Timeline for Curvature of curves through a point of a surface smoothly embedded in Euclidean space
Current License: CC BY-SA 4.0
11 events
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| Oct 10, 2023 at 4:09 | 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. | |
| Sep 10, 2023 at 3:23 | answer | added | Anton Petrunin | timeline score: 1 | |
| Feb 23, 2023 at 18:04 | comment | added | Daniel Asimov | Plus everything is local, so orientation is not an issue. | |
| Feb 23, 2023 at 5:23 | comment | added | Daniel Asimov | At any given point P of the surface 𝜮: What I call 𝜃 is always well-defined (relative to a choice of reference angle). After all, the surface is embedded, so it has a well-defined 2-plane T in n-space as its tangent plane, as well as a well-defined normal (n-2)-space N. So the sum of N and any tangent line to 𝜮 at P is a hyperplane perpendicular to T, intersecting it in a line. The angle between any two such lines is well-defined, and there are 2π worth of tangent rays. | |
| Feb 22, 2023 at 5:19 | comment | added | Willie Wong | For that matter, the principal extrinsic curvatures are also not always well-defined. The same problem should still persist in the holomorphic setting. | |
| Feb 22, 2023 at 5:12 | comment | added | Willie Wong | Since hyperplanes are totally geodesic: if you denote by $II_p(v,w)$ the vector valued second fundamental form of $\Sigma$ at the point $p$, and $v$ the tangent vector to $C$, then the curvature vector (the acceleration of $C$ with unit-speed parametrization) is exactly $II_p(v,v)$. You don't always as nice a formula as $k_1\cos^2(\theta) + k_2\sin^2(\theta)$, since the different (vector) components of $k$ may not be simultaneously diagonalizable (so what you call $\theta$ may be not well-defined). | |
| Feb 22, 2023 at 1:19 | comment | added | Daniel Asimov | I certainly expect the answer is essentially some symmetric combination of second derivatives. | |
| Feb 22, 2023 at 0:59 | comment | added | alesia | You might want to look at the second fundamental form, which becomes vector valued in codimension more than 1 | |
| Feb 22, 2023 at 0:22 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
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| Feb 22, 2023 at 0:00 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
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| Feb 21, 2023 at 23:26 | history | asked | Daniel Asimov | CC BY-SA 4.0 |