Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 25I / c

Table of contents
- Solution c

Solution

Take the ring torus with minor radius

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.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version