Hyperbolic Geometry Note #2: On Möbius transformations as isometries of the upper half plane

IMG_1347 Möbius transformations are conformal (i.e., angle-preserving), orientation-preserving, area-preserving isometries in the upper half-plane model of hyperbolic space. It piqued my interest to discover that the form of the metric in this model (which I discussed in Hyperbolic Geometry Note #1) is specifically chosen to ensure that Möbius transformations are indeed isometries. It turns out that in metric spaces one can sometimes `fiddle’ with the form of the metric in order to obtain geometries with particular properties, and the upper half-plane model is an example of this. In this note I want to explore in detail how this `fiddling’ with the metric works.

For z \in \mathbb{H}, a Möbius transformation would take the form

\gamma(z) = \frac{az + b}{cz + d}

where the coefficients a, b, c, and d are real numbers satisfying the condition ad - bc > 0. This is an isometry in \mathbb{H} (if one uses the metric discussed in my previous note) because it preserves lengths, i.e., for distinct z, z^{\prime} \in \mathbb{H} we have

d_{\mathbb{H}}(\gamma(z), \gamma(z^{\prime})) = d_{\mathbb{H}}(z, z^{\prime})

I want to first prove this in detail before going on to show how the form of the metric guarantees that this result will hold. To prove the isometry property we note that if \sigma is a path from z to z^{\prime} then (\gamma \circ \sigma)(z) \equiv \gamma(\sigma(z)) is a path from \gamma(z) to \gamma(z^{\prime}). The shortest such path (whose length would by definition be the hyperbolic distance between the two points) must also be of this form, so all we need to prove is that

\text{Length}_{\mathbb{H}}(\gamma \circ \sigma) = \text{Length}_{\mathbb{H}}(\sigma)

where (as discussed in my previous note) the hyperbolic length of a path \sigma is obtained by integrating the function f(z) = 1/\text{Im}(z) along \sigma, so

\text{Length}_{\mathbb{H}}(\sigma) = \int_{\sigma}\frac{1}{\text{Im}(z)}dt = \int_a^b \frac{|\sigma^{\prime}(t)|}{\text{Im}(\sigma(t))}dt

We need two auxiliary results concerning Möbius transformations, namely, expressions for |\gamma^{\prime}(z)| and \text{Im}(\gamma(z)). For any z \in \mathbb{H} we have

|\gamma^{\prime}(z)| = \big| \frac{(cz + d)a - (az + b)c}{(cz + d)^2} \big| = \big| \frac{ad - bc}{(cz + d)^2} \big| = \frac{ad - bc}{|cz + d|^2}

and if we write z = u + iv we have

\text{Im}(\gamma(z)) = \frac{\gamma(z) - \overline{\gamma(z)}}{2i}

= \frac{1}{2i}\big\{\frac{az + b}{cz + d} - \frac{a\overline{z} + b}{c\overline{z} + d}\big\}

= \frac{1}{2i}\frac{(az+b)(c\overline{z}+d) - (a\overline{z}+b)(cz+d)}{(cz+d)(c\overline{z}+d)}

= \frac{1}{2i}\frac{azd + bc\overline{z} + bd - a\overline{z}d - bcz -bd}{|cz + d|^2}

= \frac{1}{2i}\frac{2iv(ad - bc)}{|cz + d|^2}

= \frac{ad - bc}{|cz + d|^2}v

= \frac{ad - bc}{|cz + d|^2} \text{Im}(z)

Using these two results and the chain rule we obtain

\text{Length}_{\mathbb{H}}(\gamma \circ \sigma) = \int \frac{|(\gamma \circ \sigma)^{\prime}(t)|}{\text{Im}(\gamma \circ \sigma)(t)}dt

= \int \frac{|(\gamma^{\prime}(\sigma(t))||\sigma^{\prime}(t)|}{\text{Im}(\gamma \circ \sigma)(t))}dt

= \int \frac{ad - bc}{|c\sigma(t) + d|^2} |\sigma^{\prime}(t)|\frac{|c\sigma(t) + d|^2}{ad - bc}\frac{1}{\text{Im}(\sigma(t))}dt

= \int \frac{|\sigma^{\prime}(t)|}{\text{Im}(\sigma(t))}dt = \text{Length}_{\mathbb{H}}(\sigma)

which proves the isometry result.

