The primary goal of these notes is to record the relationship between the three models of hyperbolic space given by

- The hyperboloid model
- The Poincaré model
- The Klein model

Most of these information is easily found elsewhere, but so this is mostly just some notes that make it easy for myself to find.

## Hyperboloid model

As someone who frequently works in pseudo-Riemannian geometry, the most familiar model of hyperbolic space $\mathbb{H}^d$ for me is as the *hyperquadraic* $\{ t^2 - |x|^2 = 1, t > 0\}$ in *Minkowski space* $\mathbb{R}^{1,d}$, with the induced metric from the Minkowski metric.

This model can be parametrized in several different ways.

### Polar coordinates

First note that the hyperboloid passes through the point $(1,0)$ in Minkowski space. Let $o$ denote this point, and let $\rho$ denote the geodesic distance from $o$. And let $\omega\in \mathbb{S}^{d-1}$. We can parametrize $\mathbb{H} \setminus \{o\}$ using $\mathbb{R}_+ \times \mathbb{S}^{d-1}$ as the usual polar coordinates.

The mapping is given by (with $t,x$ the coordinates on $\mathbb{R}^{1,d}$): \begin{align} t &= \cosh\rho \newline x & = \sinh \rho \cdot \omega \end{align} The induced metric takes the form \begin{equation} ds^2 = d\rho^2 + \sinh(\rho)^2 d\omega^2 \end{equation}

### Cartesian coordinates

Notice that $\mathbb{H}$ can also be parametrized by $\mathbb{R}^d$, using that it is a graph over $\{0\} \times\mathbb{R}^d$ in Minkowski space.

The mapping is $x \mapsto (\sqrt{1 + |x|^2}, x)$, and the induced metric is \begin{equation} ds^2 = \sum_{i = 1}^d dx^i \otimes dx^i - \sum_{i,j = 1}^d \frac{x^i x^j}{1 + |x|^2} dx^i \otimes dx^j \end{equation}

### Geodesics

A well known theorem in pseudo Riemannian geometry states that the geodesics on a hyperquadric is in one-to-one correspondance to two dimensional planes through the origin in $\mathbb{R}^{m,n}$. In the case of the hyperboloid, every geodesic on $\mathbb{H}$ is given as the intersection of $\mathbb{H}$ with a time-like plane through the origin of $\mathbb{R}^{1,d}$.

## Klein model

The Klein model parametrizes hyperbolic space using the unit disc. Given a point $p$ on the hyperboloid $\mathbb{H}$, the line connecting $p$ to the origin of $\mathbb{R}^{1,d}$ intersects the plane $\{t = 1\}$ exactly once. The mapping is given by
\begin{equation}
(t,x) \mapsto \frac{x}{t} \in B(0,1).
\end{equation}
The inverse mapping can be found by solving $t^2 - x^2 = 1$ with $x / t = y$. This gives
\begin{align}
t &= \frac{1}{\sqrt{1-y^2}} \newline

x &= \frac{y}{\sqrt{1 - y^2}}
\end{align}
The induced metric is therefore
\begin{equation}
ds^2 = \sum_{i = 1}^d \frac{dy^i \otimes dy^i}{1- y^2} + \sum_{i,j = 1}^d \frac{y^i y^j }{(1-y^2 )^2} dy^i \otimes dy^j.
\end{equation}

The geodesics of the Klein model are straight lines: this can be seen by the characterization of the geodesics given above.

## Poincaré model

The Poincaré model is a different projection. Given a point $p$ on $\mathbb{H}$, we look for the line segment connecting $p$ to the point $(-1,0)$ in $\mathbb{R}^{1,d}$. This line segment intersects the $\{t = 0\}$ hyperplane exactly once within the unit ball.

The formula can be more easily by regarding $\mathbb{H}$ as the translated hyperbolid $(s-1)^2 - |x|^2 = 1$, and this translation means the mapping is to the plane $\{s = 1\}$. Therefore, a point $(t,x)$ in $\mathbb{H}$ gets mapped to $x / (t+1)$ in the unit ball.

The reverse mapping is found by solving $t^2 - x^2 = 1$ and $x / (t+1) = y$. We find then \[ t^2 - (t+1)^2 y^2 = 1 \implies (1-y^2) t^2 - 2y^2 t - (1 + y^2) = 0 \] or \begin{align} t &= \frac{1 + y^2}{1- y^2} \newline x &= \frac{2y}{1 - y^2} \end{align} This gives that the metric is given by \begin{equation} ds^2 = \frac{4 dy^i \otimes dy^i}{(1 -y^2 )^2} \end{equation}

## Horospheres

Of special importance in the Poincaré model is the horocycles: these are spheres contained within the unit ball tangent to the boundary. In other words, given $y_0 \in B(0,1)$, these are the points \[ \{ |y - y_0|^2 = (1 - |y_0|)^2 \}. \] The horospheres are important because the geodesics orthogonal to them "focus at infinity".

