[Math] Next book in learning Differential Geometry

Question

I have just finished the book "Manfredo P. do Carmo – Differential Geometry of Curves and Surfaces". My aim is to reach to graduate level to do research, but articles are not only too advanced to study after Carmo's book, but also I don't think that they are readable by just studying Carmo's book at all for a self-learner like me. Please someone tell me a book for Differential Geometry more advanced than Carmo's book but readable esp. for self-learning . Thanks a lot.

Answer 1

Here's my answer to this question at length.

http://www.geometry.org/tex/conc/differential_geometry_books.html

In summary, if you are looking at the pure mathematics style of DG, you would be looking at do Carmo's "Riemmanian geometry", three books by John M. Lee, and for something more advanced, the books by Serge Lang, Bishop and Crittenden and Peter Petersen. And for the really advanced level, there's Schoen and Yau "Lectures on Differential Geometry", which lists many hundreds of open problems to work on at the postgraduate level.

There are also a few items on this web site which address the same question, some of them several years ago.