Preview of kreyszig's book:
https://books.google.com/books/about/Differential_Geometry.html?id=B7yxgFaQKNAC&printsec=frontcover&source=kp_read_button#v=onepage&q&f=false
As seen in the link, the presentations in Kreyszig's are limited to 3 dimension. Only calculus, and a bit of linear algebra, and some ODE knowledge are needed to read kreyszig's book. My purpose for studying DG is to understand General Relativity which involve 4 dimension.
My math background is, calulus,123,ODE,linear algebra,basic complex integration. I have read an introductory book on general topology,and functional analysis also by kreyszig. So far, I havn't read any text on abstract algebra. I have also briefly read Real Analysis by terrence tao. Other than all these, I have a little knowledge about PDEs and asymptotic analysis.
If I want to learn some more general concepts and theorems about DG, on my level, what book do you recommend? Or perhaps, I should study Abstract algebra first?
Best Answer
After reading the question, I took a look at Kreyszig's book, and I would say it seems to be written in a classical style, and it certainly covers the classical material on surface geometry in considerable depth. It's probably a pretty good introduction to differential and Riemannian geometry; I particularly like the way it seems his notation corresponds with that used in treatments of higher-dimensional spaces, e.g. he brings in the Christoffel symbols $\Gamma^\alpha_{\mu \nu}$ which helps familiarize the reader with this concept which occurs ubiquitously in treatments of higher-dimensional spaces.
Having said these things, I would say that his notation and treatment are a little old-fashioned, and that a good place to go next might be Barrett O'Niell's Elementary Differential Geometry, which covers a lot of the same material (or at least appears to at first glance), but with a more modern notation and perspective, so it can help bring the reader familiar with the classics up to speed on the present-day way of doing things.
That's pretty much the way I learned the subject--starting with old-school books like Kreyszig and then, having become familiar with the concepts, moving on to O'Neill, which I encountered in my first course in differential geometry as an undergrad and CalTech.
If one is interested in General Relativity, and wants to learn the math as he goes along, I think the logical place to go is Misner, Thorne and Wheeler's Gravitation. Written by three giants in the field, including one Nobel laureate (Thorne), it explains all the differential geometry, from a modern perspective, as it goes along. It's is really quite readable, and uses modern notation. Of course, the book focuses on Loretzian geometries, space-time models, but still manages to cover a lot of odinary $2$-surface geometry in the process. If I wanted to get into GR as swiftly as possible, this is the book I'd use.
There are plenty of good books on differential and Riemannian geometry, some of them quite specialized and advanced. A brief scan of the 'net with google will turn up more titles than you can read in a few years. But I would like to mention one more in closing, and that is John Milnor's Morse Theory. He presents a very thorough, concise and readable introduction to general Riemannian geometry in the first few chapters, then devotes considerable time to the Morse theory of geodesics, which in fact forms a major advanced subject area. He also applies the results he presents to some of the classical groups, $O(n)$ and $U(n)$ and such, and in so doing derives some major topological results (e.g., Bott periodicity) via geometric technique. A very good introduction to the field with some major applications.
I wouldn't worry too much about abstract algebra as a prerequisite. Sure, it comes into GR at some point, but only, as far as I have seen, in very specialized applications.
Well, best of luck with it.