Timeline for Nash embedding for 3 manifolds
Current License: CC BY-SA 4.0
15 events
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| Dec 21, 2021 at 23:52 | comment | added | Joseph O'Rourke | Deane references Gromov's book, and I followed up, in Nash embedding theorem for 2D manifolds. See also Bill Thurston's insightful posting. All this: 2D, not 3D, manifolds. | |
| Dec 21, 2021 at 22:33 | comment | added | Deane Yang | @mme, so the question is whether there is a better but not necessarily optimal bound for nice $3$-manifolds? To be honest, I haven't looked carefully at this in a long time, so I'm not necessarily more expert. My memory is only that Gromov improved the bounds on the dimension and it might be worth looking at his book. | |
| Dec 21, 2021 at 22:25 | history | edited | Ian Gershon Teixeira | CC BY-SA 4.0 |
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| Dec 21, 2021 at 22:23 | comment | added | mme | @Deane With better dimension bounds by a factor of about 1/2, I thought, which is what I was trying to communicate: OP's bound is R^{27}. Obviously you are more expert than I; I pulled this statement and the attribution to Gunther from Thm 1.0.3 of "Isometric Embedding of Riemannian Manifolds in Euclidean Spaces". | |
| Dec 21, 2021 at 22:22 | comment | added | Ian Gershon Teixeira | Is there a compact 3 manifold that saturates the bound $ R^{14} $ (for example round projective plane saturates the bound $ R^5 $ for 2 manifolds)? Or is it unclear whether or not the bound is tight? Are there certain 3 manifolds that we can definitively say cannot be isometrically embedded in $ R^7 $? I'm looking for bound information like that.// I'll update the bound in the question thanks @mme | |
| Dec 21, 2021 at 22:16 | comment | added | Deane Yang | I meant an optimal bound. Nash'stheorem shows that any Riemannian manifold can be isometrically embedded into Euclidean space of high enough dimension. Gunther gave a simpler proof. | |
| Dec 21, 2021 at 22:13 | comment | added | mme | @Deane Surely you mean "is locally..." // OP: Gunther's version of Nash embedding gives you an isometric embedding of a compact 3-manifold into R^14. | |
| Dec 21, 2021 at 22:07 | comment | added | Deane Yang | A $3$-manifold with constant sectional or Ricci curvature is flat $3$-space, a sphere, or hyperbolic space. You know the answer for the first two. I don't believe the answer is known even for hyperbolic $3$--space. The answer is unknown even for hyperbolic $2$-space. math.stackexchange.com/questions/1528046/… | |
| Dec 21, 2021 at 20:39 | comment | added | Ian Gershon Teixeira | @DeaneYang It occurred to me the question might be too hard. Do you think I might get more useful replies if I specialize the question to really nice manifolds, for example constant curvature? | |
| Dec 21, 2021 at 19:51 | comment | added | Deane Yang | This is a very difficult question answer, even for 2-manifolds and 3-manifolds. There are some partial results in Gromov's book, Partial Differential Relations. | |
| Dec 21, 2021 at 18:36 | history | edited | Ian Gershon Teixeira |
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| Dec 21, 2021 at 18:28 | history | edited | Ian Gershon Teixeira |
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| Dec 21, 2021 at 12:35 | history | edited | YCor | CC BY-SA 4.0 |
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| Dec 21, 2021 at 12:25 | history | edited | Ian Gershon Teixeira |
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| Dec 18, 2021 at 14:29 | history | asked | Ian Gershon Teixeira | CC BY-SA 4.0 |