# Semi-discrete isothermic surfaces

## Abstract

A Darboux transformation for polarized space curves is introduced and its properties are studied, in particular, Bianchi permutability. Semi-discrete isothermic surfaces are described as sequences of Darboux transforms of polarized curves in the conformal *n*-sphere and their transformation theory is studied. Semi-discrete surfaces of constant mean curvature are studied as an application of the transformation theory.

### Keywords

Isothermic surface Discrete isothermic net Calapso transformation Darboux transformation Bianchi permutability Lawson correspondence Bäcklund transformation Constant mean curvature### Mathematics Subject Classification 2010

53A10 53C42 53A30 37K25 37K35## 1 Introduction

Integrable discretizations of surfaces or submanifolds are intimately related to the transformations of the smooth theory—permutability theorems govern consistency of the discretization in this setting. This line of thought has probably been most clearly formulated in [4] or [5]: “In this setting, discrete surfaces appear as two-dimensional layers of multidimensional discrete nets, and their transformations correspond to shifts in the transversal lattice directions. A characteristic feature of the theory is that all lattice directions are on equal footing with respect to the defining geometric properties.”

For example, given two Darboux transforms of an isothermic surface, a fourth surface can be constructed by algebraic means so that the four surfaces form a quadrilateral of surfaces with Darboux transforms along the edges of the quadrilateral; moreover, the (spectral) parameters of the transformations on opposite edges are equal. Thus, repeated transformations and application of this permutability theorem generate a lattice of isothermic surfaces—trailing the effect on a single point then generates a 2-dimensional grid in space: a discrete surface with similar properties as a smooth isothermic surface. In particular, the discrete surfaces obtained in this way admit very similar transformations to their smooth analogues, hence generating multi-dimensional lattices that display the same properties in all lattice directions, cf [4]: “Discrete surfaces and their transformations become indistinguishable.”

Our principal aim here is to explore this relation between analogous smooth and discrete theories in a setting where the interplay between them becomes most tangible, that is, we unify the smooth and discrete aspects in a single geometric object: a semi-discrete isothermic surface.

Thus, to construct the semi-discrete isothermic surfaces considered in this text, we follow the idea outlined above, see also [13]: we obtain a notion of Darboux transformation for space curves by observing how the Darboux transformation of an isothermic surface acts on a single curvature line, cf [18]—then we construct semi-discrete isothermic surfaces as sequences of Darboux transforms of curves.

The first part of the text is concerned with the analysis of the Darboux transformation of curves in Euclidean and conformal *n*-space: we introduce the Darboux transformation as a special type of Ribaucour transformation, cf [3] and [16, Sect 8.2], that preserves a quadratic differential as an additional structure on a curve—this Darboux transformation acts on curves equipped with a quadratic differential, reminiscent of the fact that the Darboux transformation of an isothermic surface preserves a holomorphic quadratic differential that is associated to every isothermic surface. This notion of Darboux transformation for *polarized* curves generalizes the notions of [18, Sect 2.6].

In particular, Darboux transformations are given by solutions of a Riccati equation that includes a (spectral) parameter—we discuss a linearization of this Riccati equation using Möbius geometric techniques, by means of a connection on a suitable vector bundle, reflecting the conformal invariance of the transformation. Darboux transforms are then obtained from parallel sections of this connection, that is, as solutions of Darboux’s linear system. A version of the aforementioned Bianchi permutability theorem for the Darboux transformation of curves is then derived using standard gauge theoretic arguments.

Semi-discrete isothermic surfaces are introduced in the second part of the text, as sequences of Darboux transforms of curves. As detailed above in the fully discrete case, the Darboux transformation for semi-discrete isothermic surfaces can then be consistently defined because of the corresponding Bianchi permutability theorem for the Darboux transformation of curves. Further, we discuss how the Christoffel and Calapso transformations of semi-discrete isothermic surfaces occur from corresponding transformations for curves and suitable permutability theorems.

In summary, we do not only obtain the full transformation theory for semi-discrete isothermic surfaces, but, relying on the aforementioned linearization of the Riccati equation that describes the Darboux transformation, we also obtain a gauge theoretic characterization of semi-discrete isothermic surfaces, as are known in the smooth and fully discrete cases, cf [9].

