Text of differential geometry

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!!!