The geometry of physics. An introduction. 3rd ed. (English) Zbl 1250.58001

Cambridge: Cambridge University Press (ISBN 978-1-107-60260-1/pbk; 978-1-139-15414-7/ebook). lxii, 686 p. (2011).
At first glance, this book is not for mathematicians, because all mathematical results have long been known. This book is not for physicists, because it does not describe the experimental methods. However, if such books exist then they are needed. And this is proved by the fact that such books are published in multiple editions and they are in physical and in mathematical libraries simultaneously. This book is a great review-handbook in which the author tries at the elementary level to show the achievements of mathematicians in the field of physics and also to explain some physical ideas for mathematicians. However, physicists and mathematicians speak different languages and this is not the first author who tries to reconcile these two factions (see Mizner, Schutz, Nakahara and etc.).
The purpose of this book is to offer advanced mathematical tools to physicists, and give for mathematicians a little bit of physical intuition. In other words, a priori, it is assumed that a physicist knows physics and wants to understand the math; respectively a mathematician knows all formulas and is extremely interested in applications. Unfortunately for physicists, mathematical tool is not enough even to understand the book itself. For instance, Chapter 5 is called ‘The Poincaré Lemma and Potentials’, but there is no formula for finding the potential of some exact form. In Appendix B the cohomology of the chain complex is considered, but for physicists it would be interesting to know what ker (im) of a linear operator means and how to calculate it. We can say that the attempt to confuse physicists in the section about cohomology is successful. However, if the physicist will learn the basic concepts of cohomology from other sources, then further reading of this book will be very instructive.
Mathematicians also expects hard work to read this book. The reader will not get physical intuition because physical formulas are taken from somewhere and given to him from above as existing. This is despite the fact that the conservation laws are the basics of physics, and all mathematical tools to the construction of these conservation laws in this book are available.
Despite a predictable stream of criticism, the author boldly took up the lost cause, and for the past three editions he has successfully developed his project by adding new material to the book.
This book consists of the following basic parts: manifolds and vector fields, tensors and exterior forms, integration of differential forms, the Lie derivative, the Poincaré lemma and potentials, holonomic and nonholonomic constraints, \(\mathbb R^3\) and Minkowski space, the geometry of surfaces in \(\mathbb R^3\), covariable differentiation and curvature, geodesics, relativity, curvature and topology, Betti numbers and de Rham’s theorem, harmonic forms, Lie groups, vector bundles in geometry and physics, fiber bundles and topological quantization, connection and associated bundles, the Dirac operator, Yang-Mills fields, Betti numbers and covering spaces, Chern forms and homotopy groups.
Such a wealth of information cannot be untapped and this book is hard to put on the shelf, because it must always be on the table of a beginning researcher in theoretical physics.
[Editorial supplement: A main addition introduced in this third edition is the inclusion of an overview: ‘An informal overview of Cartan’s exteriar differential forms, illustrated with an application to Cauchy’s stress tensor’ which can be read before starting the text (before Chapter 1).
For Part II see [ibid., 694 p. (2004; Zbl 1049.58001)].
An extensive review to the first edition [ibid., 654 p. (2004; Zbl 0888.58077)] is given by V. Perlick].


58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis
58Z05 Applications of global analysis to the sciences
53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
58A15 Exterior differential systems (Cartan theory)
81T13 Yang-Mills and other gauge theories in quantum field theory
76A02 Foundations of fluid mechanics
74Bxx Elastic materials
78A02 Foundations in optics and electromagnetic theory
80A05 Foundations of thermodynamics and heat transfer
53Z05 Applications of differential geometry to physics