Need help finding a good book on Riemann Geometry I want to learn more about calculus on manifolds and Riemann Geometry. I have been reading the book Geometry, Topology and Physics by Nakahara. But I find that it is difficult to read due to the lack of theorems and proofs.
I would like to find a book that has a clear structure and is on a level suitable for a Ph.D student in Mathematics.
Thanks in advance!
 A: Riemannian Geometry by M. do Carmo is a great book that takes a variational approach, but I feel it is somewhat old-fashioned.
Riemannian Geometry by Peter Petersen is another great book that takes a very modern approach and contains some specialized topics like convergence theory.
Geometric Analysis by Peter Li is a great book that focuses on the PDE aspects of the theory, and it is based on notes freely available on his website so you can get a taste of it.
Riemannian Geometry by W. Klingenberg is another reference that has helped me out a lot over the years.
If you want to go back in time and learn from a master, consider 
Differential Geometry in the Large by Heinz Hopf. It is still worth a read!
People always mention the M. Spivak books when users ask for Geometry references, and they are wonderful, well written, great books. But I never find myself referencing this book when doing research, even though it is on my shelf. I personally found it to be a little long winded, although there is some nice stuff in volume 5 about isometric immersions.
