Differential geometry for a physicist?

[tNa]

Hi, I'm looking to apply for theoretical physics PhDs in the coming year and have been recommended getting a little more mathematics under my belt. I have already done a bit of differential geometry in my G.R. course but I don't think it went into enough depth to be useful outside of what we covered. I'm hoping to develop at least a qualitative understanding of some of the ideas currently used in field theories, string theories etc. so I have something to talk about at interviews : p

With this in mind I was wondering if Geometrical Methods of Mathematical Physics by Schutz would be a good starting point for getting a little more abstract maths? Are there any other books people would recommend to get started?

Thanks very much.

 

[Fredrik]

The three books by John M. Lee with the word "manifolds" in the title, especially "Introduction to smooth manifolds".

I also like Isham's book.

 

[nicksauce]

I am a huge supporter of GMoMP by Schutz and would highly recommend it as a good starting point.

 

[Goldbeetle]

Accompany Schutz with something more in-depth. I suggest the book by Boothby.

 

[n!kofeyn]

just to clarify, are you wanting to learn more of the abstract theory or more concrete differential geometry? it's kind of a broad subject/title, and many books fall under different titles like smooth manifolds, differential forms, differential geometry, riemannian geometry, etc.

if you want to learn the abstract math, then i would recommend an introduction to manifolds by loring tu. it is much better than john m. lee's book, which is what i first learned this material from, and both of these books are more modern than boothby's text. lee is extremely wordy, and his proofs aren't as clean as tu's. tu's book also doesn't require as much background as lee's book, as lee sort of expects you to have read through his previous topology book. if you haven't had a topology course, lee's book will not be an easy read, and it is more fitting for a graduate course in the subject. tu purposely puts the topology on the back burner, and his book is more accessible for an undergraduate. although it's amazing that he covers most of what lee covers without sacrificing anything.

i would also recommend just searching amazon for "differential forms" as well. any of these books are good, but they come in many different flavors. differential forms are an integrable part (no pun intended) of modern differential geometry. any of these books that started pushing you towards coordinate independence will be helpful to you.

there's also books that focus more on the applications which i list out in this https://www.physicsforums.com/showthread.php?t=350943".

 

[George Jones]

As n!kofeyn has stated, contents of differential geometry references vary widely. Another book worth looking at is Differential Geometry and Lie Groups for Physicists by Marian Fecko,

https://www.amazon.com/dp/0521845076/?tag=pfamazon01-20.

This book is not as rigorous as the books by Lee and Tu, but it more rigorous and comprehensive than the book by Schutz. Fecko treats linear connections and associated curvature, and connections and curvature for bundles. Consequently, Fecko can be used for a more in-depth treatment of the math underlying both GR and gauge firld theories than traditionally is presented in physics courses.

Fecko has an unusual format. From its Preface,

A specific feature of this book is its strong emphasis on developing the general theory through a large number of simple exercises (more than a thousand of them), in which the reader analyzes "in a hands-on fashion" various details of a "theory" as well as plenty of concrete examples (the proof of the pudding is in the eating).

The book is reviewed at the Canadian Association of Physicists website,

http://www.cap.ca/BRMS/Reviews/Rev857_554.pdf.

From the review

There are no problems at the end of each chapter, but that's because by the time you reached the end of the chapter, you feel like you've done your homework already, proving or solving every little numbered exercise, of which there can be between one and half a dozen per page. Fortunately, each chapter ends with a summary and a list of relevant equations, with references back to the text. ...

A somewhat idiosyncratic flavour of this text is reflected in the numbering: there are no numbered equations, it's the exercises that are numbered, and referred to later.

Personal observations based on my limited experience with my copy of the book:

1) often very clear, but sometimes a bit unclear;
2) some examples of mathematical imprecision/looseness, but these examples are not more densely distributed than in, say, Nakahara;
3) the simple examples are often effective.

 

[Ben Niehoff]

There is also the canonical book by Nakahara. I think it is pretty good. It has many pictures, which helps a lot.