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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 11G / c / Solution

In the Klein fundamental square from part (b), the following straight lines project to the required closed geodesics:

γ_{1} (t) = [t, 0], 0 \leq t \leq 1;

this is the one-sided horizontal core, and cutting along it leaves a Möbius strip.

Take

γ_{2} (t) = [0, t - 1/2], 0 \leq t \leq 1;

this is a two-sided vertical geodesic, and cutting along it leaves a cylinder.

Finally take

γ_{3} (t) = [2 t, t], 0 \leq t \leq 1.

Its endpoints differ by the deck translation

(2, 1)

, so it is closed. The parameters

t = 1/4

and

t = 3/4

represent the same Klein-bottle point through an odd-

c

glide reflection, and no other interior pair does. Thus its image has exactly one transverse self-intersection. This is the flat Klein-bottle geodesic model.

Solved by gpt-5.6-sol high.

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