Timeline for Embed the hyperbolic plane into Euclidean spaces
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jul 9, 2019 at 15:41 | comment | added | Willie Wong | @011000: there is no relation between the $C^1$ embedding (Nash-Kuiper) and the construction by Sabitov. Sabitov sets out an explicit ansatz of a generalized surface of revolution, and essentially just solved the equations by hand and checked the regularity by hand. | |
| Jul 9, 2019 at 15:11 | comment | added | Willie Wong | As of the very comprehensive 2001 survey of Borisenko both questions seem to be open. For Q1 if you replace "embed" by "immerse" then the result holds due to Rozendorn. (The original paper is in Russian (see Borisenko for reference) but the construction is outlined in Rozendorn's 1992 survey.) And Ian Agol already mentioned the result of Sabitov for Q2 which is only $C^{0,1}$ globally but piecewise analytic. | |
| Jul 9, 2019 at 7:33 | comment | added | quarague | The embedding of hyperbolic space in $\mathbb{R}^3$ with Minkowski metric is explained on the wikipedia page for hyperbolic space and also in most text books that introduce hyperbolic space. | |
| Jul 9, 2019 at 4:25 | comment | added | 011000 | I care more about the smooth embedding and immersion, so I edit the question. I know the Nash embedding theorem, but I never heard how to embed hyperbolic space in R^3 with the Minkowski metric. Could you please explain it more precisely? What's more, are there any relationship between the C^1 embedding in R^3 and the piecewise analytic immersion in R^4? Thanks for every helpful answers. | |
| Jul 9, 2019 at 4:08 | history | edited | 011000 | CC BY-SA 4.0 |
I ask the smooth embedding and immersion, not just C^1.
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| Jul 8, 2019 at 15:43 | comment | added | Ian Agol | A piecewise analytic map (but apparently not smooth) is given here: mathscinet.ams.org/mathscinet-getitem?mr=1025303 | |
| Jul 8, 2019 at 9:28 | comment | added | Moishe Kohan | See math.stackexchange.com/questions/1528046/… and references therein. | |
| Jul 8, 2019 at 8:51 | comment | added | quarague | I suppose you already know that one can embed hyperbolic space in $\mathbb{R}^3$ with the Minkowski metric and that the Nash embedding theorem gives an embedding for any surface into Euclidean space. The dimension bound on wikipedia gives just $n \le 51$ so that is pretty far off the $4$ or $5$ dimensions you are hoping for. | |
| Jul 8, 2019 at 4:15 | review | First posts | |||
| Jul 8, 2019 at 7:20 | |||||
| Jul 8, 2019 at 4:11 | history | asked | 011000 | CC BY-SA 4.0 |