Does calculus exists in hyperbolic geometry? Definition of $C^n$ class.

Question

I am trying to find a very good book about differential hyperbolic geometry because I am interested in the definition of $C^1$ class on hyperbolic surfaces. I am trying to write this definition from the definition of differential function in euclidean geometry, but I do not know if this definition holds on hyperbolic surfaces.

Answer 1

Locally, i.e., in a neighborhod of any point $p\in M$, any $d$-dimensional manifold $M$ looks like ${\mathbb R}^d$ in the neighborhood of ${\bf 0}$. It makes sense to talk about $C^r$ functions $f:\>M\to{\mathbb R}$ as soon as it is guaranteed that the transition functions between different charts on $M$ are $C^r$. All this is available before we are considering Riemannian metrics on $M$ which may turn $M$ into a hyperbolic manifold.