A comment on full discretized isothermic tori in Euclidean spaces

Abstract

Using discretized orthogonal systems (curvature line systems) with periodicity, created using Darboux transformations and their permutability, we have discrete and semi-discrete k-dimensional isothermic tori which are full in n-dimensional Euclidean space, for any natural numbers k between 2 nd n.

Figures (6)

• Figure 1: Fully and semi discrete $2$-subnets of orthogonal nets.
• Figure 2: Semi-discrete isothermic $2$-tori with coarse meshes consisting of circles in the smooth direction in $\mathbb{R}^3$, with polarizing function $m=1$. On the left, the thickest-drawn curve is the original circle $x$, the two adjacent circles are Darboux transforms with spectral parameter $\mu=3$ for the upper one and $\mu=8$ for the lower one, and the final bottom circle completes the Bianchi cube via permutability. A more complicated example still with smooth circles is shown on the right.
• Figure 3: Semi-discrete isothermic $2$-tori with coarse meshes in $\mathbb{R}^3$ whose smooth curves are not all circular and are obtained by Darboux transforms at resonance points.
• Figure 4: Three semi-discrete tori in $\mathbb{R}^3$ with finer meshes. Each uses a circle as inital curve, but the transformed curves are non-circular in the two right figures, using resonance points.
• Figure 5: Three semi-discrete tori in $\mathbb{R}^3$ with finer meshes. The left two figures use a circle as inital curve, with non-circular transformed curves via resonance points. The right figure starts from a figure-eight elastic curve and uses non-resonant spectral parameters to obtain non-trivial Darboux transforms.

Theorems & Definitions (9)

• Theorem 1.1
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Example 2.1
• Remark 2.4
• proof
• Remark 3.1
• Remark 3.2