# Best books for Differential Geometry?

1. Dec 4, 2004

### majutsu

What are the best books for learning differential geometry well? Any recommendations appreciated.

2. Dec 4, 2004

### Pyrrhus

I used Boyce and Diprima - Elementary Differential Equations - 7th edition.

3. Dec 4, 2004

### Dr Transport

I must have missed that part of the book......

4. Dec 5, 2004

### mathwonk

the best book is michael spivak, comprehensive guide to differential geometry, especially volumes 1 and 2.

they are available from "publish or perish", just google that name, at about 50 dollars a volume.

noel j. hicks is nice too, notes on differential geometry. also a book by manfredo docarmo.

or go to the library and see which one you can read and enjoy and understand. learning is a reciprocal endeavor. i cannot tell you which book is best for you.

5. Dec 5, 2004

### samoth1

McCleary's "Geometry From a Differential Viewpoint" is a good beginners or undergrad book. I can't help with anything comprehensive, though.

Good to see you over here Majutsu!

6. Dec 5, 2004

### coalquay404

I've always found Nakahara's "Geometry, Topology, and Physics" to be a good read. Choquet-Bruhat's "Analysis, Manifolds, and Physics" is another must-have.

7. Dec 5, 2004

### NeutronStar

I can't say anything about what might be "best" since I'm just starting out myself. But I did just get a Dover book "Differential Geomerty" by Erwin Kreyszig. You can read the TOC and excepts at amazon.com at the following link: (just look up "Differential Geomerty Kreyszig")

http://www.amazon.com

I'm just starting in both the book and the subject. So far I've found Kreyszig's book inviting because it starts off fairly simple and promises to remain in Euclidean 3-space for the bulk of the book.

It has exercises along the way with detailed worked-out answers in the back of the book. It looks like it's going to be a good introduction.

Like I say, I'm just starting in the book, and I have nothing to compare it to. I just know that it looks like something I can follow through to the end in a self-study program. For $10 I think it's a pretty good buy! I might mention that I also got the Schaum's Outline on Differential Geometry along with it for an additional 11 bucks. The Schaum's Outline appears to complement Kreyszig's chapters fairly well. For a$21 investment I'd have to say that I'm pleased with these materials thus far. But I do need to remind you that I'm just starting with these books.

I would also suggest that you already have some knowledge of vectors and of vectors represented as matrices (i.e. Linear Algebra). And of course undergraduate calculus is also assumed. Kreyszig promises to develop the theory of tensors along with the differential geometry. So no prerequisite knowledge of tensors is required.

Again, I'm only starting to study these books myself so I can only tell you that I'm comfortable with the way that they start out. Also from browsing through the chapters it appears that Kreyszig has a good style of pedagogy.

Only time will tell now.

For the price you can't go wrong.

8. Dec 5, 2004

### majutsu

Thanks for all the responses. Hi Samoth!

I love the Schaum's especially for Linear Algebra, and will probably get the differential geometry book, although I hear it's only classical differential geometry. Similarly, they say Kreyszig's book, with the coordinate p.o.v. is limiting in the long run, but is the only way some people can get this stuff. I love the idea of the Spivak books, but the price frightens me. I went with the Boothby book for now. I am very interested in the Chern book, but I hear he's a little slanted in his presentation.

Mathwonk, I know you cannot know the best way I learn, but maybe it would help to know what I am looking for. I am very interested in learning diff. geometry for its applications in physics, but I am a mathematician at heart. I don't want a cookbook approach (like Schaum's) that leaves me having to go back and learn the same material like a mathematician someday. Also, I don't want to learn it so abstractly that I can't calculate or work with it. I am most interested in mathematical physics itself as a subject.

Thanks for all your help and ideas so far. This is very useful.

9. Dec 5, 2004

### quetzalcoatl9

differential geometry is a fascinating subject. i would suggest that you get as many books on the subject as you can, and skim through them all to see what you like best.

some that have worked for me are Bishop's "Tensor Analysis on Manifolds", the classic book "Gravitation" and Wald's "General Relativity".

Do Carmo's book is also good, and takes a very geometric approach. Frankel's "Geometry of Physics" is nice because it explains things in a more modern way (multilinear mappings and fields as sections of the tangent/cotangent bundle) and is fairly rigorous.

i also could not emphasize enough, to find someone who is knowledgeable of the subject for when you have questions, since some ideas in diff. geom. are not always obvious (at least to me).

10. Dec 5, 2004

### Pyrrhus

Oh yes, sorry, that's what happens when you are tired :rofl:

11. Dec 5, 2004

### NeutronStar

I never view an intuitive approach to anything as being limiting. It would only be limiting if the student allows it to be. I think Kreyszig does a wonderful job of explaining both the abstract and the intuitive povs. Being a physicist I always welcome a phenomenological pov. I see it as simply adding insight to the abstractions. This is especially true for an introductory course which Kreyszig is offering.

I have this book too, and I agree that it is also very well laid out. This book is much more abstract than Kreyszig's book. If you're looking for an entirely abstract pov I would recommend this book. I actually bought this book along with Kreyszig's book. I plan on going through Bishop's book after I finish Kreyszig.

I'm in love with Dover books because you can usually buy about ten of them for the price of one regular textbook. I always do better when getting ten different povs rather than just one.

12. Dec 7, 2004

### rdt2

I've been trying to collect the best of the www on differential geometry. There's a list here that might be useful:

http://www.mech.gla.ac.uk/BB/viewtopic.phtml?t=109

I haven't visited some of these sites for a while so some links might have died.

Last edited: Dec 7, 2004
13. Dec 7, 2004

### majutsu

thanks, all. Good stuff. RDT2 thanks for the links.

14. Dec 8, 2004

### fourier jr

yeah, the "great american differential geometry textbook" lol

15. Dec 15, 2004

### mathwonk

i just discovered there are two books by docarmo, the mroe elementary one on differential geometry of curves and surfaces, and then the next one on riemannian geometry.

majutsu, after reading your orientation to the material i do not change my suggestions at all. that is exactly how i understood them.

16. Dec 16, 2004

### pmb_phy

I have Tensors, Differential Forms, and Variational Principles
by David Lovelock, Hanno Rund which is nice. MTW is a nice text too as is Schutz's text. The last two are GR texts. Since GR is one of the major applications of tensors then its a nice way to learn the subject.

Pete

17. Feb 7, 2005

### Ruslan_Sharipov

18. Feb 7, 2005

### majutsu

Thanks everyone for great suggestions and ideas. I am still reading . . . :) I actually got detoured, because I wanted to brush up PDE and the fine points of advanced calculus before DG, so that's where I'm at the last 2 months or so.

19. Feb 10, 2005

### mathwonk

20. Jan 11, 2008

### uni

Hi,
Mechanics in differential geometry by Yves Talpaert (Springer) is a comprehensive and very useful book, both in differential geometry and physics.
Greg