Skip to main content
Added summary of some comments
Source Link
Daniel Asimov
  • 2.9k
  • 24
  • 26

Here a hexagonal 2-torus TH means any Riemannian flat torus obtained by identifying opposite edges of a regular hexagon in the plane.

It's easy to see that TH can be smoothly (C) isometrically embedded in ℝ6, and for that matter (if suitably scaled) in the unit sphere S5, and also in ℂℙ2.

But is it known whether TH can be smoothly isometrically embedded in ℝ5 or ℝ4 ?

(Also of interest is whether TH can be smoothly isometrically immersed in ℝ5 or ℝ4.)

Edit: Thanks to the Gromov theorem mentioned by Michael Albanese, we know that TH isometrically embeds smoothly in ℝ5, so the question remaining is whether it embeds in ℝ4. Pace Ian Agol's comment, I am waiting to confirm whether TH actually has a smooth isometric embedding in S^3 (up to scaling) ... and hence in ℝ4.

Here a hexagonal 2-torus TH means any Riemannian flat torus obtained by identifying opposite edges of a regular hexagon in the plane.

It's easy to see that TH can be smoothly (C) isometrically embedded in ℝ6, and for that matter (if suitably scaled) in the unit sphere S5, and also in ℂℙ2.

But is it known whether TH can be smoothly isometrically embedded in ℝ5 or ℝ4 ?

(Also of interest is whether TH can be smoothly isometrically immersed in ℝ5 or ℝ4.)

Here a hexagonal 2-torus TH means any Riemannian flat torus obtained by identifying opposite edges of a regular hexagon in the plane.

It's easy to see that TH can be smoothly (C) isometrically embedded in ℝ6, and for that matter (if suitably scaled) in the unit sphere S5, and also in ℂℙ2.

But is it known whether TH can be smoothly isometrically embedded in ℝ5 or ℝ4 ?

(Also of interest is whether TH can be smoothly isometrically immersed in ℝ5 or ℝ4.)

Edit: Thanks to the Gromov theorem mentioned by Michael Albanese, we know that TH isometrically embeds smoothly in ℝ5, so the question remaining is whether it embeds in ℝ4. Pace Ian Agol's comment, I am waiting to confirm whether TH actually has a smooth isometric embedding in S^3 (up to scaling) ... and hence in ℝ4.

appropriately —> suitably
Source Link
Daniel Asimov
  • 2.9k
  • 24
  • 26

Here a hexagonal 2-torus TH means any Riemannian flat torus obtained by identifying opposite edges of a regular hexagon in the plane.

It's easy to see that TH can be smoothly (C) isometrically embedded in ℝ6, and for that matter (if appropriatelysuitably scaled) in the unit sphere S5, and also in ℂℙ2.

But is it known whether TH can be smoothly isometrically embedded in ℝ5 or ℝ4 ?

(Also of interest is whether TH can be smoothly isometrically immersed in ℝ5 or ℝ4.)

Here a hexagonal 2-torus TH means any Riemannian flat torus obtained by identifying opposite edges of a regular hexagon in the plane.

It's easy to see that TH can be smoothly (C) isometrically embedded in ℝ6, and for that matter (if appropriately scaled) in the unit sphere S5, and also in ℂℙ2.

But is it known whether TH can be smoothly isometrically embedded in ℝ5 or ℝ4 ?

(Also of interest is whether TH can be smoothly isometrically immersed in ℝ5 or ℝ4.)

Here a hexagonal 2-torus TH means any Riemannian flat torus obtained by identifying opposite edges of a regular hexagon in the plane.

It's easy to see that TH can be smoothly (C) isometrically embedded in ℝ6, and for that matter (if suitably scaled) in the unit sphere S5, and also in ℂℙ2.

But is it known whether TH can be smoothly isometrically embedded in ℝ5 or ℝ4 ?

(Also of interest is whether TH can be smoothly isometrically immersed in ℝ5 or ℝ4.)

Source Link
Daniel Asimov
  • 2.9k
  • 24
  • 26

What's the lowest-dimensional Euclidean space in which a hexagonal 2-torus smoothly embeds isometrically?

Here a hexagonal 2-torus TH means any Riemannian flat torus obtained by identifying opposite edges of a regular hexagon in the plane.

It's easy to see that TH can be smoothly (C) isometrically embedded in ℝ6, and for that matter (if appropriately scaled) in the unit sphere S5, and also in ℂℙ2.

But is it known whether TH can be smoothly isometrically embedded in ℝ5 or ℝ4 ?

(Also of interest is whether TH can be smoothly isometrically immersed in ℝ5 or ℝ4.)