Hyperbolic Geometry - Lecture 1

Vocabulary and core definitions from the first lecture on Hyperbolic Geometry, focusing on the hyperbolic metric in the upper halfplane.

Hyperbolic plane (H)

The metric space $(H, \rho)$, where $H = \{C \mid \text{Im}(z) > 0\}$ is the upper open halfplane of complex numbers.

C¹-curve in H

A $C^1$-map $z : [0, 1] \rightarrow H$ written in the form $z(t) = x(t) + iy(t)$, where $x$ and $y$ are real functions.

Euclidean length ($l(z)$)

Calculated as $l(z) := \int_{0}^{1} |z'(t)| dt$.

Hyperbolic length ($l_h(z)$)

The value defined by

$l_h(z) := \int_{0}^{1} \frac{|z'(t)|}{\text{Im}(z(t))} dt = \int_{0}^{1} \frac{\sqrt{(x'(t))^2 + (y'(t))^2}}{y(t)} dt$.

Hyperbolic distance ($\text{dist}_h = \rho(z_1, z_2)$)

The infimum $\inf l_h(z)$ taken over all $C^1$-curves $z$ in $H$ such that $z(0) = z_1$ and $z(1) = z_2$.

Theorem $\rho$ is a metric space

States that the function $\rho : H \times H \rightarrow \mathbb{R}$ is a metric on $H$, meaning it satisfies: (1) $\rho(z_1, z_1) = 0$ and \rho(z_1, z_2) > 0 if $z_1 \neq z_2$, (2) $\rho(z_1, z_2) = \rho(z_2, z_1)$, and (3) $\rho(z_1, z_3) \leq \rho(z_1, z_2) + \rho(z_2, z_3)$.