Is the cross ratio the unique invariant under projective transformations up to multiples?

Question

I have been studying the actions of $PSL_2(\mathbb{R})$ on the hyperbolic plane recently, and the hyperbolic distance $d(z_1, z_2)$ is the absolute value of the log of absolute value of the cross ratio between $z_1, z_2$ and the two points of the h-line that goes through $z_1, z_2$. So it seems that this distance measure is the unique function that satisfies $d(z_1, z_3) = d(z_1, z_2)+d(z_2, z_3)$ if they are on the same h-line and $z_2$ is "in-between" $z_1, z_3$, this uniqueness is up to multiples.

So that uniqueness comes from the cross ratio, so I wonder if the cross ratio is the only value that display this uniqueness?

Thanks!

Answer 1

No, the hyperbolic distance is not the only function satisfying $d(z_1,z_3)=d(z_1,z_2)+d(z_2,z_3)$. You could assign arbitrary (not even neccessarily continuous) values to all the points, interpret them as “distance from some origin”, and use differences between these values as distances. Your equation would be satisfied in all these cases, even if $z_2$ were not between $z_1$ and $z_3$.

The canonical hyperbolic distance has some other relevant properties:

• It is zero if two points coincide, and positive otherwise. At least for points inside the hyperbolic plane. Which is of course a consequence of the absolute value you included.
• The function inside the absolute value is analytic.
• The function tends towards infinity as a point tends toward the boundary of the model.
• If you apply a hyperbolic rigid motion, the lengths stay the same.

I guess that last point should be the most useful in actually characterizing the distance metric, but it assumes an existing concept of isometric transformations.