Timeline for Understanding $\kappa$-cones
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| S Aug 2, 2023 at 22:02 | history | bounty ended | CommunityBot | ||
| S Aug 2, 2023 at 22:02 | history | notice removed | CommunityBot | ||
| S Jul 25, 2023 at 20:28 | history | bounty started | Justin_other_PhD | ||
| S Jul 25, 2023 at 20:28 | history | notice added | Justin_other_PhD | Authoritative reference needed | |
| Jul 18, 2023 at 18:58 | comment | added | Justin_other_PhD | @HenrikRüping Thanks a lot for the explanation of (1). For (2) by p product I mean $d_{X\times Y}((x_1,y_1),(x_2,y_2))^p:= d_X(x_1,x_2)^p + d_Y(y_1,y_2)^p$ (for $1\le p<\infty$ and when $p=\infty$ one takes max). So $p=2$ is the usual product metric. | |
| Jul 18, 2023 at 14:21 | comment | added | HenrikRüping | It turns out that this contruction makes sense, even if we start with more general spaces than Riemannian spheres. Usually taking $C_\kappa$ increeases the topological dimension by one, so I would not expect the first equation to hold in any sense. I dont know what a $p$-product is, so I cannot say anything about the second question. | |
| Jul 18, 2023 at 14:19 | comment | added | HenrikRüping | But if we knew additionally the curvature $\kappa$, there is such a construction, namely $C_\kappa$. | |
| Jul 18, 2023 at 14:18 | comment | added | HenrikRüping | I think the idea is that in all three model spaces spheres around points are isometric up to rescaling to the standard sphere. One way to see this is that the three spaces are the only simply connected Riemannian manifolds, where the isometry group acts transitively on the frame bundle. But this property passes to spheres around points. Now it would be nice, if we had a way to reconstruct the balls from the spheres. As we have already seen without knowing in which of the three spaces we are, this is impossbile (the spheres are in all 3 cases isometric). | |
| Jul 17, 2023 at 19:07 | history | edited | YCor | CC BY-SA 4.0 |
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| Jul 17, 2023 at 14:49 | history | asked | Justin_other_PhD | CC BY-SA 4.0 |