Timeline for Canonical immersion of the double torus
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jul 30, 2020 at 22:37 | comment | added | M. Winter | The sphere and the torus are highly symmetric in a way that higher-genus surfaces are not. Your 4-dimensional embedding of the torus (and the standard embedding of the sphere) are special in the sense that they realize all the symmetries of these surfaces. Since these surfaces have many symmetries, such embeddings are "rare". But, for example, the double torus has not many symmetries and so there are many somehow good embeddings, but none of these stand out as especially symmetric or favorable. | |
| Jul 30, 2020 at 15:32 | history | edited | YCor | CC BY-SA 4.0 |
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| Jul 30, 2020 at 14:42 | comment | added | Mauro Patrão | A related question is: is there any explicity (maybe algebraic) isometric embedding of a genus 2 surface endowed with a metric of curvature K = -1? | |
| Dec 20, 2016 at 13:00 | comment | added | Ben McKay | @AntonPetrunin: this is the Jacobian mapping from algebraic geometry, I think. | |
| Dec 20, 2016 at 12:55 | answer | added | Robert Bryant | timeline score: 6 | |
| Jan 21, 2015 at 18:59 | comment | added | Anton Petrunin | Each harmonic form gives you a map to $\mathbb S^1$; you can take a basis for harmonic forms and map your surface in an $n$-torus, the later can be embedded into $\mathbb R^{2{\cdot}n}$ if you want. This embedding is not isometric, but it is kind of canonical; so maybe something can be build on it. | |
| Jan 21, 2015 at 12:04 | comment | added | Joseph O'Rourke | I wonder if first embedding a lemniscate and then "fattening" it might lead to a clean embedding? | |
| Jan 21, 2015 at 11:36 | history | asked | Jjm | CC BY-SA 3.0 |