatively self-contained. Some recent research, however, such as the Augmented Lagrange Multiplier method that was independently proposed by Hestenes and Powell, deserves more emphasis than is given in this book.

The book is divided into three major parts: convex analysis, optimality condi- tions and duality, and computational methods. Detailed numerical examples and graphical illustrations are provided to aid the reader. Also, each chapter contains many exercises. These include, simple numerical problems that reinforce the ma- terial discussed in the text, problems introducing new material, and theoretical exer- cises for advanced students. At the end of each chapter, extensions, references, and material related to that covered in the text are presented. These notes, as well as an extensive bibliography, should be very useful to the reader interested in further study. Also, Chapter 1 gives several examples of problems from different engi- neering disciplines that can be viewed as nonlinear programs.

This book can be used both as a reference for nonlinear programming topics and as a text in the field of operations research, management science, industrial engi- neering, applied mathematics, and in engineering disciplines that use analytic optimi- zation techniques. The material in this book requires some mathematical maturity and a working knowledge of linear algebra and calculus. The appendix summarizes some mathematical topics frequently used in the book.

This book can be used both with graduate and undergraduate students. The pref- ace discusses various one-semester courses that can be given using the book.

STAN FROMOVITZ University of Maryland

Differential Geometry, Lie Groups, and Symmetric Spaces. By SIGURDUR HELGA- SON. Academic Press, New York, 1978. xv+628 pp. $27.00. When the Book Review Editor of this Review wrote me to enquire if I wanted to

do this book, he also enclosed a note asking me if I even thought it was appropriate to review. I took this to mean that he was afraid that readers of this journal would not find Lie group theory relevant to their professional activities. Of course, I am hardly one to concede such a point-a good part of my own work has been concerned with applications of Lie group theory to physics and control theory-but the evident fact that some segments of the applied mathematical community might think this, if only in private, has inspired me to make several comments about the relations between "pure" and "applied" mathematics.

In the 1960's, when the pure mathematicians were prospering, there was a touch of "noblesse oblige" in their attitude toward applications, which reached its extreme in the COSRIMS report [1], [2]. In the 1970's there has been a tendency all through science to emphasize financially the "applied" over the "basic", and this has, slowly but inevitably, had an effect on the balance within mathematics. I, and many others whose education was in "pure" mathematics, welcomed this renewed em- phasis on ties between mathematics and the outside world, but the pendulum is now swinging too far, and there is a danger that the health of the whole mathematical en- terprise will be irreversibly affected.

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BOOK REVIEWS 525

I wholeheartedly applaud Reuben Hersh's recent article [3], and his insistence that our ignorance of the "philosophical" aspect of intellectual discourse has had a disastrous effect on our ability to guide mathematics as a learned profession. If ev- eryone only does his job in the best way he is able, and, perhaps modestly, does not think or talk about the broader issues, the result is chaos-everyone is cancelled out, there is no consensus, students are confused, and decide to forget the hard work of learning mathematics and get an MBA to make money. If we are to follow Hersh's advice and examine the pure vs. applied question in this broader light, I believe we must go at least as far back as the 19th century. Certainly, that was the last period in which there was a strong unification, and individuals could and did readily move back and forth. Of course, some of the greatest mathematical minds of the 19th cen- tury (e.g., Gauss, Cauchy, Jacobi, Riemann, and Poincare) were equally at home in both. Felix Klein's "Lectures on the History of Mathematics in the 19th Century" [4] is probably the best place to start reading to capture the flavor of those times. The impression I gained from it was the importance of the continuity between pure and applied, the difficulty and inappropriateness in drawing a sharp line to separate them. Many theories which we think of today as the ultimate in pure mathematics were, of course, the "applied mathematics" of the time. (Certainly this is true of Lie group theory!) In a related manner, Poincare has emphasized [5] the crucial importance of the synthesis between "intuition" and "logic"; as he puts it, we need both Weir- strass and Riemann. It is clear from the context that Poincare would put applied mathematics, which he would probably call "mathematical physics", on the "intui- tion" side of the balance. It seems to me that a study of 19th century science teaches us that the most useful sort of applied mathematicians are those who bring together in their own mind scientific ideas and the appropriate mathematics. They are mathe- matical generalists, prepared to learn any branch of mathematics if it will be useful in their work. Many physicists still have this capability, and this is one major reason that physics is still throwing out mathematical ideas of extraordinary richness.

