I'd like a textbook that covers do Carmo's contents (can be more), but that isn't do Carmo. I did not like his writting style. That being said, I particularly like the styles of:
Walter Rudin (Principles of Mathematical Analysis and Real and Complex Analysis)
Bredon (Topology and Geometry)
Munkres (Topology)
And in order to further specify, I like concise textbooks that go directly to the subject and prove the results in an efficient manner.
Thank you.
On
You may check these ones:
(i) Riemannsche Geometrie im Grossen (D. Gromoll, W. Klingenberg, W. Meyer). It's a bit old and in german but it is good.
(ii) An Introduction to Riemannian Geometry (L. Godinho, J. Natario). This is pretty recent and sounds cool but it is a bit elementary. I has also applications in Relativity.
(iii) Riemannian Geometry - A Modern Introduction (I. Chavel).
There is another one also in german which is a lot good but I don't recall its title right now but I'll check and add later.
Your last comment suggests that you may like Riemannian Geometry by Petersen, but I personally find it difficult to read. From what I understand, it covers more than do Carmo does and it is useful beyond a first course in the subject.
Another book that is worth checking out is Lee's Riemmanian Manifolds; note though that it doesn't cover the latter parts of do Carmo's book in as much detail. As with his Introduction to Smooth Manifolds, Lee is incredibly clear in his exposition and I find it very easy to read.