I would like to ask for suggestions for a differential geometry text book, reaching the theory of $n$-dimensional (not only cuves and surfaces) differentiable and Riemannian manifolds in sight of applications to the study of the theory of relativity. Since, when a text proposes exercises, it often uses the results derived in them to prove following lemmas and theorems, and since I don't attend classes at the university and have not the opportunity to ask professors about solutions, I would need a text including solutions to the proposed exercises (or a text that doesn't use the results proved in the exercises to prove the theory), which are quite rare to find in "intermediate" to advanced texts. I find it very frustrating to solve exercises and to be left with the doubt I convinced myself of wrong ideas. I prefer a book lacking exercises (which can be found on the net) than a book using results of exercises in the theory part, but not showing the solutions to the exercises. I can use texts in Italian, English, Spanish, Portuguese and French. Thank you so much for any suggestions!!!
I'm quite fond of Sternberg's Curvature in Mathematics and Physics. It's available online here, and for buying, here. The book covers all that you've listed. It also doesn't contain any exercises, except in the first two chapters, which set the basis for the book - so the exercises in the first two chapters just provide you with a strong foundation.