The geometry of physics. An introduction. (English) Zbl 0888.58077

Frankel, Theodore

Cambridge: Cambridge University Press. xxii, 654 p. £65.00, $ 95.00 hbk; £19.95/pbk (1997).

C.S.Sharma (London):
The layout, the typography and the illustrations of this advanced textbook on modern mathematical methods are all very impressive and so are the topics covered in the text. The author claims to have had the benefit of lessons from Andreotti, Bott, Chern, Eells, Evans, Flanders, Hopf, Loewner, Samelson, Steenrod and Wheeler and so one expects the account to be crisp and authoritative. In this context, it is disappointing to find that inverted letters of the alphabet such as \(\exists\) and \(\forall\) are missing from the text: but that might be quite diplomatic on the part of the author because most theoretical physicists react to these as bulls to red rags. On the other hand, in the humble opinion of the reviewer, the essence of modern mathematics lies in the definitions: mathematical objects should be well-defined and contradiction-free. Taking the treatment of the Hodge duality as a bench mark, the reviewer found the fundamental definitions unsatisfactory and even absurd. What is one to make out of the equation (14.5) on page 362 which reads \[ *1=\sqrt{|g|}\varepsilon_{12\dots n}dx^1\land \dots\land dx^n=\text{vol}^n? \] Here \(\text{vol}^n\) is the volume element which is defined elsewhere without the epsilon symbol and one wonders what the epsilon symbol is doing here, perhaps it tells us that the sign is positive! For one-forms the Hodge duality is defined by contraction of the volume element by the dual of the one-form and for a \(p\)-form the Hodge duality is said to be a generalization of this process. We are not even told what the dual of a two form is in this context. Among the formulae which illustrate this generalization are \[ \alpha^p\land *\beta^p = (\alpha\land*\beta)_{12\dots n}dx^1\land \dots\land dx^n \] and \[ \alpha^p\land *\beta^p=\langle\alpha^p, \beta^p\rangle\text{vol}^n. \] The inner product for forms is perhaps defined somewhere, but the reviewer could not locate it with the help of the index. The volume element is called a pseudoform because its sign depends on the orientation of the coordinate system! The reviewer found the notation cumbersome and unsatisfactory and the logical sequence bizarre. However, the book may appeal to theoretical physicists with a more traditional training.
V.Perlick (Berlin):
It is not easy to write a book that pleases both physicists and mathematicians. Typically, mathematicians don’t like physics text-books because they are annoyed by a lack of precision. Conversely, physicists don’t like mathematics text-books because they are bored by ‘irrelevant’ subtleties among which it is difficult for them to find the results they are interested in. As a consequence, physicists and mathematicians are traditionally trained with different books, and this is certainly one of the main reasons why they often have problems to find a common language. The book under review may be viewed as a valuable contribution to bridge this gap.
On the one hand, it meets the mathematical standards of a text-book on differential geometry; on the other hand, it strongly appeals to intuition and emphasizes applications to physics. The book is elementary in the sense that it starts out with a careful introduction to manifolds, so the only mathematical prerequisites needed are basic linear algebra and elementary calculus. In later chapters, however, it gradually proceeds to rather advanced topics such as singular homology and Morse theory. As to the physical applications, the author evidently has some personal preferences. E.g., a lot of space is devoted to various aspects of gauge theory, whereas classical mechanics is treated comparatively briefly and some other topics such as string theory are missing completely. None the less, it is not much of an exaggeration to say that after working through these 654 pages the reader will have an overview of almost all applications of differential geometry on finite-dimensional manifolds to theoretical physics.
The book is subdivided into three parts. The first part, entitled ‘Manifolds, tensors, and exterior forms’ introduces the basic notions of differential geometry, with a special emphasis on differential form calculus. Applications to physics include classical mechanics, classical electrodynamics and phenomenological thermodynamics in the spirit of Carathéodory. The second part is entitled ‘Geometry and topology’. It begins with basic material on Riemannian and pseudo-Riemannian geometry and then turns to topological techniques, including homology theory. Applications to physics concentrate on relativity. The last part ‘Lie groups, bundles, and Chern forms’ centers upon the notion of symmetry. It includes a detailed chapter on the Dirac operator which is to be viewed as particularly valuable. This chapter may help to overcome the unsatisfactory situation that for a physics student it is still difficult to access the sophisticated mathematical literature on this subject, whereas the traditional physics literature restricts to calculational rules and leaves the geometric background in the dark. Other applications to physics include the Aharonov-Bohm effect, topological quantization, the Berry phase, and Yang-Mills theory. The book ends with an appendix on continuum mechanics in differential form calculus and with a list of selected references.
Each chapter begins with a quotation or with a question, raising the readers curiosity about the subject to be addressed, and it ends with a set of problems. The problems come without solutions but, in most cases, for a reader who has carefully worked through the preceding chapter, it should not be too difficult to solve them. A large number of carefully prepared figures contributes to the general impression that the book is written with great didactical skill. Lecturers will find a lot of interesting suggestions, whereas graduate and advanced undergraduate students may use the book successfully for self study. At least for the paperback version the prize of the book, in relation to the amount of material presented, is probably unparalleled. In any case, this book should not be missing in any physics or mathematics library.

Reviewer: C.S.Sharma (London); V.Perlick (Berlin)

MSC:

58Z05	Applications of global analysis to the sciences
58-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis
81T13	Yang-Mills and other gauge theories in quantum field theory
53Z05	Applications of differential geometry to physics
58A10	Differential forms in global analysis

Keywords:

geometry of physics; mathematical methods; Morse theory; singular homology; gauge theory; differential form calculus; classical mechanics; classical electrodynamics; phenomenological thermodynamics; relativity; symmetry; Dirac operator; Aharonov-Bohm effect; topological quantization; Berry phase; Yang-Mills theory