In the last two or three decades of the twentieth century the study of differential geometry and topology became very important to theoretical physicists. Of course, general relativity had brought Riemannian geometry to physics, but developments in particle physics, condensed matter physics, and statistical mechanics raised the level of interest and broadened the scope to include a great deal of geometry and topology. More recently, string theory and cosmology have brought these subjects even more to the forefront. The current book was among the first textbooks to introduce these topics to physicists. It was originally published in 1982 and has recently been reissued by Dover.
This is a book written by physicists for physics students. Its breadth is significant. It begins with basic ideas of topology and proceeds over the course of barely three hundred pages to encompass a great deal of differential geometry and algebraic topology. The mathematical prerequisites include something like advanced calculus, linear algebra and some abstract algebra, mostly group theory and some notion of group representations. The physics prerequisites are fuzzier; one could read much of the book without any background in physics, but the authors do remind readers that they may have seen connections and covariant derivatives, for example, in general relativity.
After introductory material, the authors begin with basic manifold theory, then move quickly to the fundamental group, homology, higher homotopy groups, and cohomology. The authors state important theorems — often not in full generality — and provide proofs when they believe the proofs offer some insight. The level of rigor is lower than in comparable texts by mathematicians, especially when considering (actually, not considering) questions about existence of objects of interest or how well-defined certain operations might be. But there is nothing here to offend any but the ultimate purists. This is after all a book designed to introduce the concepts to physicists, and the approach seems pedagogically sound.
The longest section of the book is on fiber bundles, connections and characteristic classes. The authors do an excellent job motivating the constructions and appealing to the reader’s intuition. There is brief chapter on Morse theory, and an even shorter one on defects and textures in an ordered medium. A longer final chapter describes applications of all the foregoing material to Yang-Mills theory and particle physics.
This book is one of three that are commonly recommended to physics students. The other two are Nakahara’s Geometry, Topology and Physics and Frankel’s The Geometry of Physics. An informal survey of physicists I know suggests that Nakahara is regarded as the standard because of its completeness, and that Nash and Sen’s book is the second choice, less complete but well regarded for its intuitive approach.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.