As an example of the developed transformation theory and gauge theoretic approach for semi-discrete isothermic surfaces, we discuss semi-discrete surfaces of constant mean curvature in space forms, thereby extending results from [19, 20]. Here we rely on the intimate relation between the Christoffel transformation and a mixed area that is used to define the mean curvature of a surface with Gauss map in a space form, cf [6, 10] for the discrete case; a characterization in terms of linear conserved quantities, cf [8, 11], is then a direct consequence of the gauge theoretic characterization of semi-discrete isothermic surfaces and its relation with the Christoffel transformation.

## 2 Darboux transformations of curves

Our idea is that semi-discrete isothermic surfaces should be obtained by restricting a sequence of Darboux transforms of an isothermic surface to a fixed curvature line. We therefore seek a notion of “Darboux transformation” for (decorated) curves in space and shall find it in much the same way that a “Bäcklund transformation” for curves of constant torsion was found by restricting Bäcklund transformations of pseudo-spherical surfaces to a single asymptotic line in [13, Sect 1.1]. In particular, the Darboux transformation will occur as a special type of Ribaucour transformation, cf [16, Sect 8.2] or [3, Sect 3]: the Ribaucour transformation of hypersurfaces or submanifolds descends to their curvature lines, where only the enveloping condition remains a non-trivial condition.

**Definition 2.1**

Two curves \(x,\hat{x}:I\rightarrow {\mathbb R}^n\) will be said to form a *Ribaucour pair* if they envelop a circle congruence, i.e., if tangents at corresponding points *x*(*s*) and \(\hat{x}(s)\) are tangent to a common circle *c*(*s*). Either curve of a Ribaucour pair is a *Ribaucour transform* of the other curve.

Using a semi-discrete version of the Clifford algebra cross ratio of [14], cf [7, Sect 2.4], we obtain an algebraic characterization of Ribaucour pairs of curves:

**Definition and Lemma 2.2**

*tangent cross ratio*

*y*, which provides an alternative method to determine the tangent cross ratio of a Ribaucour pair from a suitable ratio of scaling factors:

*n*-sphere \(S^n\), thought of as the projective light cone of the \((n+2)\)-dimensional Minkowski space \({\mathbb R}^{n+1,1}\),

*n*-sphere \(S^n\) constitutes a

*lift*of the curve into the light cone \(\mathcal{L}^{n+1}\). In particular, for lightlike vectors \(o,q\in {\mathbb R}^{n+1,1}\) with inner product \((o,q)=-1\) consider the orthogonal decomposition

*Euclidean lift*of an immersed curve (or, more generally, submanifold) in \({\mathbb R}^n\) is obtained from the isometry

*subgeometry*of the conformal geometry of \(S^n\), cf [16, Sect 1.4]. Hence identifying \({\mathbb R}^n\cong Q^n\) as a subset of the conformal

*n*-sphere \(S^n\) by means of (2.4), we will also refer to any light cone map

*lift*of a curve \(x:I\rightarrow {\mathbb R}^n\subset {\mathbb R}^{n+1,1}\) in Euclidean space. Further, identifying \(\Lambda ^2{\mathbb R}^{n+1,1}\cong {\mathfrak o}({\mathbb R}^{n+1,1})\) via

**Lemma 2.3**

*x*and \(\hat{x}\), respectively,

*x*resp \(\hat{x}\) in \({\mathbb R}^n\), (2.5) reads

*q*; hence comparing \({\mathbb R}^n\)-parts of the equation we recover (2.2), which yields the claim. Conversely, assuming (2.2), expansion of

*infinitesimal cross ratio*

*polarization*) \({ds^2\over m}\) as part of the data.

**Definition 2.4**

*polarized domain*\((I,{ds^2\over m})\), \(m:I\rightarrow {\mathbb R}^\times \), will be called a

*Darboux pair*if its infinitesimal cross ratio is a constant multiple of the reference polarization,

*Darboux transform*of the other polarized curve.

