Modern Differential Geometry For Physicists (2nd Edition): Second Edition: 61 (World Scientific Lecture Notes In Physics) Paperback – 23 Mar. 1999

Chris Isham has written several books, which I have, in this brief, clear yet complete style - it's a joy to read this "simple" style. I should have gone to Imperial, though I can't complain about the teachers at my uni

This book deserves 5 stars only when read in the proper perspective: it is not a "how-to-calculate-it" guide. Rather, it focuses on explaining the general ideas of differential geometry, and it does a splendid job. The first chapter introduces general point-set topology, by considering the idea of "nearness" in sets, first by examining metric topologies, then by generalizing to neighborhood topologies, and finally by narrowing it down to the axioms of point-set topology, which is the best motivation I've ever seen for these axioms. Some basic ideas of topology are discussed and illustrated using numerous examples. The amount of examples in this book is truly outstanding and one of its finest points. Every new idea is supplemented by at least 3-4 examples, some standard and some not so standard at all. The next chapter focuses on differentiable manifolds and the important concept of tangent spaces. The third chapter deals with vector fields, forms, and tensors. Chapters 4, 5 and 6 discuss lie groups, fiber bundles and connections. Emphasis is put on the geometrical approach (for example, in discussing the topological properties of lie groups). Isham is also a fair expositor. He always informs the reader what the limitations of his ideas are; e.g., he always explains when certain ideas can and cannot be extended to infinite dimensional spaces, and points out the difficulties of doing so.

I felt that this book was not suitable for physicists specifically those who have been educated in the UK system. The book requires a knowledge of real analysis as well as point set topology. Most physicists educated in the UK only get a course on mathematical methods and they really wouldn't appreciate the style of this book. Furthermore, it is not useful as a book one can use to learn differential geometry in a way that can be applied in physics. The topology chapter does have examples, but the chapters on differential geometry are weak on real examples. I believe that the book is written in a manner that reflects the author's own desire to be a mathematician rather than a physicist. In conclusion, I do not believe this book would teach anyone to use differential geomtry in applications with any confidence.