Timeline for Explicit triples of isomorphic Riemann surfaces
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 3, 2022 at 13:55 | history | made wiki | Post Made Community Wiki by Asaf Karagila♦ | ||
| May 3, 2022 at 12:58 | comment | added | Sam Nead | The symmetries of the embedded Riemann surface need not (and usually will not) be induced by isometries of the ambient three-manifold. This happens, for example, for the hexagonal Hopf torus embedded in the three-sphere. | |
| May 3, 2022 at 12:39 | comment | added | Francesco Polizzi | I am confused since the automorphism group of the Klein quartic is not a subgroup of $\operatorname{SO}(3)$, so this conformal model in $\mathbb{R}^3$ cannot have a rotational symmetry... | |
| May 3, 2022 at 12:33 | comment | added | Sam Nead | Every Riemann surface admits a conformal embedding into $\mathbb{R}^3$. See mathoverflow.net/questions/53999 for references and a discussion. However, those "constructions" are not explicit (and I would argue, not pretty!). | |
| May 3, 2022 at 12:30 | comment | added | Francesco Polizzi | Ok, I understand what you mean. In fact, there is no smooth embedding of the Klein quartic in $\mathbb{R}^3$, so these combinatorial models are the best one can achieve, in a sense. | |
| May 3, 2022 at 12:27 | comment | added | Sam Nead | I did edit the question, but not in that way... Greg Egan's embedding is not smooth, and it is not conformally correct. It does show the combinatorics of the KQ, but it is not "isomorphic" as a Riemann surface to the KQ. The remarks hold for Helaman Ferguson's sculpture. Don't get me wrong: both are useful and lovely - they are just not examples of | |
| May 3, 2022 at 12:19 | comment | added | Francesco Polizzi | Did you edit the question? In a previous version, you said that for (1) you also accept a visualization in $\mathbb{R}^3$ (which is provided by the Wikipedia article linked). | |
| May 3, 2022 at 12:17 | comment | added | Sam Nead | Where is the conformally correct embedding into $S^3$ given? | |
| May 3, 2022 at 12:05 | history | answered | Francesco Polizzi | CC BY-SA 4.0 |