Differential-geometry textbook with solved problems

Question

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 ?

Answer 1

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.

Answer 2

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.

Answer 3

Differential Geometry by Pressley.

Answer 4

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!

Answer 5

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.