Timeline for What is the best way to draw curvature?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 5 at 1:43 | comment | added | Mozibur Ullah | @BenMcKay: And finally, drawing means we are describing extrinsically. | |
| Aug 5 at 1:42 | comment | added | Mozibur Ullah | @Ben McKay: Plus the OP says "but I would like to find good ways to visually depict the notion of curvature". There is no mention of intrinsic here. | |
| Aug 5 at 1:39 | comment | added | Mozibur Ullah | @Ben McKay: You can always embed a Riemann manifold in flat space. | |
| Nov 20, 2020 at 8:13 | comment | added | Sebastian | @Ben McKay: It is not quite true for mean curvature: taking orientation into account, there exists for all $H\in\mathbb R$ oriented spheres with mean curvature $H$, where we take the plane for $H=0$. In fact, this leads to the mean curvature sphere congruence of a immersed surface in 3-space, which plays an important role in the investigation of Willmore surfaces. But of course, this concerns the extrinsic geometry of the surface and not the intrinsic geometry. | |
| Nov 19, 2020 at 16:58 | comment | added | Ben McKay | Approximation by ellipsoids will not give you the Gauss or the mean curvature, as there is no approximating ellipsoid with the correct values for those, on some surfaces. | |
| Nov 19, 2020 at 15:41 | comment | added | Gabe K | I don't see how gauge theory and (co)homology relate to extrinsic curvature. However, you are correct that there are many situations that extrinsic geometry is important. Nonetheless, the reason I was asking about intrinsic curvature is that it's much more difficult to visualize than extrinsic geometry. For instance, it's not hard to show that tori admit a flat metric, but most students' image of a donut is not flat (and you can't embed a flat donut smoothly in ℝ3). | |
| Nov 19, 2020 at 15:37 | comment | added | user44143 | The book is not by Hilbert alone! | |
| Nov 19, 2020 at 15:13 | history | edited | Mozibur Ullah | CC BY-SA 4.0 |
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| Nov 19, 2020 at 14:33 | history | edited | Mozibur Ullah | CC BY-SA 4.0 |
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| Nov 19, 2020 at 7:54 | comment | added | Ben McKay | The problem is really about curvature of Riemannian manifolds, not of the embedding of an embedded submanifold in Euclidean space, which is really very different. | |
| Nov 19, 2020 at 7:08 | history | answered | Mozibur Ullah | CC BY-SA 4.0 |