Recommendations for Differential Geometry and Group Theory Books

[Matterwave]

Hi guys,

I just wanted to hear some thoughts and recommendations. I am looking to learn/study up on differential geometry (including n-forms, tensors, etc) and perhaps group theory so as to better understand the mathematics behind some of the physics that I'm interested in (General Relativity, and the foundations of Quantum Mechanics with extensions perhaps into QFT).

I'm looking for 2 books maybe that could serve as a general overview/review of the important topics (graduate level or advanced undergraduate). I don't need an extremely thorough treatment of the subjects (I am, after all, no mathematician), but I want to go a little deeper than the math that I've been taught solely from learning physics.

Thanks!

 

[George Jones]

Maybe A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres,

https://www.amazon.com/dp/0521829607/?tag=pfamazon01-20.

 

[Matterwave]

That actually looks really good. The only thing is though, is that I would prefer a paperback version or something so that I can keep it around with me and read it during my free time. It's kind of hard to be chugging around such a big book all the time.

 

[micromass]

As for group theory, I highly recommend "A book of abstract algebra" by Pinter. It's a real easy read, and it covers the very fundamentals of groups.

For differential geometry, the book "Introduction to smooth manifolds" by Lee is good, but it presupposes (a little bit) of topology. If you don't know topology, then "A comprehensive introduction to differential geometry" by Spivak is excellent!

 

[Matterwave]

I looked up "A book of abstract algebra" by Pinter, but there's 2 versions...one is 100 bucks and 1 is 11 bucks...what's up with that lol! (Is the old one like a collectible or something...>.>)

Thanks for the recommendations, I'll look into them. =]

 

[Matterwave]

I saw this book "Differential Geometry" by Erwin Kreyszig. The reviews on Amazon seem quite good for it. Anyone know this? My only concern is that the introduction says he only covers curves and surfaces in 3-D Euclidean space. I would imagine generalizing to 4-D wouldn't be too bad, but going to non-Euclidean spaces may be difficult? I don't know all too much about the topic, so please advise. Thanks.

 

[micromass]

I saw this book "Differential Geometry" by Erwin Kreyszig. The reviews on Amazon seem quite good for it. Anyone know this? My only concern is that the introduction says he only covers curves and surfaces in 3-D Euclidean space. I would imagine generalizing to 4-D wouldn't be too bad, but going to non-Euclidean spaces may be difficult? I don't know all too much about the topic, so please advise. Thanks.

The good thing about Kreyszig is that he writes understandable books. In fact, he has written one of the best introductory functional analysis books. The bad thing about his differential geometry book is that it's a bit outdated, and it won't go into full generality... So, if you read Kreyszig's book, then you will probably need to read another (more modern) book afterwards... But this reading will go very fast if you already know what they're doing...

Another good book with a lot of links to physics is "Differential and Physical geometry" by Lee. Or a less comprehensive book "Intro to differential geometry and general relativity" by Warner...

Also, the book about abstract algebra is a pure math book. You won't find any links to physics in the book. Maybe this is not really what you want...

 

[Sankaku]

I looked up "A book of abstract algebra" by Pinter, but there's 2 versions...one is 100 bucks and 1 is 11 bucks...what's up with that lol! (Is the old one like a collectible or something...>.>)

The $100 one is the original hardback for when it was assigned in schools. The $11 one is a Dover reprint with the same content. It clearly shows the dishonesty of the textbook industry, but that is a different rant.

 

[Matterwave]

So, I went to the local barnes n noble, but the only one I found was the Pinter book. It was a bit too pure math for me. Looks like I'll have to get these books online...>_>

 

[Daverz]

The good thing about Kreyszig is that he writes understandable books. In fact, he has written one of the best introductory functional analysis books. The bad thing about his differential geometry book is that it's a bit outdated, and it won't go into full generality... So, if you read Kreyszig's book, then you will probably need to read another (more modern) book afterwards... But this reading will go very fast if you already know what they're doing...

It's a book on classical differential geometry, so calling it outdated is a bit unfair. I don't have this book, but I really like the shorter book by Struik, Lectures on Classical Differential Geometry (also a Dover).

 

[robphy]

Here are some paperback titles:

http://books.google.com/books?id=HAPMB2e643kC
Geometrical methods of mathematical physics (Bernard F. Schutz)

https://www.amazon.com/dp/0521539277/?tag=pfamazon01-20
The Geometry of Physics: An Introduction (T. Frankel)

http://books.google.com/books?id=WgZ3Ia0SPE8C
A course in mathematics for students of physics [2 vols] (Bamberg & Sternberg)

http://books.google.com/books?id=wp2A7ZBUwDgC
Mathematical Physics (Geroch)

 

[Matterwave]

Thanks for the recommendations guys! Now if only I can find a bookstore that has these books hahaha...I tend to like to look them over before I buy them...>.>

 

[Matterwave]

Rob, between the Schutz and Frankel Diff geometry books, which one would you recommend? They both look good to me; they both look to be about as thorough/mathematically rigorous as I wanted, and they both seem to be quite connected to physics (which was what I was looking for). As far as the writing style or clarity (lack of typos, etc included), which one would you go with?

 

[George Jones]

In my opinion, Frankel is much better than Schutz, but also somewhat more difficult.

 

[robphy]

I think Schutz is easier to read... and is probably a better place to start.

 

[Matterwave]

Oh noes, 2 differing opinions! What am I going to do? >.>

Maybe I'll go with Schutz because it's 500 pages shorter...and easier for me to access.

George, do you have a reason why Frankel is far better? If it's just more thorough or rigorous or something, I'm fine with reading something not super rigorous because this is just a supplement to physics for me, but if he writes much clearer or presents the material in a much more clear and elegant fashion or something, I may have to think about it.

EDIT: For group theory, I think I'll go with Group Theory and Quantum Mechanics by Tinkham. It was recommended to me by a physics friend, and the reviews on amazon are all quite good.

 

[George Jones]

George, do you have a reason why Frankel is far better? If it's just more thorough or rigorous or something, I'm fine with reading something not super rigorous because this is just a supplement to physics for me, but if he writes much clearer or presents the material in a much more clear and elegant fashion or something, I may have to think about it.

Schutz is not rigourous enough for my tastes, but if rigour isn't much of a concern, I agree with robphy, Schutz is probably a better place to start. Schutz is more rgourous than many mathematical physics books. 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,

but I think studying Schutz is more realistic for what it sounds like you want to do.

Unfortunately, I think that Fecko is at its worst pedagogically in its first chapter. This book is not as rigorous as the books by Lee (which I own, and I which have found useful), but it more rigorous and comprehensive than the book by Schutz. 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 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 other mathematical physics books;
3) the simple examples are often effective.

 

[Matterwave]

Ok, thanks George.

 

[Matterwave]

So I ordered "Group Theory and Quantum Mechanics" by Tinkham, "Geometrical Methods of Mathematical Physics" by Schutz, and "Quantum Mechanics and Path Integrals" by Feynman (for my own enjoyment haha).

Thanks for all your help guys! =]

 

[Daverz]

A nice small book that covers n-forms and tensors is https://www.amazon.com/dp/0486640396/?tag=pfamazon01-20.

A book that tries to build intuition about much of the same material is https://www.amazon.com/dp/0521539277/?tag=pfamazon01-20, though at 720 pages it's a bit of a thumper.