I am currently reading the book "Semi-Riemannian geometry with applications to relativity" by Barret O'Neill, and I am trying to do the exercises suggested at the end of each chapter, as I find this the best way to really understand the mathematics behind.
The aim of most exercises is to prove a statement. For example (Ex 13 Chapter 1)
For a function $f\in\mathcal{F}(M)$ its differential $df:T(M)\rightarrow\mathbb{M}$ and its differential map $df:T(M)\rightarrow T(\mathbb{R}^1)$ differ only by a canonical isomorphism.
I find myself a little bit lost when trying to solve these kind of problems as often I do not know how to start. I would like to find a book with similar exercises and with the solutions explained (manifold theory, tensors, and semi-riemannian manifold and submanifolds, just as the first 4 chapters of O'Neills book). I hope once I understand some exercises, I will be able to do the rest on my own