Timeline for What is the least dimension of the Euclidean space into which every Riemannian flat n-torus embeds isometrically?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 8, 2023 at 17:40 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Added C^∞ condition
|
| Mar 24, 2022 at 20:18 | comment | added | Daniel Asimov | I just noticed that the paper by C. Tompkins, "Isometric embedding of flat manifolds in Euclidean space" (Duke Math. J., v. 5, 1939) contains a proof that for any compact Riemannian flat n-manifold M, its isometric embedding dimension e(M) satisfies the inequality e(M) ≥ 2n. Hence f(n) ≥ 2n. | |
| Mar 23, 2022 at 20:17 | answer | added | Robert Bryant | timeline score: 9 | |
| Mar 23, 2022 at 17:37 | comment | added | Daniel Asimov | Actually, since the n-torus is diffeomorphic to a submanifold of (n+1)-space, Nash-Kuiper shows that all Riemannian n-tori (flat or not) isometrically embed continuously differentiably in (n+1)-space. So it's really the case of smooth embeddings that is of interest. | |
| Mar 23, 2022 at 0:44 | comment | added | Daniel Asimov | Ben McKay — I was thinking only of smooth isometric embeddings, but knowing about any isometric embeddings would be interesting. | |
| Mar 22, 2022 at 22:04 | comment | added | Robert Bryant | A parameter count suggests that $f(n)\le n(n{+}1)$, and it's not hard to verify this for $n=1$ and $n=2$. This bound is probably not sharp, though. | |
| Mar 22, 2022 at 21:31 | comment | added | Ryan Budney | I suspect there is not much research on either problem, but I would be delighted to be wrong. | |
| Mar 22, 2022 at 20:28 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Math Jaxed
|
| Mar 22, 2022 at 18:54 | comment | added | Ben McKay | Smooth isometric embedding, or just isometry of metric spaces? | |
| Mar 22, 2022 at 18:30 | history | asked | Daniel Asimov | CC BY-SA 4.0 |