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.
