Timeline for Umbilic points on Euclidean hypersurfaces
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 26, 2016 at 18:51 | vote | accept | User0.9999999..... | ||
| Jul 25, 2016 at 12:35 | answer | added | Robert Bryant | timeline score: 4 | |
| Feb 28, 2016 at 16:39 | comment | added | Robert Bryant | Yes, it's too much to ask that all the curvatures be equal. When $n>2$, the general ellipsoid in $\mathbb{R}^{n+1}$ will have no points where all the principal curvatures are equal. (Just do the computation for $n=3$ and you will see why.) Only for $n=1$, $3$, or $7$ is there any chance of having an $n$-sphere (convex or not) in $\mathbb{R}^{n+1}$ with $n$ distinct principal curvatures at each point. (For $n=1$, this is trivial, of course.) There are examples (not convex and not embedded) of $3$-spheres immersed in $\mathbb{R}^4$ that have $3$ distinct principal curvatures at each point. | |
| Feb 26, 2016 at 0:29 | comment | added | User0.9999999..... | I'm looking for points where all curvatures are equal. I guess that is too much to ask? | |
| Feb 25, 2016 at 16:22 | comment | added | Robert Bryant | In higher dimensions, do you want to define 'umbilic point' to be a point where two of the eigenvalues are equal or a point where all of the eigenvalues are equal? | |
| Feb 24, 2016 at 22:42 | history | asked | User0.9999999..... | CC BY-SA 3.0 |