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I'm looking for a textbook in differential geometry which inside has exercises with (at least) final answers.

Since it's my first course in differential geometry it doesn't have to cover material (we finished the course with gauss-bonnet theorem)but rather to have hard-leveled problems. What book would you recommend to me ?

Amzoti
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This book (Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers) contains detailed solutions to 375 core exercises on differentiable manifolds, Lie groups, fiber bundles, and Riemannian geometry. It should be helpful to anyone who needs to aquire a solid working technique in these fields and to students who have problems to illustrate concepts, methods, and theorems by examples.

C.F.G
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Starting out as a PhD student in Diff Geom and Mathematical Physics there were two textbook s that got me through the first terms and up to speed. They were

Differential Forms and Connections by R.W.R Darling and Geometry and Topology in Physics by Mikio Nakahara.

The first textbook introduces the tools of modern differential geometry, exterior calculus, manifolds, vector bundles and connections, to advanced undergraduate and beginning graduate students in mathematics, physics and engineering through the use of numerous concrete examples and (non-worked through) solutions provided.

The second textbook focuses on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems.

  • Darling's book does not come with solutions... – soap Oct 15 '18 at 11:35
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    @Soap A lot of the answers here are without worked solutions; the point being with alot of graduate level textbooks, one should be able to master the content and work though the problems not requiring a worked solution. darling's textbook gave me the right pitch and level to be able to attack many of the problems within the textbook, and all I cared about at the end was did I get the correct answer as I knew I could apply the theory. –  Oct 15 '18 at 11:41
  • Fair enough :) You may want to change your answer, though? Because you said that it has solutions provided – soap Oct 15 '18 at 12:14
  • @Soap OK, well it has solutions, just not worked solutions. But I shall make that clear. –  Oct 15 '18 at 12:46
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Differential Geometry by Pressley.

Faraad Armwood
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There is a Schaum's Outline available. Has all the answers and covers most of the topic. Very good resource although, you may miss just a few topics or hard problems that an exhaustive treatment would have.

Presley's book (Springer, also available Amazon, not linking so I don't get spam controlled) covers the topic and has high reviews and all the solutions (and some hints prior to solutions, which can help).

Kreyszig's book also has all the answers at end. It does require that you learn simple tensor calculus as you go. But is restricted to 3D space (but gives you the tools to go more complicated later). Here is a review that is generally positive but worries that tensor calc is too much for undergrads. FWIW, the Amazon reviews are extremely positive.

Good luck and happy self studying!

Siong Thye Goh
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You may be pleased with Toponogov's book, Differential Geometry of Curves and Surfaces - A Concise Guide.

Quoting from the preface:

A distinctive feature of the book is a large collection (80 to 90) of nonstandard and original problems that introduce the student into the real world of geometry. Most of these problems are new and are not to be found in other textbooks or books of problems. [...]

Almost all the problems are given with their solutions, although the hope of the author is that an honest student will solve them without assistance, and only in exceptional cases will look at the text for a solution.

Mark Fantini
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  • I took a look at that book, and most of the problems aren't solved, which may sometimes problematic (e.g the geodesic of right helicoid [p. 150] are not that easy to find). Where do the solutions hide? of course the solutions are not neccesary but that's a net to verify I'm not proving wrong theorems/lemmas. –  Oct 16 '14 at 05:16
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    I'd say that you need to learn to check your arguments —for geometry and everything else— and comparing your argument to another is not really the correct way to do that. – Mariano Suárez-Álvarez Apr 28 '17 at 20:30