In a recent book review in this Review [6], Hector Sussmann quotes Joe Keller as blaming the fiasco of catastrophe theory on the desire of pure mathematicians to do applications, while knowing only mathematics. While this is indeed an acute ob- servation (more typical, in fact, of the 1960's), it is one-sided; judging from my obser- vations, a much more serious issue now is the narrowness of the mathematical base of most applied mathematicians and those scientists and engineers who use mathe- matics in their work. Clearly, a basic problem is the lack of high-level exposition and surveys which would make the whole spectrum of mathematical thought as widely accessible as possible. Certainly, there seems to be much more emphasis on these in the Soviet Union, and, for whatever reasons, Soviet scientists and mathematicians seem to have kept in much closer contact.

Lie group theory itself is an excellent case-history for the study of the inter- action between pure and applied mathematics. It was developed by Lie in the 1870's and 1880's, based on a close study of two seemingly different situations, which Lie's genius saw were interrelated: The curves in projective space that are acted on transi- tively by groups of projective transformations (Klein and Lie were students of Plucker, the founder of projective differential geometry, as well as an experimental physicist of first rank!), and the way that explicit solutions of certain differential equations are related to groups of symmetries. Lie correctly sensed that there was a vast new theory underlying these special situations-the excitement is very evident in his papers [7]. (Lie was, according to both Klein and Poincare, an intuitive person, presumably very much a product of the romantic nationalist movement of the times.

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526 BOOK REVIEWS

He states many times that his goal is to carry over the ideas of his fellow Norwegian, Abel, from algebra to differential equations.) Now, there were certain subtleties in understanding how groups could act on differential equations. In order to elucidate them, Lie developed two superstructures, the theory of "infinitesimal transforma- tions", which we call " Lie algebras", and " mapping elements", which C. Ehresmann developed in the 1940's in terms of what he called the "theory of jets". Lie's ideas were picked up by Elie Cartan, who developed them brilliantly. Most of the basic Lie theory that we use today was completed by 1926. However, this material was almost completely unknown to the world, "pure" or "applied", until comparatively re- cently. It is still difficult to understand, especially for an applied person who has no natural feeling or sympathy for the version of geometry that underlies the work of both Lie and Cartan. For example, a recent monograph [8] by two eminent applied mathematicians does not even display the structure which Lie gave the subject in 1884 [7], not to mention the later work of Cartan and Ehresmann. I mention this not to berate the authors of this otherwise admirable book, but only to draw the moral that it is essential that applied mathematicians be prepared intellectually to keep up with the advances in all branches of mathematics which are related to their subject.

The book under review is a revision of Differential Geometry and Symmetric Spaces, first published in 1960, which finally made widely available to the mathemat- ical-scientific world the work of Cartan. (Lie's work on the relation to differential equations played a very minor role-the equivalent book in this field is yet to be written.) It triggered a vast array of further research, and has been widely used by physicists in applications of group theory. What was so impressive about Helgason's 1960 book was its skill and force in developing the needed background in manifold theory and algebra, and pressing on to give crisp, but accessible expositions and proofs of many major results. It is still required reading for anyone seriously in- terested in Lie group theory, pure or applied. The first eight chapters of this new book have the same titles as the earlier work, with some additions. Chapters 9 and 10 are mostly new, containing a more extensive treatment of the classification theory utilizing some unpublished work of V. Kac. The old Chapter 9, devoted to the study of analysis on symmetric spaces, has disappeared; the author promises as a sequel which will treat this topic at booklength. Another valuable feature is a more exten- sive system of notes, guiding the reader through the history and further literature.

REFERENCES [1] Th1e Mathematical Sciences, A Report, National Academy of Sciences, Washington, DC, 1969. [2] R. HERMANN, Comments on the COSRIMS Reports, Amer. Math. Monthly, 17 (1970), pp. 517-521. [3] R. HERSH, Some Proposals for Reviving the Philosophy of Mathematics, Adv. Math., 31 (1979), pp.

31-50. [4] F. KLEIN, Development of Mathematics in the 19th Centur, Math Sci Press, Brookline, MA, 1979. [5] H. POINCARE, The Value of Science, Dover, New York, 1958. [6] H. SUSSMANN, Review of "Catastrophe Theory, Selected Papers", by E. C. Zeeman, SIAM Review,

21(1979), pp. 258-276. [7] SOPHUS LIE, Uber Differentialinariainten, Mathematische Annalen 24 (1884), pp. 537-578. (trans-

lated in "Lie Groups: History, Frontiers and Applications", Vol. 3, Math. Sci. Press, Brookline, MA, 1976).

[8] G. BLUMAN AND J. COLE, Similiarity Methods for Differential Equations, Springer-Verlag, New York, 1974.

ROBERT HERMANN Association for Physical and Systems Mathematics

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