Timeline for Bisector of two points in a Riemannian manifold has measure $0$
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 29, 2023 at 17:44 | vote | accept | Saúl RM | ||
| Jan 4, 2023 at 0:05 | answer | added | Saúl RM | timeline score: 1 | |
| Jan 3, 2023 at 23:15 | comment | added | Saúl RM | @Neal As a length space, that is, $d(x,y)$ is the infimum of lengths of curves from $x$ to $y$. To be honest I didn't know if there was other "natural metric" we could give to a Riemannian manifold | |
| Jan 3, 2023 at 17:00 | comment | added | Neal | Here $d(x,y)$ is distance in the sense of a length space, or in the sense of metric space? One can imagine there's an open subset of a manifold wild enough that the bisector in the length-space sense might attain positive measure. | |
| Jan 3, 2023 at 16:41 | answer | added | Leo Moos | timeline score: 3 | |
| Jan 3, 2023 at 15:31 | comment | added | Saúl RM | I see. Thanks, if I see how to make something like that work I will add it. To be honest I am more interested in a reference/short proof when $M$ is closed (I asked the question more generally because it is probably still true). Maybe if I find nothing after some time I will try to write more in detail the proof sketch that I mention in the question | |
| Jan 3, 2023 at 15:18 | comment | added | YCor | I don't have a full argument in mind. The idea would be to arranges things so that distances outside the given ball are large enough to avoid distort the distance in the smaller ball. | |
| Jan 3, 2023 at 13:47 | comment | added | Saúl RM | @YCor How can we make sure that the closed manifold contains an isometric copy of $U$? (I see how to make the Riemannian metric coincide with that of $U$, but not the distance) | |
| Jan 3, 2023 at 13:31 | comment | added | YCor | It looks as a quite local problem and the general case should follow from the closed case given $p,q$ and $r\in B(p,q)$, choose open subsets $U\subset V$ diffeomorphic to $\mathbf{R}^d$ containing $p,q,r$ with $U$ included in a compact subset of $V$, and use it to construct a closed manifold (diffeomorphic to the $d$-sphere) that contains an isometric copy of $V$. | |
| Jan 3, 2023 at 13:27 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals
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| Jan 3, 2023 at 13:22 | history | asked | Saúl RM | CC BY-SA 4.0 |