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TLDR: Is there any differential geometry book that starts from scratch to graduaate level topics on differentiable manifolds?

I am a student self-studying math. By now I finished Abbott analysis, Munkres topology, Rudin RCA, and Aluffi Chapter 0. Now I am finding texts about differential geometry, which is self-contained so there are no additional prerequisites on undergraduate differential geometry, are suitable for self-studying, and covers topics on graduate level smooth manifolds(topics covered in books such as Lee's Smooth Manifolds).

I found some books by Serge Lang, which seems to meet the majority of the requirements. But personally I don't prefer Lang's style, so I am finding some other books.

SU Lee
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  • Manfredo Do Carmo, Differential Geometry of Curves and Surfaces 1976 is a favorite of mine. – Rene Schipperus Jan 26 '22 at 15:51
  • Take a look at this book. It is meant as a first book on the subject, is suitable for undergraduates and has answers to many of the exercises. – John Douma Jan 26 '22 at 15:55
  • Annexes in Lee's are also nice to quickly review everything you have to know for reading his (great) book. – Balloon Jan 26 '22 at 15:56
  • I can also recommend Do Carmo, especially at the beginning. It is really focused on $\mathbb{R}^n$ and especially on $\mathbb{R}^3$ though. This makes it intuitively great, but won't cover "further" topics. If you have some knowledge in abstract algebra (groups, rings, modules, tensor products etc.), I can also recommend Barden & Thomas, An Introduction to differentiable manifolds. Its more challenging, but will reward you with some "deeper" phenomena. – Timotheus Hauptinius Jan 26 '22 at 16:04
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    I started learning differential geometry with the book An introduction to manifolds of Loring Tu. It is self-contained. Once you have a basic knowledge you can read more advanced books. P.S.: Im also a self-learner. I tried to read Aluffi's book but I get bored about 40% or so :p – Masacroso Jan 26 '22 at 16:26
  • Does this answer your question? Where to start learning Differential Geometry/Differential Topology? – John Douma Jan 26 '22 at 17:48

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