*x*and \(\hat{x}\), so is the notion of Darboux transformation. An example of the Darboux transformation in Euclidean geometry is given by a tractrix construction, cf [18, Def 2.41]: if \(y:I\rightarrow {\mathbb R}^n\) denotes an arc-length parametrized curve then

*arc-length polarization*\({ds^2\over m}=|dx_\pm |^2\); namely, with a principal normal field \(n:I\rightarrow S^{n-1}\) of

*y*and the corresponding curvature \(\kappa \) we obtain

*n*-sphere \(S^n={\mathbb P}(\mathcal{L}^{n+1})\), where the Riccati equation (2.9) linearizes and becomes

*Darboux’s linear system*, cf [15], that is, yields the zero curvature representation of the associated integrable system. In particular, for a polarized curve

*any*light cone lift of the curve, we introduce a 1-parameter family of (flat) connections by

**Definition and Corollary 2.5**

*isothermic family of connections*of a polarized curve. The Darboux transforms of a polarized curve \(\langle \xi \rangle :(I,{ds^2\over m})\rightarrow S^n\), with respect to a parameter \(\mu \in {\mathbb R}\), are given by \({D\over ds}^{t}|_{t=\mu }\)-parallel sections, that is, by solutions of

*Darboux’s linear system*

Consequently, any polarized curve admits a \((1+n)\)-parameter family of Darboux transforms as any choice of the spectral parameter \(t=\mu \) and of an initial point in \(S^n\) yields a unique Darboux transform via a \({D\over ds}^{t}|_{t=\mu }\)-parallel light cone section. On the other hand, this new description (2.11) of the Darboux transformation lacks the symmetry of (2.7): thus our next goal is to understand how Darboux’s linear system changes under the Darboux transformation.

**Lemma 2.6**

**Lemma 2.7**

\(\gamma \circ \pi =0\) since \(\gamma (\xi )=0\);

\(\gamma \circ \varpi =\pi \circ \gamma \circ \varpi \) since \(\gamma (\langle \xi \rangle ^\perp )\subset \langle \xi \rangle \);

\(\gamma \circ \hat{\pi }=(\pi +\varpi )\circ \gamma \circ \hat{\pi }\) since \(\gamma ({\mathbb R}^{n+1,1})\subset \langle \xi \rangle ^\perp \); further

\(\pi \circ \gamma \circ \hat{\pi }=0\) since \(\gamma \in \Omega ^1({\mathfrak o}({\mathbb R}^{n+1,1}))\) is skew symmetric, so that \(\gamma (\hat{\xi })\perp \hat{\xi }\).

It is now readily derived that the well known Bianchi permutability theorem for the Darboux transformation of isothermic surfaces descends to a permutability theorem for the Darboux transformation of polarized curves, cf [1, Sect 3] or [9, Sect 4.2]:

**Theorem 2.8**

**Theorem 2.9**

Given three Darboux transforms \(\langle \xi _i\rangle \), \(i=0,1,2\), of a polarized curve \(\langle \xi \rangle :(I,{ds^2\over m})\rightarrow S^n\) with different parameters, \(\mu _i\ne \mu _j\) for \(i\ne j\), there is a simultaneous Darboux transform \(\langle \xi _{012}\rangle \), with parameters \(\mu _k\), of the simultaneous Darboux transforms \(\langle \xi _{ij}\rangle \) of \(\langle \xi _i\rangle \) and \(\langle \xi _j\rangle \), where \(i,j,k\in \{0,1,2\}\) are pairwise distinct.

## 3 Semi-discrete isothermic surfaces

We are now prepared to define semi-discrete isothermic surfaces, as sequences of Darboux transforms of curves—where we will parametrize the sequence by a graph \(G=(V,E)\) and oriented edges \((ij)\in E\) will be denoted by the ordered pair of their endpoints \(i,j\in V\).

**Definition 3.1**

A map \(\langle \xi \rangle :\Sigma \rightarrow S^n\), \((i,s)\mapsto \langle \xi _i(s)\rangle \), on a semi-discrete domain \(\Sigma =G\times I\) will be called a *semi-discrete isothermic surface* if there is a polarization \({ds^2\over m}\) on *I* so that any adjacent curves \(\langle \xi _i\rangle \) and \(\langle \xi _j\rangle \), \((ij)\in E\), form a Darboux pair of curves on the polarized interval \((I,{ds^2\over m})\).

*Moutard lift*

*G*and “vertical” edges added between corresponding points of the two copies.

Similarly, a “Christoffel duality” for polarized curves in \({\mathbb R}^n\), together with a suitable permutability theorem, yields a Christoffel duality for semi-discrete isothermic surfaces, cf [19, Thm 4.3], as long as there are no cycles in the discrete part *G* of the domain \(\Sigma \) of the semi-discrete isothermic surface.

