Reference on projective space

Question

Given $V$ a finite dimensional vector space we could consider $\mathbb{P}(V)$ with its classical topology given by the quotient map $V\setminus \{0\} \to \mathbb{P}(V)$ that assigns to a vector its line. Choosing coordinates in $V$ it's easy to endow to $\mathbb{P}(V)$ a smooth manifold structure. I'm looking for a reference on the coordinate free version of this differentiable structure. I'm aware that this construction uses hyperplanes $H$ to construct an atlas for $\mathbb{P}(V)$ that go from lines $\ell$ such that $\ell + H = V$ to euclidean space.

Answer 1

You might find the following books useful:

1. Tu
2. Naber
3. Rudolph & Schmidt