Good problem book in differential geometry
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.
John M. Lee's Introduction to Smooth Manifolds is also a great book with lots of exercises.
The following list contains books on Differential Geometry with many, many exercises with solutions:
- Tu, Introduction to Manifolds
- Jänich, Vector Analysis
- Gadea et al, Analysis and Algebra on Differential Manifolds, Workbook with Solutions
- Lafontaine, An Introduction to Differential Manifolds
- Mishchenko & Fomenko, Problems in Differential Geometry and Topology
- Aubin, A Course in Differential Geometry
- Barden & Thomas, An Introduction to Differential Manifolds
- Morita, Geometry of Differential Forms
Hope it helps.
Barrett O'Neill's book "Elementary Differential Geometry" is really good. He has tons of exercises and he's a mathematical physicists/ diff. geometer so there you go. He also has a text on black holes and another text called "Semi-Riemannian Geometry" (tons of physics applications, problems).
$\textbf{Update}$: Loring Tu also has a differential geometry book now.
Spivak's Introduction to Differential Geometry
http://strangebeautiful.com/other-texts/spivak-intro-diff-geom-v1-3ed.pdf