**Definition 3.2**

*Christoffel dual*if

This duality for polarized curves yields the first equation of [19, (10)], the second yields permutability with the Darboux transformation:

**Theorem 3.3**

To obtain the Calapso transformation, or “conformal deformation”, of a polarized curve we introduce a (family of) gauge transformation(s) that trivialize the (flat) connections of its isothermic family of connections (2.10):

**Definition 3.4**

*Calapso transformations*of \(\langle \xi \rangle \), each curve \(\langle \xi ^t\rangle :=\langle T^t\xi \rangle \) is a

*Calapso transform*of \(\langle \xi \rangle \).

Note that the Calapso transformations \(T^t\) can be chosen to take values in the orthogonal group since the connections \({D\over ds}^{t}\) are metric connections; then they are unique up to post-composition by a (lift of a) Möbius transformation.

To carry this Calapso deformation for curves over to one for semi-discrete isothermic surfaces we will again require a permutability theorem; to this end, we will need to gain control of the Calapso transformations of the Calapso and Darboux transforms of a polarized curve.

**Lemma 3.5**

- (i)
\(\tilde{T}^tT^\tau =T^{\tau +t}\) for a Calapso transform \(\langle \tilde{\xi }\rangle =\langle T^\tau \xi \rangle \) of \(\langle \xi \rangle \);

- (ii)
\(\hat{T}^t\Gamma _{\langle \xi \rangle }^{\langle \hat{\xi }\rangle }(1-{t\over \mu })=T^t\) for a Darboux transform \(\langle \hat{\xi }\rangle \) of \(\langle \xi \rangle \), where \({D\over ds}^{t}\big |_{t=\mu }\hat{\xi }=0\).

**Theorem 3.6**

Suppose that \(\langle \xi \rangle ,\langle \hat{\xi }\rangle :(I,{ds^2\over m})\rightarrow S^n\) form a Darboux pair with parameter \(\mu \in {\mathbb R}^\times \), and let \(\tau \ne \mu \). Then their Calapso transforms \(\langle T^\tau \xi \rangle \) and \(\langle \hat{T}^\tau \hat{\xi }\rangle \) form, if suitably positioned, a Darboux pair with parameter \(\mu -\tau \).

Thus we have learned that the transformations of curves extend to sequences of Darboux transforms of curves, that is, to semi-discrete isothermic surfaces, by means of suitable permutability theorems:

**Theorem and Definition 3.7**

The Darboux, Christoffel and Calapso transformations for polarized curves extend to corresponding transformations of cycle-free semi-discrete isothermic surfaces.

An alternative approach to the transformations of semi-discrete isothermic surfaces is more directly based on a family of semi-discrete flat connections, similar to those of Definition and Corollary 2.5:

**Definition and Theorem 3.8**

*semi-discrete connection*on a vector bundle

*X*over a domain \(\Sigma =G\times I\), \(G=(V,E)\), is a pair \((\Gamma ,\nabla )\), consisting of connections over each smooth component, \(\nabla _i\) on \(X_i\) for each \(i\in V\), and vector bundle isomorphisms between components, \(\Gamma _{ij}:X_j\rightarrow X_i\) for each \((ij)\in E\); a semi-discrete connection is

*flat*if all \(\nabla _i\) are flat and are gauge equivalent via \(\Gamma _{ij}\),

*m*and \(\mu \) on

*I*and

*E*, respectively, so that the associated

*isothermic loop of connections*\((\Gamma ^t,\nabla ^t)_{t\in {\mathbb R}}\) consists of flat connections, where

This characterization of semi-discrete isothermic surfaces is an immediate consequence of Definition 3.1 in conjunction with Lemma 2.7. To obtain the Darboux and Calapso transformations only the flatness assertion is required, which already follows from the simpler statement of Lemma 2.6: then the following characterizations of the Darboux and Calapso transformations follow directly from their definitions and the corresponding characterizations Definition and Corollary 2.5 and Definition 3.4 for polarized curves, cf Lemma 3.5.