It turns out that if we want this isometry result to hold, we must define the hyperbolic length using the function 1/\text{Im}(z) as above. To see why this is the case, let \rho: \mathbb{H} \rightarrow \mathbb{R} be a continuous positive function. Define the \rho-length of a path \sigma: [a, b]  \rightarrow \mathbb{H} in the usual way to be

\text{Length}_{\rho}(\sigma) = \int_{\sigma} \rho = \int_a^b \rho(\sigma(t))|\sigma^{\prime}(t)|dt

Now suppose that \text{Length}_{\rho} is invariant under Möbius transformations of \mathbb{H}, i.e., if \gamma is a Möbius transformation we have

\text{Length}_{\rho}(\gamma \circ \sigma) = \text{Length}_{\rho}(\sigma)

On the left hand side we have

\text{Length}_{\rho}(\gamma \ \circ \ \sigma) = \int_a^b \rho((\gamma \ \circ \ \sigma)(t))|(\gamma \ \circ \ \sigma)^{\prime}(t)|dt = \int_a^b \rho((\gamma \ \circ \ \sigma)(t))|\gamma^{\prime}(\sigma(t))||\sigma^{\prime}(t)|dt

while on the right hand side we have

\text{Length}_{\rho}(\sigma) = \int_a^b \rho(\sigma(t))|\sigma^{\prime}(t)|dt

These two expressions can only be (identically) equal if for all t we have

\rho((\gamma \circ \sigma)(t))|\gamma^{\prime}(\sigma(t))| = \rho(\sigma(t))

Writing z = \sigma(t), we can express this condition on the function \rho as

\rho(\gamma(z)) |\gamma^{\prime}(z)| = \rho(z)

Now suppose we take \gamma(z) = z + b in this last equation, with b any real number and z = x + iy. We get

\rho(z + b) |1| = \rho(z)

or

\rho(z + b) = \rho(z)

This shows that the real part of the argument of \rho does not matter, because (for example) we can set b = -x and get

\rho(iy) = \rho(z)

We deduce that \rho(z) depends only on the imaginary part of z, so we can write

\rho(z) = \rho(y)

Now suppose we take \gamma(z) = kz with k > 0 in the equation

\rho(\gamma(z)) |\gamma^{\prime}(z)| = \rho(z)

We get

\rho(kz) |k| = \rho(z)

which (since \rho depends only on the imaginary part of its argument) we can write as

k \rho(ky) = \rho(y)

This equation can only hold for all y in the upper half plane if on the left hand side we have

\rho (ky) = \frac{c}{ky}

and on the right hand side we have

\rho(y) = \frac{c}{y}

where c is some positive constant. Therefore we see that if Möbius transformations of \mathbb{H} are to be isometries, it must be the case that the function used in calculations of distances between points in \mathbb{H} must be (up to a normalising constant c > 0) the reciprocal 1/\text{Im}(z).

It is this result that really characterises the upper half-plane model of hyperbolic space, leading to the result that the geodesics in the upper half plane are either vertical lines or semicircles with endpoints on the real axis. The reasoning (in brief) is as follows.

geodesics

It can easily be shown that the imaginary axis in \mathbb{H} is a geodesic: any path joining two distinct points ia and ib has a hyperbolic length greater than or equal to \log b/a, with equality to \log b/a only when the path is a straight line along the imaginary axis. It can also easily be shown that a path can be found between any two points in \mathbb{H} which is either a vertical line or a semicircle with endpoints on the real axis, and that both of these can be mapped bijectively to the imaginary axis by a suitable Möbius transformation. (This Möbius transformation is either a simple translation if the path between the two points is a vertical line, or a transformation of the form

\gamma(z) = \frac{z - \zeta_1}{z - \zeta_2}

if the path is a semicircle with real endpoints \zeta_1 < \zeta_2). Since Möbius transformations are isometries, this must mean that such vertical lines and semicircles are also geodesics. It can finally be shown that there are no other such paths in \mathbb{H}, so the Poincaré half-plane model of hyperbolic space is characterised by the fact that geodesics in \mathbb{H} are either vertical lines or semicircles with endpoints on the real axis.

