Analysis, Manifolds and Physics Revised Edition: Volume I

Writing a review for something that everybody knows its high quality would be a waste of time, but perhaps not anymore - younger people should know the 'standard candles'. Unless you are in a place where all this material you can attend from lectures, this is the book that if you are (or want to be) a mathematical physicist must try to read 'a little every day', hoping that eventually things will start focusing and you will catch up. It should be considered in a sense as THE modern analogue of Synge & Schild's Tensor Calculus - it has the same selection of topics but now all on manifolds: Analysis on Manifolds, Riemannian geometry, Integration, Connections, plus distributions and aplications to PDEs and selected topics of infinite-dim geometry. So you have here a source-book that will not only allow you to formulate, in a modern way, physical laws (differential geometry) but also help you to study them (PDEs). It is a profitable reading for someone who is somewhat versatile with elementary abstract mathematics, say at the level of Geroch's Mathematical Physics (algebra, topology, measure theory, functional analysis), but once you get going, you never stop! Start off from chapter 3 and get back (or look up Geroch) if you need an explanation of a word you don't understand or have forgotten. After you have a basic understanding a Riemannian geometry from this book you'll hopefully be able to reach Mme Choquet's new book on GR and the Einstein Equations, it is a continuation and uses the same notation, or volume 2 of the book under review on various topics in mathematical physics. Most importantly, keep your cool and don't get intimidated!

This is the book that inspired me as an undergraduate to learn and appreciate, to a large extent, how physics and mathematics cohabit so beautifully. I continue to see, to this day, as a graduate student, how many of the recent developments in theoretical physics have been inspired by new mathematics and conversely. I still refer to this book on occasion, since it is laid out in a style that is amenable to mathematicians (such as myself!). It's an excellent read, and I highly recommend it (even as bed time reading!) Best regards, A.

This was the most reliable (and compact) source of mathematical information that I could find. I used it (almost exclusively) to teach myself what I needed to know to do my PhD thesis in General Relativity with Torsion fields. It is very overwhelming to look at the book the first time. But give it time. It is well written and inviting. The only draw back for me was the very small print.

I use this book constantly. At first I thought it was good only as a reference, but now I know it is possible to actually LEARN any of its subjects from scratch. I particularly like its chapters on manifolds, Lie Groups and bundles. Connections on a principal bundle is extremely well done, with a translation to physics (gauge theories) performed in detail in an exercise (so to speak). Some time ago David Ruelle said that the physicists needed a presentation of recent mathematics in the form of Bourbaki's "Fascicule des resultats", a synthesis of the subject with complete definitions, examples and theorems clearly stated, but with the proofs ommited. This is it, except that in this book most demonstrations are not ommited, only those too complicated. The whole book is extremely readable, if you concentrate, turn out the TV, etc. A precious book.