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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

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My former advisor at ucla Peter Petersen wrote the GTM Riemannian Geometry book. I had a copy of it when it was in manuscript form, some 30 years ago.

I never mastered the subject. It seems to be exceedingly difficult to do. Nevertheless, all the "yellow peril" books are known for being very high quality.

I always said that my advisor, Peter, seemed to be quite sound. Even though he couldn't help me write my thesis and get my phd.


There is also the book by the "french guys", Galot, Hulin and Lafontaine.

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