Struggling with Differential Geometry Before Graduate School?

[siyphsc]

I am going to be entering the university of texas graduate physics department in August. I am currently signed up for the class "Topics in Geometry and Quantum Physics" (http://www.ma.utexas.edu/users/dafr/M392C/index.html) and am pretty worried about the summer reading. I am having a hard time with Spivak's "Comprehensive Introduction to Differential Geometry." Well, I am able to read the material (after a lot of thought/concentration) but still am unable to solve any problems. I was wondering if anyone had any good, more basic recommendations for learning differential geometry. Furthermore, should I be spending time trying to answer the problems, or should I just read the text? I hope this class won't be too far over my head.
Thanks in advance.

 

[blerg]

check out do Carmo's "Differential Geometry of Curves and Surfaces" for a simpler introduction to differential geometry

in response to your second question, doing problems is a fundamental part of mathematics. if you can't do the problems, you don't fully understand the material. it is great that you recognized that you're inability to do the problems is a problem.

 

[arunma]

I'd also recommend the book by Manfredo P. do Carmo. It's accessible to even an undergrad (I took it my sophomore year), so you should have no trouble with it. My only warning before going out and buying it or something is that it's meant for mathematicians. The text builds up formalism, and most of the problems involve proving stuff. Not sure how theoretical you want to go, but do keep that in mind.

 

[n!kofeyn]

While I agree that do Carmo's book is the standard in graduate level differential geometry, reading this book will not prepare you for what the professor is asking you to make sure you know (I actually read the link you posted).

He is asking you to be knowledgeable with calculus on manifolds, which is what volume 1 of Spivak's Comprehensive Introduction book is about (I read its table of contents). I would not mess with this book any longer. I have the following two suggestions:

https://www.amazon.com/dp/0387480986/?tag=pfamazon01-20 by Loring Tu
My opinion is to pick this book up immediately and starting working through it. It covers the same material as Spivak's comprehensive volume 1 and Lee's book, but is much more concise and clear. Its excersises are also helpful, especially since there are selected solutions in the back. I wish we would have used this book for the class I took for calculus on manifolds, and it is now a great read and reference for my qualifying exam preparation. It was published in 2008, so the notation is modern (notation is a big mess with this subject, especially with the older books).

https://www.amazon.com/dp/0387954481/?tag=pfamazon01-20 by John M. Lee
This is a book your professor lists and it and Loring Tu's book cover basically the same material with similar notation. This is also the book that I just used in a semester long class and it was a good book. It is much longer than Tu's book though and tends to get bogged down at times, which won't be great for summer reading. I think Tu's book will get you going much quicker. It seems Lee also assumes you are more comortable with topology than Tu, especially since Lee assumes you have read his previous book on topology. :)

In short, get Loring Tu's book and use Lee's and Spivak's books as backup since your professor listed those (most likely he hasn't heard of Tu's book which is why he didn't list it. You could e-mail him just to make sure, but it definitely covers the same material). It is shorter than both Spivak and Lee, has more doable exercises (which are labeled by what topic they fall under), and it is more modern than Spivak (Lee is a modern book as well). I think Tu's book gets you going much faster than Lee, which is good given your summer time constraints. By the way, Loring Tu is the same author that helped write Differential Forms in Algebraic Topology that your professor lists on the site as well. The Warner book he also lists is old like Spivak's, and I've heard that its notation is terrible.

Do not follow the recommendations to read do Carmo for this course, as this isn't the material he is asking you to know. Are you transferring to UT's graduate program (I just guessed given that this seems like an upper level topics course)? Either way congratulations on acceptance. What topology or differential geometry have you had before?

 

[arunma]

Hmm, in light of what n!kofeyn (who, unlike myself, did not fail to notice that you posted a link), I'll have to recind my do Carmo suggestion.

 

[siyphsc]

Thanks for all of your suggestions! I think I will get a copy of Tu's book in the library today.
To answer your questions, n!kofeyn, I will be a first year graduate student. I decided to take the class above because its offered by Dan Freed, who is a pretty famous mathematical physicist (he wrote a book with ed witten!)...I feel like this course may be a 'once in a lifetime' type thing. The course should also help me understand another class I'm taking next semester on a deeper level (qft). I know it will be a lot of work, but I think it will be worth it. Perhaps I will take it pass/fail. The only topology I have learned was what was covered in my Real Analysis I & II courses (Rudin, Stein). As for differential geometry, I don't really have any formal coursework which may be why I was struggling with Spivak.

 

[n!kofeyn]

Sounds cool. It's nice that you don't have to take qualifying exams, because we have to take them after the first year, plus an oral exam after the third year or so, in addition to the dissertation defense. Although, it does help solidify the material. My plan is to try and learn some QFT myself. I'm a math student, but I would like to possibly study some mathematical physics. I'll be taking a quantum mechanics course next year to get me some background so I can move on to QFT. Hopefully my math training will help me out quite a bit.

Well like I said though, the material in the first volume of Spivak isn't the stereotypical differential geometry, it's calculus on manifolds. The book https://www.amazon.com/dp/1852331526/?tag=pfamazon01-20 by Pressley is a great starter to true differential geometry (it's a good precursor to do Carmo's book). It has solutions to every exercise which is good for self-study.