Hyperbolic Geometry Note #1: Strange behaviour of length calculations in the Poincaré half-plane model of hyperbolic space

halfplane1 In the Poincaré half-plane model of hyperbolic space, the upper half-plane \mathbb{H} is the set of complex numbers z with positive imaginary part:

\mathbb{H} = \{z \in \mathbb{C}: \text{Im}(z) > 0 \}

The boundary of \mathbb{H}, \delta \mathbb{H}, is the real axis plus the point \infty. Formally,

\delta \mathbb{H} = \{z \in \mathbb{C}: \text{Im}(z) = 0 \} \cup \{ \infty \} \equiv \mathbb{R} \cup \{ \infty \}.

\delta \mathbb{H} is often referred to as the circle at infinity because it is topologically equivalent to a circle. This is the two-dimensional analogue of the topological equivalence of the 2-sphere and the extended complex plane (`the Riemann sphere’) which I wrote about in a previous note. As in that note, the argument is based on stereographic projection. We represent the unit circle in the complex plane \mathbb{C} as the set

K = \{z \in \mathbb{C}: |z| = 1 \}

and we construct a homeomorphism

\pi: K \rightarrow \delta \mathbb{H}

by imagining a line passing through i at the `north pole’ of the circle and another point z \in K-\{i\}.

circle at infinity

This line meets the real axis at a unique point \pi(z). We define \pi(i) = \infty. The map \pi is a homeomorphism from K to \delta \mathbb{H} = \mathbb{R} \cup \{ \infty \}, which is a topological way of saying that K and \delta \mathbb{H} are `the same’.

What I want to focus on in this note is really the `at infinity’ part of the circle at infinity. It is said to be `at infinity’ because the upper half-plane model of hyperbolic space has the curious property that points on \delta \mathbb{H} are at an infinite distance from any point in \mathbb{H}. This arises from the fact that the hyperbolic length of a path in the upper half-plane \mathbb{H} is obtained by integrating the functon f(z) = 1/\text{Im}(z) along that path.

To clarify, let \sigma be a (piecewise differentiable) path in \mathbb{H} with parametrisation

\sigma(t): [a, b] \rightarrow \mathbb{H}

As usual in complex analysis, if f: \mathbb{H} \rightarrow \mathbb{R} is a continuous function, then the integral of f along the path \sigma is defined to be

\int_{\sigma}f = \int_a^b f(\sigma(t)) |\sigma^{\prime}(t)|dt

where |\cdot| denotes the usual modulus of a complex number, i.e.,

|\sigma^{\prime}(t)| = \sqrt{(\text{Re}\sigma^{\prime}(t))^2 + (\text{Im}\sigma^{\prime}(t))^2}

(Note that no matter what parametrisation is chosen for the path, the value of this integral will be the same so one usually tries to find a parametrisation that is as easy to integrate as possible).

As stated earlier, the hyperbolic length of the path \sigma is obtained by integrating the function f(z) = 1/\text{Im}(z) along \sigma, so

\text{Length}_{\mathbb{H}}(\sigma) = \int_{\sigma}\frac{1}{\text{Im}(z)}dt = \int_a^b \frac{|\sigma^{\prime}(t)|}{\text{Im}(\sigma(t))}dt

I want to highlight two curious results that follow from this setup. First, consider the points u + iv and u + kiv in \mathbb{H} where 0 < k < 1 and v > 0.

vertical

A parametrisation of the `vertical’ path between them in the diagram could be

\sigma(t) = u + kiv + (1 - k)ivt

t \in [0, 1]

We then have

\int_{\sigma}\frac{1}{\text{Im}(z)}dt = \int_0^1 \frac{|\sigma^{\prime}(t)|}{\text{Im}(\sigma(t))}dt = \int_0^1 \frac{|(1 - k)iv|}{kv + (1 - k)vt}dt = \int_0^1 \frac{(1 - k)v}{kv + (1 - k)vt}dt

= [\log(kv + (1 - k)vt)]_0^1 = \log v - \log kv = \log \big(\frac{v}{kv}\big) = \log \big(\frac{1}{k}\big)

Since the length of the path is \log(1/k) we see that as the lower point gets closer and closer to the real axis (i.e., as k \rightarrow 0), the length of the path goes to \infty. This is the motivation for the `at infinity’ nomenclature for the boundary \delta \mathbb{H}.

