Here a hexagonal 2-torus T<sub>H</sub> means any Riemannian flat torus obtained by identifying opposite edges of a regular hexagon in the plane.

It's easy to see that T<sub>H</sub> can be smoothly (C<sup>∞</sup>) isometrically embedded in ℝ<sup>6</sup>, and for that matter (if appropriately scaled) in the unit sphere S<sup>5</sup>, and also in ℂℙ<sup>2</sup>.

But is it known whether T<sub>H</sub> can be smoothly isometrically embedded in ℝ<sup>5</sup> or ℝ<sup>4</sup> ?

(Also of interest is whether T<sub>H</sub> can be smoothly isometrically immersed in ℝ<sup>5</sup> or ℝ<sup>4</sup>.)