Complex distance between two hyperbolic lines

# Complex distance between two hyperbolic lines

An oriented hyperbolic line in hyperbolic $3$-space $H^3$ is uniquely determined by its “initial” ideal point (by ideal point, I mean a point on the sphere at infinity $S^2_\infty$) and its “final” ideal point. We can thus write, by abuse of language, that an oriented hyperbolic line $L$ “is”: $L = (a, b)$, where $a$, resp. $b$, is its initial, resp. final, point.

Now suppose we are given two oriented hyperbolic lines, say $L_i = (a_i, b_i)$, for $i = 1, 2$. We could define the cross ratio of $L_1$ and $L_2$ to be the cross-ratio of the $4$ corresponding ideal points, namely

$$\operatorname{CR}(L_1, L_2) = \frac{(a_1 – b_2) (a_2 – b_1)}{(a_1 – b_1)(a_2 – b_2)},$$

where we are identifying the sphere at infinity $S^2_\infty$ with the Riemann sphere via stereographic projection from the “North” pole $(0, 0, 1)^T$. Thus, each of the $a_i$, $b_i$, for $i = 1, 2$, is a complex number or possibly $\infty$, with the usual conventions for $\infty$. When in doubt, the reader should use homogeneous coordinates on $\mathbb{P}^1_\mathbb{C}$.

Let $\delta$ be the hyperbolic distance between lines $L_1$ and $L_2$: it is the hyperbolic distance between the points $p_1 \in L_1$ and $p_2 \in L_2$, now thinking of each $L_i$ as an oriented hyperbolic line, rather than a pair of ideal points, that are closest to each other. If $L_1$ and $L_2$ intersect at a finite point in hyperbolic $3$-space, then their hyperbolic distance $\delta$ vanishes. Let $v_1$ be the unit vector based at $p_1$ which is tangent to $L_1$ and oriented along the orientation of $L_1$. Let $v’_1$ be the parallel transport of $v_1$ along the geodesic line segment $p_1 p_2$; in particular, $v’_1$ is based at $p_2$. Let $\alpha$ be the angle, using the right-hand rule (thinking of the unit tangent vector to $p_1 p_2$ at $p_2$, as a kind of normal vector), between $v’_1$ and $v_2$. Note that $\alpha$ is well defined up to a multiple of $2 \pi$.

One can also check that, even if we permute $L_1$ and $L_2$, both $\delta$ and the angle $\alpha$ remain the same. We then define the complex distance between $L_1$ and $L_2$ to be

$$d_\mathbb{C}(L_1, L_2) = \delta + i \frac{\alpha}{2} \in \mathbb{C}/(i \pi \mathbb{Z}).$$

In turns out, and it can be proved, that

$$\operatorname{CR}(L_1, L_2) = \operatorname{cosh}^2(d_\mathbb{C}(L_1, L_2)).$$

Note: I have independently rediscovered this formula, but, even though not very well known, this formula is not new. Indeed, as Sam Nead mentioned on MathOverflow (and I paraphrase him here), the complex distance between two hyperbolic lines appears on p. 355 in Marden’s book Outer circles: an introduction to hyperbolic 3-manifolds. In the second edition of the book, with the title, Hyperbolic manifolds: an introduction in 2 and 3 dimensions, the formula appears on page 432.

However, I am not sure whether Sam Nead’s reference, i.e. Marden’s book, either editions, mentions the formula for the cross-ratio in terms of the complex distance. I have no doubt it is somewhere in the literature, though. If someone knows of some references containing such formulas, then please email me to inform me about them.