I am trying to gain more insight about the hyperbolic spaces, and in particular the Poincaré ball model. I got the basic intuition between the idea and how basic computations such as distances and the metrics can be defined in the Poincaré ball model. However, some aspects of the idea are still a mystery for me.
Assume that we have two points on a $d$-dimensional Poincaré ball; i.e., $\mathbf{x} \in \mathbb{R}^d$ and $\mathbf{y} \in \mathbb{R}^d$. Both $\mathbf{x}$ and $\mathbf{y}$ have norms less than one. What I would like to know is: "how is the dot product between these two points defined in the Poincaré ball model?"