Textbooks on differential geometry

Question

I want the best textbook on differential geometry and I want it to be comprehensive..I mean I want it to start from scratch til it reaches the most advanced topics..if possible I want it to have a mathematically rigorous and theoretical approach but still not ignore the subject's various applications in physics...I don't care if it's 10000 pages.

If that's not possible then I want a list of textbooks to read in succession with the same properties above.

I looked on the web a lot but quickly got lost because there are so many suggestions and most people simply want introductions and I'm not prepared to buy a huge textbook based on someone's review of it only to discover that it wasn't what I hoped for.

So..if someone recommends a certain textbook..Kindly say what's good or not good about it..of course I'm not expecting to find a perfect one I only want to find one that's not very different from what I have in mind.

Thanks in advance.

Answer 1

Kobayashi and Nomizu, Foundations of Differential Geometry, vol. I & II has it all. Not an easy read. Best of luck with it.

Doesn't have much on physics, though. For that you might try Hawking and Ellis, Large Scale Structure of Space-Time; lots of Riemannian geometry from the ground up and applied to General Relativity. Also not an easy read. And yes, it is the Hawking.

These works might be available online, perhaps even legitimately so.

Answer 2

For differential geometry, I've studied a book earlier this year. It was published by Mir Publishers Moscow. It is 'Problems in Differential Geometry and Topology', written by A.S.Mishchenko, Yu.P. Solovyev and A.T. Fomenko. The book is fine, but just a little bit problem oriented. Also you can go for 'Differential Topology and Geometry' by Gérard P. Joubert, Robert P. Moussu and Robert H. Roussarie.