Timeline for Intrinsic vs Extrinsic geometry of convex surfaces
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 20, 2017 at 11:34 | comment | added | Robert Bryant | @MikhailKatz: Yes, I do know the answer. I'll enter it there. | |
| Oct 19, 2017 at 11:04 | comment | added | Mikhail Katz | Hi @Robert, would you know the answer to this (in the noncomplete case)? | |
| Oct 16, 2017 at 11:18 | comment | added | Mohammad Ghomi | OK, fair enough. I agree that if we look at it this way, then one can establish the existence of at least one umbilic in a purely intrinsic way. | |
| Oct 16, 2017 at 11:09 | comment | added | Robert Bryant | But isn't this really a matter of semantics? The umbilics are the points where the quadratic form $Q$ that satisfies the Gauss and Codazzi equations (which are 'intrinsic' since they depend only on the metric $g$, interpreted as the first fundamental form) is a multiple of $g$. (Globally on $S^2$, $Q$ is unique up to sign when the Gauss curvature of $g$ is positive.) You don't have to call the eigenvectors of $Q$ relative to $g$ 'principal directions' if you don't want to. Since there are many local solutions to the Gauss and Codazzi equations, there is no local argument forcing umbilics. | |
| Oct 16, 2017 at 1:25 | comment | added | Mohammad Ghomi | Yes, I referenced the topological argument above, but I guess that I was not including that as "intrinsic" because it refers to principal directions which are not intrinsic. The hope (and I understand that it could be a pretty long shot) would be to recognize the umbilic in terms of the metric. | |
| Oct 16, 2017 at 1:19 | history | answered | Robert Bryant | CC BY-SA 3.0 |