Timeline for Conformal structure determined by principal curvatures
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 18, 2011 at 8:05 | answer | added | Dan Fox | timeline score: 3 | |
| Apr 18, 2011 at 3:25 | comment | added | Deane Yang | It is worth noting that everything works equally well in higher dimensions. The second fundamental form of a strongly convex hypersurface in Euclidean space determines a second Riemannian metric on the hypersurface, in addition to the one induced by the Euclidean inner product. If you forget the Euclidean structure and use only the affine structure of the ambient space, then the second fundamental form is still well-defined up to a scalar factor and you get a conformal structure. Both of these have been studied extensively. The latter is used to derive local invariants of a Finsler manifold. | |
| Apr 17, 2011 at 13:28 | vote | accept | marc | ||
| Apr 17, 2011 at 12:32 | answer | added | Bill Thurston | timeline score: 8 | |
| Apr 17, 2011 at 1:04 | comment | added | Ramsay | This metric is sometimes used in constructing an approximating inscribed polyhedron for the surface. There are some optimality results, e.g. K. Clarkson (2006) "Building triangulations using epsilon nets" and the works of P. M. Gruber cited therin. | |
| Apr 16, 2011 at 21:04 | comment | added | Anton Petrunin | In other words you want to use second fundamental form as a metric tensor and see what happens. I do not know nice geometric meaning for the obtained space and would be surprised if there is a nice one. For conformal structure --- there is only one conformal structrure on the sphere... | |
| Apr 16, 2011 at 19:59 | comment | added | Will Jagy | What do you do at umbilic points? | |
| Apr 16, 2011 at 18:39 | history | edited | marc | CC BY-SA 3.0 |
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| Apr 16, 2011 at 18:06 | history | asked | marc | CC BY-SA 3.0 |