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
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| Oct 8, 2023 at 17:29 | comment | added | Daniel Asimov | I hadn't realized that rescaling and passing to a cover could get from one flat n-torus to (arbitrarily close to) any flat n-torus. I need to think about that. ... ... Okay, I think I see how. | |
| Oct 8, 2023 at 17:23 | comment | added | Anton Petrunin | One can do a bit better: $f(n)\leqslant\tfrac{n\cdot(n+3)}2$. In 2B of "Isometric Immersions with Controlled Curvatures", Gomov constructs a free isometric embedding of a flat $n$-torus in the $\tfrac{n\cdot(n+3)}2$-dimensional Euclidean space. By rescaling this example and passing to a cover, one can get a flat torus arbitrarily close to a given one. Now, by applying Nash's deformation, one gets an embedding of any flat torus. | |
| Mar 23, 2022 at 20:17 | history | answered | Robert Bryant | CC BY-SA 4.0 |