The metric tensor for the Poincaré ball model of hyperbolic geometry is

$$ g_{ij} = \frac{\delta_{ij}}{(1 - \lvert \mathbf{r} \rvert^2)^2} $$

where $\mathbf{r}$ is the position in the ambient Euclidean space.

An example of a uniform tiling of a 2-dimensional hyperbolic space is shown below:

enter image description here

My question is as follows:

Given that we use $\mathbf{r} \in \mathbb{R}^n$ where $\lvert \mathbf{r} \rvert^2 < 1$ to model an $n$-dimensional hyperbolic space, what are the isometries of this space in terms of $\mathbf{r}$? How do I express the analogues of translation and rotation in terms of transformations on $\mathbf{r}$ (i.e. rearranging the points in the image above to show a translation or rotation of the entire hyperbolic space)?

So far I have seen examples involving Mobius transformations, but how can these be generalized beyond the plane? Is there a general matrix acting on $\mathbf{r}$ that can express these transformations? I may not be understanding this clearly, so please correct me where necessary.

This paper states that any hyperbolic isometry is induced by a matrix of one of the following types:

$$ \left( \begin{array}{cc} e^t & 0 \\ 0 & e^{-t} \end{array} \right) $$

$$ \left( \begin{array}{cc} 1 & \pm 1 \\ 0 & 1 \end{array} \right) $$

$$ \left( \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \right) $$

for geodesic translation by hyperbolic distance $2t$, horocylic translation, and rotation by angle $2 \theta$.


According to Wikipedia, if hyperbolic space is translated such that the origin in the unit Poincaré disk is translated to $\mathbf{v}$, $\mathbf{x}$ is translated to

$$\frac{ ( 1 + 2 \mathbf{v} \cdot \mathbf{x} + \left| \mathbf{x} \right| ^2 ) \mathbf{v} + ( 1 - \left| \mathbf{v} \right| ^2 ) \mathbf{x}}{ 1 + 2 \mathbf{v} \cdot \mathbf{x} + \left| \mathbf{v} \right| ^2 \left| \mathbf{x} \right| ^2 }$$

  • 1
    $\begingroup$ The Wikipedia article that you linked to states the form of isometries. Nothing new: rotation around the center, inversion in a sphere orthogonal to the given one; and compositions of these things. $\endgroup$ – user147263 Aug 25 '15 at 4:53
  • $\begingroup$ @NormalHuman Are inversions the hyperbolic "reflections"? $\endgroup$ – Akiva Weinberger Aug 30 '15 at 4:55
  • $\begingroup$ Yes. They are involutions fixing a sphere of hyperplane. $\endgroup$ – user147263 Aug 30 '15 at 4:58

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