The second curious result involves the lengths of paths between two points at the `same height’ in \mathbb{H}, say -2 + i and 2 + i.

triangle

A parametrisation of the `horizontal’ path between them in the leftmost diagram above could be

\sigma_1(t) = 2t + i

t \in [-1, 1]

We then have

\int_{\sigma_1}\frac{1}{\text{Im}(z)}dt = \int_{-1}^1 \frac{|\sigma_1^{\prime}(t)|}{\text{Im}(\sigma_1(t))}dt = \int_{-1}^1 \frac{2}{1}dt = 2 \cdot 2 = 4

so the length of the horizontal path between the two points is 4. Now consider the situation shown in the remaining two pictures above. These involve piecewise linear paths between the two points that go diagonally from -2 + i to a point ki and then diagonally from ki to 2 + i. I am leaving k unspecified so that we can study the lengths of the piecewise linear paths as a function of k.

A parametrisation of the piecewise linear paths is given by

\sigma_2(t) = \big[ (2 - i)t + ki(1 + t) \big]I_{[-1, 0]}(t) + \big[ (2 + i)t + ki(1 - t) \big]I_{[0, 1]}(t)

where I_A(t) denotes the indicator function for set A. We then have

\int_{\sigma_2}\frac{1}{\text{Im}(z)}dt = \int_{-1}^1 \frac{|\sigma_2^{\prime}(t)|}{\text{Im}(\sigma_2(t))}dt = \int_{-1}^0\frac{|2 + (k - 1)i|}{k(1 + t) - t}dt + \int_0^1\frac{|2 + (1 - k)i|}{k(1 - t) + t}dt

= \int_{-1}^0 \frac{\sqrt{4 + (k-1)^2}}{k + t(k-1)}dt + \int_0^1 \frac{\sqrt{4 + (k-1)^2}}{k - t(k-1)}dt

= \sqrt{4 + (k-1)^2} \bigg\{ \int_{-1}^0 \frac{1}{k + t(k-1)}dt + \int_0^1 \frac{1}{k - t(k-1)}dt \bigg\}

= \sqrt{4 + (k-1)^2} \frac{1}{k-1}\bigg\{ \int_{-1}^0 \frac{k-1}{k + t(k-1)}dt - \int_0^1 \frac{-(k-1)}{k - t(k-1)}dt \bigg\}

= \frac{\sqrt{4 + (k-1)^2}}{k-1} \bigg\{ \big[\log(k + t(k-1))\big]_{-1}^0 - \big[\log(k - t(k-1))\big]_0^1 \bigg\}

= \frac{\sqrt{4 + (k-1)^2}}{k-1} \bigg\{ \log k - (- \log k) \bigg\}

= 2\frac{\sqrt{4 + (k-1)^2}}{k-1} \log k

This final expression gives the lengths of the piecewise linear paths as a function of k. If we plot this function of k we see something completely at odds with Euclidean geometry:

GraphFINAL

We see that for values of k between 1 and 6.5 the lengths of the piecewise linear paths are actually less than the length of the straight horizontal line between the two points -2 + i and 2 + i. The minimum possible path length between the two points is achieved by one of the piecewise linear paths (with a value of k somewhere between 2.25 and 2.5) rather than by the straight horizontal path between them. This shows that hyperbolic lengths behave very differently from the lengths we are used to in Euclidean geometry.

Clearly, this approach to calculating distances between points is at odds with the triangle inequality, so it cannot be used directly as the metric for measuring distances between points in hyperbolic space. Instead, given z, z^{\prime} \in \mathbb{H}, the hyperbolic distance d_{\mathbb{H}}(z, z^{\prime}) between z and z^{\prime} is defined as

d_{\mathbb{H}}(z, z^{\prime}) = \text{inf}\{\text{Length}_{\mathbb{H}}(\sigma) : \sigma \text{ is a piecewise differentiable path with endpoints } z \text{ and } z^{\prime} \}

So what we do is consider all piecewise differentiable paths between z and z^{\prime}, calculate the hyperbolic length of each such path, then take the shortest. It turns out that this infimum is always actually achieved by some path (a geodesic) and this path is unique. The graph I plotted above is an illustration of this.

The metric d_{\mathbb{H}} clearly does satisfy the triangle inequality, i.e.,

d_{\mathbb{H}}(x, z) \leq d_{\mathbb{H}}(x, y) + d_{\mathbb{H}}(y, z)

for all x, y, z \in \mathbb{H}. The distance between two points according to this metric must be increased if one goes via a third point, because it is by definition the infimum of all hyperbolic lengths between the points, so those involving any third point must have already been discounted as being longer.