**Theorem 3.9**

- (i)the Darboux transforms \(\langle \hat{\xi }\rangle \) of \(\langle \xi \rangle \), with respect to a parameter \(\mu \), are given by \((\Gamma ^\mu ,\nabla ^\mu )\)-parallel sections, that is, by solutions of Darboux’s linear system$$\begin{aligned} \forall (ij)\in E:\Gamma ^\mu _{ij}\hat{\xi }_j = \hat{\xi }_i \hbox { and }\forall i\in V:\nabla ^\mu _i\hat{\xi }_i = 0; \end{aligned}$$
- (ii)the Calapso transforms \(\langle \xi ^t\rangle \) of \(\langle \xi \rangle \) are given as images of \(\langle \xi \rangle \) under trivializing gauge transformations,$$\begin{aligned} \langle \xi ^t\rangle = \langle T^t\xi \rangle , \hbox { where }T^t\cdot (\Gamma ^t,\nabla ^t) = \left( \mathrm{id},{d\over ds}\right) \Leftrightarrow {\left\{ \begin{array}{ll} \forall (ij)\in E: T^t_i\circ \Gamma ^t_{ij} = T^t_j, \\ \forall i\in V: T^t_i\circ \nabla ^t_i = {d\over ds}\circ T^t_i. \end{array}\right. } \end{aligned}$$

*special isothermic surfaces*as those semi-discrete isothermic surfaces that admit a

*polynomial conserved quantity*, i.e., a polynomial map

*linear conserved quantity*\(p(t)=zt+q\) yields \(q\equiv const\), \(z\perp \xi \) and, cf [12, (2.3)],

*z*, as before; note that \( \pi _j(d_{ij}z+{1\over \mu _{ij}}(\pi _j-\pi _i)q) = \pi _j(z_j+{1\over \mu _{ij}}\,q) \) since \(z\perp \xi \). Thus normalizing \(\xi =x\) so that \((x,q)\equiv -1\) we learn that

*z*is, up to scale, a Christoffel transform of

*x*in \({\mathbb R}^{n+1,1}\):

*normalized linear conserved quantity*. Conversely, \(p(t)=zt+q\) yields a (normalized) linear conserved quantity of \(\langle \xi \rangle =\langle x\rangle \) as soon as

*mixed area element*

*z*and

*x*define

*parallel nets*, that is, with suitable functions \(a_i\) and \(\alpha _{ij}\) we have

*x*and

*z*are not homothetic. Integrability of

*z*then yields

*x*must be a

*conjugate net*in \({\mathbb R}^{n+1,1}\), cf [21, Sect 2]; hence adjacent curves \(x_i\) and \(x_j\) form Ribaucour pairs, since

*x*maps into the conformal

*n*-sphere, and

*x*is

*circular*or a semi-discrete

*curvature line net*, cf [19, Definition 1.1]. Now, we use (3.6) to analyze vanishing of the mixed area,

*z*, which now reads

*x*is isotropic, \((x,x)\equiv 0\), we have \(x_{ij}\perp d_{ij}x\), hence using \(d_{ij}({x^{\prime }\over \nu })\parallel d_{ij}x\) from (3.8) we obtain

*x*, as

Note that these results descend to Euclidean ambient geometry; however, corresponding proofs in a purely Euclidean setting will require some arguments to be adapted, for example, by using the tangent cross ratio instead of isotropy of *x* to obtain *m* and \(\mu \) as functions of one parameter.

*mixed area mean curvature*in space forms,

*tangent plane congruence*for

*x*in \(Q^3\), cf [19, Prop 5.2] and [20, Definition 8] for the Euclidean case and see [10, Definition 2.3] for arbitrary ambient space form geometries in the fully discrete case: namely, consider

*x*has a suitable Christoffel dual \(z:\Sigma \rightarrow S^{3,1}\), if and only if \(\langle \xi \rangle =\langle x\rangle \) has a normalized linear conserved quantity \(p(t)=zt+q\), cf [6, Lemma 4.1] and [10, Thm 2.8] resp [8, Prop 2.5] and [11, Sect 5].

## Notes

### Acknowledgments

We would like to thank M Yasumoto for fruitful and enjoyable discussions around the subject. This work has been partially supported by the Austrian Science Fund (FWF) and the Japan Society for the Promotion of Science (JSPS) through the FWF/JSPS Joint Project Grant I1671-N26 “Transformations and Singularities”.

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