Differential Geometry and Lie Groups for Physicists by Marián Fecko (PDF)

Description

Differential geometry plays an increasingly important role in modern theoretical physics and applied mathematics. This textbook gives an introduction to geometrical topics useful in theoretical physics and applied mathematics, covering: manifolds, tensor fields, differential forms, connections, symplectic geometry, actions of Lie groups, bundles, spinors, and so on. Written in an informal style, the author places a strong emphasis on developing the understanding of the general theory through more than 1000 simple exercises, with complete solutions or detailed hints. The book will prepare readers for studying modern treatments of Lagrangian and Hamiltonian mechanics, electromagnetism, gauge fields, relativity and gravitation. Differential Geometry and Lie Groups for Physicists is well suited for courses in physics, mathematics and engineering for advanced undergraduate or graduate students, and can also be used for active self-study. The required mathematical background knowledge does not go beyond the level of standard introductory undergraduate mathematics courses.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The best differential geometry book to bridge the gap between physics and mathematics texts.Exposition: Instead of concentrating on proving things it allows you experience building up the theory and the explanations are superbly done. Also the hints to the problems make them doable with sufficient effort no matter how difficult they may be. I am now convinced that there is no better book for the self study of differential geometry. Also, having build up the theory I managed to retain so much information topics I had attempted to study in the past.Breadth: For physics it contains almost everything you could ever ask for and even things you didn’t know you wanted.Depth: Although one can argue that you can go more indepth. I have never witnessed a book written for physicists that is both comprehensible enough to make it a joy to work through yet detailed enough to give you the ability to read books and papers written in mathematical physics and even other areas of mathematics. Moreover, unlike many mathematics texts teaches you both the notation of physicists to do computations and the notation of mathematicians to do proofs.Anyway buy this underated gem of a book. You won’t regret it. I even bought two extra copies of the book for my two friends who are physics PhD students like me.

⭐Best introductory book on the subject I have read. It’s more from a physicists view point and really helps getting a feel for the subject.

⭐You can’t be a spectator while reading the book. I am a physics graduate student and I must confess I was at first very annoyed that every single concept was built using exercises, I am used to sitting down reading a book for 10 minutes or 20 minutes intervals and then doing an exercise or two, but this book is very different. There are few paragraphs and literally no full pages where the author explains concepts, lays out proofs and does examples. The reader builds up every concept through little exercises. I eventually calmed down and realized happily that the exercises were making me think of the little and subtle concepts I would have missed had everything been laid out for me. The wonderful thing is that the exercises are very short , the author puts helpful hints or even the full solution. For example in ” Geometry of physics” or ” Geometry, Topology and Physics” there are explanations of what flow of vector fields are, there is usually a picture and a paragraph of two explaining and deriving the concepts. That is not the case with this book, everything about the flows the reader finds out by doing little exercises. This takes getting used but is so worth it.The material of this book is very wide and about as much as Theodore frankel’s book but it goes slower and is definitely better than Nakahara’s book.WARNING FOR PHYSICISTS: Be prepared to think abstractly, this might be hard if you haven’t seen advanced undergraduate maths courses

⭐This book is a gem for physicists trying to self study and repeatedly getting lost in books written by mathematicians! Although, the early chapters might need to be supplemented with some more rigorous texts (mostly, to find rigorous answers and proofs to the questions in the middle of the text (with hints to most of them)), the exposition is highly intuitive and logical. Also examples and exercises related to physics really help in developing a good conceptual grasp! The only problem is that you can’t read it on a bed, or half asleep 😛

⭐An excellent reference for self-study. Four stars not five, because contrary to its claim, a reader with an undergraduate physics background cannot read it from the start to end without referring to other books. I decided to learn some General Relativity after hearing Smolin talk better smack than Triple H, and encountering Penrose’s intriguing Road to Reality. Fecko logically and succintly weaves together many possible views of each subject he discusses. He clarified for me, for example, the links between the approaches taken by the texts of d’Inverno and Ludvigsen. Many of these links are given as well-structured exercises, so the book is best used when one has an uneasy suspicion that something might be true. Fecko also gives outstanding motivations and intuitive pictures for many definitions. Even after I had understood pull-backs and differentials, it was a delight to discover that putting a shoe on my foot was as good as putting my foot in the shoe.

⭐Before discovering the new book my Marian Fecko I thought I know all that I need about differential geometry (I co-authored a monograph on this subject myself). I had my favorite books: Kobayashi-Nomizu, Bishop-Crittenden, Sternberg, Michor, Abraham and some more. Yet “Differential Geometry and Lie Groups for Physicists” was a completely new experience. It is written with a “soul” and covers topics that are important but missing in other books. As I was working on a paper dealing with torsion, I emailed the Author with some of my ideas and questions and got an instant answer.Readers looking for explanations and geometrical interpretations of the abstract concepts will certainly find this book irreplaceable. Lie and covariant derivatives, parallel transport, Hodge operator, Cartan’s moving frame method, Laplace-Beltrami operator, Lie groups, Maxwell equations, Clifford algebras and spin bundles, SL(2,C), Dirac operator, Momentum map etc. etc. – all introduced and explained in a concise yet clear way, with exmaples and exercises.This book should find its place on the bookshelf of everyone interested in geometrical concepts required for understanding contemporary theoretical physics.I recommend this book to all students and professionals. It should find its place in every university library.Just one warning: certain mathematical symbols did not find their way to the “Index of frequently used symbols” at the end of the book. The reader trying to read the book starting from p. 600 may find it necessary to spent some time going through the earlier chapters to find out the meaning of a given symbol.

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