Best Differential Geometry Books for Independent Study

[JasonJo]

I have studied from baby Do Carmo (Differential Geometry of Curves and Surfaces) and have a background in pointset topology. What is the next best book to use? I'm looking for a good diff geo book to independent study from next semester. My professor recommended Introduction to Differentiable Manifolds and Lie Groups by Frank Warner, but it seems really advanced: https://www.amazon.com/dp/0387908943/?tag=pfamazon01-20
along with Comparison Theorems by Ebin and Cheeger (yet to be published, but will be published before next semester).
I was thinking about Do Carmo's Riemannian Geometry, but I'm not sure. There does not seem to be such a clear lineage when it comes to differential geometry books.

 

[Daverz]

I think Do Carmo's Riemannian Geometry is doable in a semester. It's a very readable book.

Some other books you might look at:

https://www.amazon.com/dp/0821839888/?tag=pfamazon01-20

Lots of cool material and fresh approaches to old material in Kuhn's book.

https://www.amazon.com/dp/0521468000/?tag=pfamazon01-20

I haven't looked through an actual copy of this Dover book on differential topology, but it looks interesting:

https://www.amazon.com/dp/0486462447/?tag=pfamazon01-20
http://web.doverpublications.com/cgi-bin/toc.pl/0486462447

 

[quasar987]

My university uses the book by Noel J. Hicks for the predoc exams. It is old and out of print, but I found a link to it on wikipedia!

I've flipped through it entirely and its looks like the best intro to abstract diff. geom book I've seen and I've seen them all. I look foward to learning from it.

 

[jdstokes]

I also studied a course from baby Do Carmo and I have also been looking for a more advanced text (for the purposes of general relativity and gauge theories, however).

A nice book with physics intuition is The geometry of physics by Frankel.

 

[Daverz]

I also studied a course from baby Do Carmo and I have also been looking for a more advanced text (for the purposes of general relativity and gauge theories, however).

I'd take a look at Göckeler & Schücker.

https://www.amazon.com/dp/0521378214/?tag=pfamazon01-20

 

[mathwonk]

noel j hicks is indeed an excellent book (and scarce, but i also found one to replace my old lost copy), but warner is also very good and very readable in my opinion. and it is relatively elementary in what it assumes, compared to what it teaches you. e.g. it uses tensors but first explains them in a very elementary and low level way.

of course if you liked do carmo, it seems a no brainer to continue with him. in the end i always like spivak best for the ultimate diff geom book, especially vol 2.