Need help finding a good book on Riemann Geometry

Solution 1:

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.