 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4/25i/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

1

and major radius

R_{n} = n + 2

. With angular coordinate

θ

around the generating circle, its Gaussian curvature is

K_{n} (θ) = \frac{c o s θ}{R _{n} + c o s θ},

and hence

in f_{S_{n}} K_{n} = - \frac{1}{R _{n} - 1} ⟶ 0.

Thus the displayed $lim sup$ condition holds, while every $S_{n}$ has genus one and is not diffeomorphic to the sphere.

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