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 , a Möbius transformation would take the form

where the coefficients , , , and are real numbers satisfying the condition . This is an isometry in (if one uses the metric discussed in my previous note) because it preserves lengths, i.e., for distinct we have

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 is a path from to then is a path from to . 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

where (as discussed in my previous note) the hyperbolic length of a path is obtained by integrating the function along , so

We need two auxiliary results concerning Möbius transformations, namely, expressions for and . For any we have

and if we write we have

Using these two results and the chain rule we obtain

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 as above. To see why this is the case, let be a continuous positive function. Define the -length of a path in the usual way to be

Now suppose that is invariant under Möbius transformations of , i.e., if is a Möbius transformation we have

On the left hand side we have

while on the right hand side we have

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

Writing , we can express this condition on the function as

Now suppose we take in this last equation, with any real number and . We get

or

This shows that the real part of the argument of does not matter, because (for example) we can set and get

We deduce that depends only on the imaginary part of , so we can write

Now suppose we take with in the equation

We get

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

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

and on the right hand side we have

where is some positive constant. Therefore we see that if Möbius transformations of are to be isometries, it must be the case that the function used in calculations of distances between points in must be (up to a normalising constant ) the reciprocal .

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.

It can easily be shown that the imaginary axis in is a geodesic: any path joining two distinct points and has a hyperbolic length greater than or equal to , with equality to 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 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

if the path is a semicircle with real endpoints ). 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 , so the Poincaré half-plane model of hyperbolic space is characterised by the fact that geodesics in are either vertical lines or semicircles with endpoints on the real axis.