27 Nov
2020
27 Nov
'20
11:25 a.m.
Yup, that's exactly it. A Klein bottle can be seen as a cylinder S^1 x [0,1] after the ends S^1 x {0} and S^1 x {1} have been identified by e.g. the map (cos(t), sin(t)) —> (cos(t), -sin(t)), and this is just what Allan's Klein bottle does. (More generally, the family of great circles on S^2 could rotate any odd multiple of 180º before returning to where it started, and their totality would be a Klein bottle.) —Dan
On Friday/27November/2020, at 10:15 AM, Allan Wechsler <acwacw@gmail.com> wrote:
For #1, take a family of great circles on S^2 that all intersect in the same two poles, like the meridional lines on a globe. Have the circle rotate 180 degrees, while its partner, a single point, goes all the way around S^1.