I'm doing a PhD thesis about Riemannian geometry and i would like to improve my knowledge. I know all the basic defintions and concepts (I have the "semiRiemannian geometry with applications to relativity" from Barret O'neill and i have studied it during the degree and the master ). So i'm looking for books with the following to characteritics: First, a book which go forward with respect the content of O'neill's book. Secondly, it would be great that the author/s stop in the basic (or advanced) concepts to think about it and not just give the defintions and going on.
Thanks for advance