 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ib/paper-1/3e/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

D = {z \in C : ∣ z ∣ < 1}, d s^{2} = \frac{4∣ d z ∣ ^{2}}{( 1 - ∣ z ∣ ^{2} ) ^{2}} .

- 1

For

a \in D

and

∣ η ∣ = 1

T_{a, η} (z) = η \frac{z - a}{1 - a z}

is an isometry of the disc, and every orientation-preserving disc isometry has this form. In particular,

T_{z_{1}, 1}

sends

z_{1}

to zero and

z_{2}

to a point of modulus

ρ = \frac{z _{2} - z _{1}}{1 - z _{1} z _{2}} .

Rotations are also isometries, so it remains only to integrate the metric along a radial geodesic from

0

ρ

d (0, ρ) = \int_{0}^{ρ} \frac{2 d r}{1 - r ^{2}} = 2 artanh ρ .

d (z_{1}, z_{2}) = 2 artanh \frac{z _{2} - z _{1}}{1 - z _{1} z _{2}} = lo g \frac{1 + ρ}{1 - ρ} .

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