I know that there are a lot of questions in this topic, but none of them has been useful to me. So that's mine; I need to describe the geodesics of $\mathbb{H}^n$ in the two models of it:
$ \mathbb{H}^n$ in the hyperboloid model $I^+ = \{ (x_0,...,x_n) \in \mathbb{R}^{n+1} | -x_0^2+ x_1^2 +...+ x_n^2= -1, \ x_0>0 \} $ with metric $g_1^{n+1}$ (standard metric on $\mathbb{R}^{n+1} $ of signature 1).
The Poincaré ball model $\mathbb{D}^n= \{ X \in \mathbb{R}^n s.t. |X| < 1 \}$ with metric $hyp$ given by $hyp= \frac{euc}{(1-|X|^2)^2} $
In the first model I know that geodesics are given by intersection of $I^+$ with plane passing through the origin. I need a few details; start by taking a point $p \in I^+ $ and a tangent vector $ u \in T_pI^+$ and let $\gamma_u$ be the geodesic starting at $p$ with velocity $u$. Let $\pi_u = span \{ Op,u\} \subset \mathbb{E}^n_1 = (\mathbb{R}^{n+1},g_1^{n+1})$.
Now (as suggested by the professor) we can consider (does it exists?) a reflection $$ s_u: \mathbb{E}_1^{n+1} \to \mathbb{E}_1^{n+1}$$ that fixes $\pi_u$ and act as $-id$ on $\pi_u^{\perp,g_1^{n+1}}$. Since $s_u$ is an isometry and $s_u(p)=p, \ s_u(u)=u $ then $ s_u( \gamma _u) = \gamma_u $. So $\gamma_u$ is contained in $\pi_u$.
Why does exactly $\gamma_u= I^+ \cap \pi_u$ ?
For the second model, I'm a bit confused. First I want to describe geodesics passing through $O$ and then get every other geodesics by acting with the Moebius group ( wich is $Isom(\mathbb{D}^n)$ and acts transitively). Intuitively I get that geodesics passing through $N$ in $I^+$ are sent by stereographic projection into diameters of $\mathbb{D}^n$. Do I get in this way every geodesics in $\mathbb{D}^n$ passing through $O$? How can I get a bit more rigorous?
At this point, as I said, in order to get other geodesics is needed to know how the Moebius group acts on $\mathbb{D}^n$. I know that Moebius tranformations send lines and circles in lines and circles and that they are conformal maps. Let now $ O\neq p \in \mathbb{D}^n$ and $ u \in T_p^1(\mathbb{D}^n)$. There exists an isometry $F$ such that $d_pF(v)=u$ and $F(O)=p$ with $v \in T_O^1(\mathbb{D}^n)$. So $F(\gamma_v)= \gamma_u$. But $\gamma_u$ is a diameter of direction $u$ and, for what I said earlier it can only be sent into another diameter or into a circle intersecting orthogonally $\partial \mathbb{D}^n$. Is this correct? Is it lacking of precision somewhere?
Thanks to everybody for the attention.