S. I. Goldberg, Curvature and Homology (Academic Press, 1962), xvii + 315 pp., 68s
Proceedings of the Edinburgh Mathematical Society
yellow back " series in 1955. The translators have given a competent and accurate rendering and only very occasionally is one aware of the construction of the German sentence which lies behind the translation. The pitfalls confronting the translator are many and the reader will smile at rather than criticise the few into which the present translators have stumbled. For instance the Bibliography refers to the book by M. Hameresh (1962) as one of the older books on the subject. Again, Gruppenkeim
... Again, Gruppenkeim is translated in the text as local Lie group but in the index as group embryo. The phrase " restrict things to the subgroup " which occurs in the two branching theorems is unpleasant and this lapse is not attributable to the German version. This instance is not, however, typical of the translation. Many of the footnotes in the German edition appear to the reviewer to have been inserted by way of explanation at the proof stage. It is a pity that the opportunity was not taken of incorporating these in the text of the English edition. Apart from four pages on Freudenthal's treatment of the spin representations of the rotation group there is no additional material in the English version though in a few places the argument has been modified. This book is the best presentation of the theory of matrix representations of groups to date and it assumes very little previous knowledge on the part of the reader. Preliminary chapters deal with the relevant portions of the theory of groups and matrices. The general representation theory is followed by a detailed treatment of the symmetric group, the full linear group, the real linear group, the unimodular group, the real unimodular group, the unitary group, the unimodular unitary group, the orthogonal group, the rotation group and the Lorentz group. These are the groups of greatest interest to physicists whose requirements have been specially considered in the contruction of the book. The volume does not, however, include actual applications of the theory to physical problems. The pure mathematician, on the other hand, will regret that no account is given of the theory of modular representations though the author can justly claim that this theory lies outside the scope of his book. The printing of the text is excellent, but the same cannot be said of some of the diagrams, in particular figures 11, 12 and 15, which appear to have been reduced excessively from the drawings. No actual misprints were observed but a few letters appear to have dropped out of the type here and there. The book will form a valuable addition to the algebraist's library. D. E. RUTHERFORD GOLDBERG, s. i., Curvature and Homology (Academic Press, 1962), xvii+315 pp., 68s. This book is a graduate text and the reader is assumed to have some knowledge of Riemannian geometry, Lie groups, and the elements of analytic and algebraic topology. It is essentially a monograph which collects together relevant material available elsewhere in books, in collections of lecture notes and in scattered research papers. The contents are indicated by the headings of the chapters and appendices: I. Riemannian manifolds; II. Topology of differentiable manifolds; III. Curvature and homology of Riemannian manifolds; IV. Compact Lie groups; V. Complex manifolds; VI. Curvature and homology of Kaehler manifolds; VII. Groups of transformations of Kaehler and almost-Kaehler manifolds; (A) de Rham's theorems; (B) The cup product; (Q The Hodge existence theorem; (D) Partition of unity.