Hopefully you enjoy the calculus on manifolds material. It was tough to learn but it was fun because it was such powerful stuff. I'm currently studying the material for my qualifying exams, so if you have some questions let me know. It will help me out and hopefully you out as well. Haha.

 

[mordechai9]

Sorry to resurrect this old thread but I wanted to point out something useful. I am working through Spivak in a class right now. Basically, the most important thing is to have a decent background in topology. I would recommend going through a good solid introduction on topology (Munkres or whatever I guess is most common) before ever attempting Spivak. Trust me, it will make your life MUCH much easier. Actually, that's a major understatement -- this background is absolutely necessary.

 

[n!kofeyn]

Sorry to resurrect this old thread but I wanted to point out something useful. I am working through Spivak in a class right now. Basically, the most important thing is to have a decent background in topology. I would recommend going through a good solid introduction on topology (Munkres or whatever I guess is most common) before ever attempting Spivak. Trust me, it will make your life MUCH much easier. Actually, that's a major understatement -- this background is absolutely necessary.

I disagree that a topology book such as Munkres is absolutely necessary to learn calculus on manifolds. For example, in the book I mentioned above by Loring Tu, he specifically notes this as a feature of his book in that he doesn't accentuate the point-set topology like other texts. He relegates that material to the appendices and presents it only when needed. Lee's book on the other hand is very bogged down in topology at times, which makes it a sloppy read. One can learn calculus on manifolds with just a working or basic knowledge of topology. I looked through the homework assignments on the link that the original poster posted, and the problems are not really topological in nature in terms of point-set type topology. To my knowledge the manifolds book by Warner doesn't accentuate the point-set topology either.

Plus, the original poster needed to get into calculus on manifolds as quick as possible. I don't consider Spivak's book (I'm not even for sure which one you are referring to) to be the best book anyway. Tu's book is the most clear, concise, and focused I've seen from the math side. I have seen some books on gauge theory or topological physics that describe manifold theory on a good intuitive level.

I would like to hear from the original poster on how the class is going. The class seems very heavy-hitting, as those homework assignments look challenging.

 

[mordechai9]

n!kofeyn -- Let me preface the following by saying that I fully respect your views and it sounds like you know a lot more about the subject than I do, so I don't presume to be telling you anything you don't know. Furthermore I don't consider myself a very talented mathematician so after all, what I consider a prerequisite, maybe somebody else considers absolutely unecessary.

That being said, let me point out that I was referring needing topology in preparing for Spivak's book (comprehensive introduction volume 1, as stated in the OP). It seems as though you missed this declaration. I will reiterate it more explicitly. If you expect to understand Spivak's book, you will need to have an introduction to topology. The first three chapters are extremely topology oriented... I think this must be obvious to anyone who has read the book. Features like topological continuity and homeomorphisms are probably the dominant feature -- an these are topological notions (in the way that Spivak employs them.) There is also a fair share of analysis (inverse function theorem, as a primary example) -- I'm not denying that.

Secondly, I own the book by Tu and I have been reading through it alongside Spivak -- actually a big part of my motivation for purchasing it was thanks to your original reply in this topic. Actually I have Lee's book as well. Now, I agree with you that perhaps Tu uses less topology. However, he has a substantial topology appendix in the back of the book. He refers to this appendix often. Let me quote you a little passage in the preface of his book: "While reading the first four chapters, the student should at the same time study Appendix A to acquire the point-set topology that will be assumed starting in Chapter 5." Let me point out that Tu's book has a total of 28 chapters.

I'm not sure why you disagreed so voluminously -- but I think you might be at the stage where you are so advanced that you take for granted that you ever needed this stuff to understand the material. But as a novice, I think my opinion is extremely well founded and I hope that you can see better now why I made my post.

 

[n!kofeyn]

I don't really think I disagreed voluminously. My post should be taken in context with what the original poster needed. The original poster is an incoming graduate student in UT-Austin's physics program and was needing advice to get to the level that the prerequisites of the class called for. Spivak's book, among others, was merely a suggestion by the professor to use during summer reading if you check the link posted. Since the poster was having trouble, I suggested the books I did, all in the context of what the poster needed.

If learning the subject of smooth manifolds from the ground up, then yes, a firm foundation in basic topology would be helpful. But, the original poster needed to get caught up to calculus on manifolds as quickly as possible, since they are taking a rather advanced course which utilized that material heavily. So spending time going through a book like Munkres would have been counterproductive at that stage in the summer.

I am by no means an expert in this. I myself just recently learned the subject of smooth manifolds, so that should be taken into account. I just know the difference between reading a book like Lee's and a book like Tu's. Lee sort of expects you to have read his previous book on topological manifolds and seems to sort of use topological language that isn't always needed, which gets in the way at times. When I read Tu, I don't get this feeling, and I think that he is using only what topology is necessary, which is probably the path that would be most efficient for physicists anyways. So his appendix would be good enough to go through. I personally think that's the best way to learn calculus on manifolds. Once you learn point-set topology, that's it. There really isn't much more there, but smooth manifold theory is very rich. I mean, it would even be ideal if you have had a rather thorough, theoretical course in linear algebra and multivariate calculus (especially with practice in the calculation aspects of differential forms) before attempting smooth manifolds.