Timeline for Is higher order mean curvature extrinsic or intrinsic
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 28, 2017 at 13:06 | comment | added | Mohammed Mohammed | When r is even, if we change the direction of the unit normal vector, the Hr change the sign, this is because we have an even product of principal curvature in their definition but in the cas of r odd Hr don't change the sign. this is true I know that instrinsic quantities depend only of the first fundamental form but I see that Hr depend on principal curvatures so the second fundamental form. this is what make aa doubt to my conjecture | |
| Oct 28, 2017 at 8:45 | answer | added | Anthony Carapetis | timeline score: 5 | |
| Oct 28, 2017 at 5:05 | comment | added | Anthony Carapetis | From the clues of mean and Gaussian curvature I'm guessing $H_r$ is the $r^\text{th}$ elementary symmetric polynomial of the principal curvatures? | |
| Oct 28, 2017 at 2:23 | comment | added | Deane Yang | It is known that the only intrinsic local differential invariants of a Riemannian manifold are the metric, the curvature tensor, and its covariant derivatives. For a hypersurface, the Gauss equations say that the latter are all quadratic functions of the second fundamental form and its covariant derivatives, obtained by differentiating the Gauss equations. More invariants can be obtained by contracting indices using the metric tensor. I do not know what you mean by "higher order mean curvature", but it would have to fit within my explanation above to be intrinsic. | |
| Oct 27, 2017 at 23:37 | history | asked | Mohammed Mohammed | CC BY-SA 3.0 |