 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-4/25f/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The geodesic curvature is the tangential component of

α''

, namely

k_{g} = ⟨ α'', N \times α' ⟩

k_{g} = 0

. Gauss-Bonnet theorem for a compact region says

\int K + \int_{\partial} k_{g} + \sum exterior angles = 2 π χ

For

K = 0

, a flat cylinder has infinitely many parallel closed geodesics.

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