The change of variables given above means that in the hyperboloid model the horospheres are given by
\[
|x|^2 - 2 y_0 \cdot x (t+1) = (1- 2|y_0|)(1+t)^2
\]
However, we also have that $|x|^2 = t^2 - 1$. So we can reduce the equation to
\[
t - 1 - 2 y_0 \cdot x + ( 2 |y_0|- 1)(t + 1) = 0
\]
which we can rewrite as
\[ |y_0| t - y_0 \cdot x = 1 - |y_0| \]
which is the equation for a *null* hyperplane. In other words, the horospheres in $\mathbb{H}$ are precisely the intersection of a null ($d$-dimensional) hyperplane with the hyperboloid.

The "focus at infinity" property of the orthogonal geodesics are actually pretty easy to prove in the hyperboloidal model.
Let $\Pi$ be a null hyperplane, its (Lorentzian) unit normal we write as $n$. (Note that $n$ is therefore also tangent to $\Pi$.
Let $p\in \Pi \cap \mathbb{H}$, and so is a unit time-like vector in $\mathbb{R}^{1,d}$. Denote by $P$ the plane spanned by $\{p,n\}$.
I claim the unique vector that is tangent to both $P$ and $\mathbb{H}$ (hence the generator of the geodesic associated to $P$) is orthogonal to $\mathsf{T}\Pi \cap \mathsf{T}\mathbb{H}$.
Observe that the vector in question must be a linear combination of $p$ and $n$. Now, by construction $p$ is orthogonal to any element of $\mathsf{T}\mathbb{H}$. And as discussed earlier $n$ is orthogonal to any element of $\mathsf{T}\Pi$. Therefore their linear combination must be orthogonal to any element of the intersection.
Therefore geodesics orthogonal to $\Pi\cap \mathbb{H}$ in $\mathbb{H}$ are *precisely* those geodesics whose tangent vector asymptotically converge to $n$ (regarded as a curve in $\mathbb{R}^{1,d}$).

## More about the horosphere

Within the set $\{ t > |x| \}$, we can define the coordinate system $(\tau,\rho, \omega)$ using the transformation
\[ t = \tau \cosh(\rho), \qquad x = \tau \sinh(\rho) \omega \]
$\mathbb{H}$ corresponds to the set $\{ \tau = 1\}$.
Minkowski space can be (locally) described as a warped metric in this coordinate system, with metric given as
\[ ds^2 = - d\tau^2 + \tau^2 (d\rho^2 + \sinh(\rho)^2 d\omega^2). \]
Therefore the wave operator $\Box = -\partial_t^2 + \sum \partial_{x^i }^2$ can be decomposed in this coordinate system as
\[ \Box = - \partial_{\tau}^2 - \frac{d}{\tau} \partial_{\tau} + \frac{1}{\tau^2} \triangle_{\mathbb{H}}. \]
The partial differentiation by $\partial_{\tau}$ is the action by the vector field
\[ \partial_{\tau} = \frac{t}{\tau} \partial_t + \frac{x}{\tau} \partial_x .\]
Now, let $n$ be a past-directed null vector in $\mathbb{R}^{1,d}$.
The function $\eta_n: (t,x) \mapsto \langle (t,x), n\rangle$ has its level sets the null hyperplanes, and easily one sees $\Box f(\eta_n) = 0$ for *any* $f:\mathbb{R}\to \mathbb{C}$.

On the other hand, since $\eta_n$ is a linear function, we have that $\partial_{\tau} \eta_n = \frac{1}{\tau} \eta_n$. And hence $\partial_{\tau}^2 \eta_n = 0$. This therefore implies \begin{equation} \triangle_{\mathbb{H}} f(\eta_n) = \tau^2 \Box f(\eta_n) + f''(\eta_n) \eta_n^2 + f'(\eta_n) d \eta_n \end{equation} In particular, we see that if \begin{equation} f_z(x) = x^z \end{equation} for any $z\in \mathbb{C}$ (note that by construction $\eta_n$ is the inner product of a future pointing unit time-like vector against a pass-directed null vector and so is always positive), immediately we have \begin{equation} \triangle_{\mathbb{H}} f_z(\eta_n) = z(z+ d-1) \end{equation} and so these functions are formally eigenfunctions of the Laplacian on the hyperboloid.