# Three hyperbolic metrics

Up to a constant factor, there is just one conformally invariant Riemannian metric ${\rho}$ on the disk ${\mathbb{D}=\{z\in {\mathbb C}\colon |z|<1\}}$. Indeed, on the tangent space at ${0}$ the metric must be a multiple of the Euclidean one, due to rotational invariance. Normalizing so that at ${0}$ both metrics coincide, we can use the invariance under conformal automorphisms (Möbius transformations)

$\displaystyle \psi_a(z) = \frac{z+a}{1+\bar a z},\quad |a|<1$

to find that on the tangent space at ${a}$ the metric ${\rho}$ differs from Euclidean only by the factor of ${|\psi'_a(a)| (1-|a|^2)^{-1}}$. This can be written as ${d\rho(z) = (1-|z|^2)^{-1}|dz|}$, indicating what we integrate to find the length of curves with respect to ${\rho}$. This is the Poincaré disk model of the hyperbolic plane. The geodesics of ${\rho}$ are precisely the circles orthogonal to ${\partial\mathbb{D}}$, and diameters.

We could model the hyperbolic plane on any other proper simply-connected domain ${\Omega\subset {\mathbb C}}$, just by pulling back ${\rho}$ under the Riemann map ${\phi\colon\Omega\rightarrow\mathbb{D}}$. Explicitly, ${d\rho_{\Omega}(z)=(1-|\phi(z)|^{-2})\,|\phi'(z)|\,|dz|}$. But is this really explicit? We have no closed form for ${\phi}$ except for very special domains ${\Omega}$. The density ${\rho_\Omega}$ can be quite tricky: see this MathOverflow question.

On the disk ${\mathbb{D}}$, the density of ${\rho}$ is roughly the reciprocal of the distance to the boundary. The Koebe distortion theorem yields the same for all simply connected domains:

$\displaystyle \mathrm{dist}\,(z,\partial \Omega)^{-1} \le \frac{|d\rho_\Omega(z)|}{|dz|}\le 4\, \mathrm{dist}\,(z,\partial \Omega)^{-1}$

This suggests replacing the hyperbolic metric ${\rho_\Omega}$ with the quasihyperbolic metric ${d\delta_\Omega(z)=\mathrm{dist}\,(z,\partial \Omega)^{-1}\,|dz|}$. The density of ${\delta_\Omega}$ is about as simple as we could wish for, but the shape of its geodesics is not as obvious; indeed, a number of papers were written on this subject in the last 35 years.

Can we have a hyperbolic-type metric with explicit geodesics and explicit density? The Hilbert metric delivers both, at least in bounded convex domains. The idea is simple: instead of dividing the length of each tangent vector ${v\in T_a}$ by ${\mathrm{dist}\,(a,\partial \Omega)}$, we divide it by ${\mathrm{dist}_v\,(a,\partial \Omega)}$, which is the distance measured in the direction of ${v}$. In other words, this is how long you could walk in the direction ${v}$ before hitting the boundary. It is clear that the length of any curve going to the boundary is infinite due to the divergence of ${\int_{0^+} \frac{dx}{x}}$, same as for hyperbolic and quasihyperbolic metrics. Since it is awkward to have a “metric” that is not symmetric (the lengths of ${v}$ and ${-v}$ are not the same), we symmetrize:

$\displaystyle \|v\|_{\Omega,a} =\frac12 \left(\frac{1}{\mathrm{dist}_v\,(a,\partial \Omega)} + \frac{1}{\mathrm{dist}_{-v}\,(a,\partial \Omega)} \right) |v|_{{\mathbb R}^2}$

Now the distances between points are defined in the usual way, as the infimum of lengths of connecting curves ${\gamma \colon [a,b]\rightarrow\Omega}$,

$\displaystyle L(\gamma) = \int_a^b \|\gamma'(t)\|_{\Omega,\gamma(t)}\,dt$

This is the Hilbert metric ${d_\Omega}$. Despite all its non-Euclideanness, the geodesics of ${d_\Omega}$ are line segments. This (nontrivial) fact makes it easy to calculate the distance between two points ${a,b\in\Omega}$: besides the points themselves, one only needs to consider the pair of points ${a',b'}$ where the line through ${a,b}$ meets ${\partial \Omega}$. The integral ends up being (half of) the logarithm of the cross-ratio of these four points. Assuming ${a',a,b,b'}$ are situated in the listed order, the relevant cross-ratio is

$\displaystyle \frac{|a'-b|\, |a-b'|}{|a'-a|\,|b-b'|}$

which is greater than ${1}$, making the logarithm positive.

I personally find the integral of reciprocals of distances more intuitively accessible than the logarithm of cross-ratio. Either version of the definition makes it clear that the Hilbert metric is invariant under projective transformations, a property not shared by the metrics ${\rho}$ and ${\delta}$.

It turns out that ${d_{\mathbb D}}$ is also a model of the hyperbolic plane, but with geodesics being line segments rather than circular arcs. Along each diameter ${d_{\mathbb D}}$ coincides with ${\rho}$, because at Euclidean distance ${r}$ from the center it scales vectors by

$\displaystyle \frac{1}{2}\left(\frac{1}{1-r}+\frac{1}{1+r}\right) = \frac{1}{1-r^2}$

Behold the magic of partial fractions!