Timeline for Why 2-tori with Gauss curvature $\geq -1$ cannot collapse to segment?
Current License: CC BY-SA 4.0
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| Aug 21, 2019 at 20:14 | comment | added | John Harvey | Essentially it's Theorem 0.5 of Shioya-Yamaguchi, but in lower dimension. Working through that argument will show how it works in dim 2. To expand a little, a non-compact non-negatively curved 2-dimensional limit space will deformation retract onto a 'soul', which is a point or a circle. In the former case the space is a disk, while in the latter it is a line bundle over the circle. The trivial line bundle, though, has two ends, so it is the Möbius strip. However, that is not orientable. Therefore the space is $D^2 \cup (S^1 \times I) \cup D^2$, i.e. it is $S^2$. Hope that helps. | |
| Aug 19, 2019 at 9:55 | comment | added | asv | Thank you. This is a little bit too concise for me, particularly the last two sentences of the first approach. | |
| Aug 18, 2019 at 13:38 | history | answered | John Harvey | CC BY-SA 4.0 |