Books on differential geometry

What level should the books be?? Undergrad, grad? Do you know topology, analysis, linear algebra,...??

 

You can't go wrong with these two books:

https://www.amazon.com/Elements-Dif...qid=1349469211&sr=8-1&keywords=Millman+parker

https://www.amazon.com/Differential...=UTF8&qid=1349469231&sr=1-1&keywords=Do+Carmo

 

I have this one on my iPad, and it's very good:

http://www.math.uga.edu/%7Eshifrin/ShifrinDiffGeo.pdf [Broken]

(from http://www.math.uga.edu/~shifrin/)

 

I also like do Carmo for the basic curves and surfaces stuff. Once you learn a little topology (not much) you should check out Spivak (especially Volumes 1 and 2). Spivak's differential geometry books are awesome.

 

The book by Do Carmo was pretty good. I used it as a supplement in my differential geometry course in college, although it was a little bit advanced for me because it was the first time that I had ever seen any of the material.

A great introductory book is Barrett O'Neil's Elementary Differential Geometry. It has a lot of worked problems and solutions in the back so you can check yourself.

For more problems, try the Schaum's Outline of Differential Geometry

There isn't a perfect overlap of content and notation between these books, but I found that it was more than enough to do well in differential geometry. One thing to remember (which I learned the hard way!) is that 80% of the difficulty is understanding notation! So, the more resources, the better.

 

If you're coming at this from a physicist's point of view, both Nakahara's and Chris Isham's books on differential geometry are great. (Google is your friend, etc.)