Timeline for On triangle comparison in Riemannian manifolds with upper sectional curvature bound
Current License: CC BY-SA 3.0
14 events
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| Mar 23, 2018 at 11:50 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Feb 21, 2018 at 10:54 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jan 22, 2018 at 9:59 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Dec 23, 2017 at 9:19 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Nov 23, 2017 at 8:38 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Oct 24, 2017 at 7:27 | comment | added | Changyu Guo | @valeri: Thanks very much for the interesting example. | |
| Oct 24, 2017 at 7:01 | comment | added | valeri | to Changyu Guo. Actually, I can modify previous arguments to get counterexample; simply connected compact surface where inequality does not hold. Take thin cylinder instead of torus and glue to its ends first cone-like surfaces (of nonpositive curvature) smoothly connecting end-circles to much bigger circles on unit spheres. We obtaine surface with $k=1$, but $d(Q_t,R_t)\le c(k,t)d(Q,R)$ does not hold for any bounded $c(k,t)$, | |
| Oct 24, 2017 at 0:31 | comment | added | Anton Petrunin | Since all triangles are thin, the sphere is the worse case. On the sphere (as well as on the plane), your inequality does not hold --- so try to correct your question. | |
| Oct 23, 2017 at 21:07 | comment | added | valeri | to Changyu Guo. Then you might need also that the closed curve $PQR$ to bound some disk (be null homotopic), otherwise it might be wrong. Take very thin torus where $PQR$ is the meridian, where $Q, R$ are very close and almost opposite to $P$. The quotient $d(Q_t,R_t)/ d(Q,R)$ for a fixed $t=1/2$ may be unbouded when $d(Q,R)$ goes to zero. | |
| Oct 23, 2017 at 18:26 | answer | added | Raziel | timeline score: 1 | |
| Oct 23, 2017 at 17:48 | comment | added | Changyu Guo | To valeri: Thanks. You are absolutely correct. | |
| Oct 23, 2017 at 17:47 | history | edited | Changyu Guo | CC BY-SA 3.0 |
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| Oct 23, 2017 at 13:33 | comment | added | valeri | is it $d(Q_t,R_t)\le c(k,t)d(Q,R)$? | |
| Oct 23, 2017 at 12:09 | history | asked | Changyu Guo | CC BY-SA 3.0 |