Open Communications in Nonlinear Mathematical Physics ]ocnmp[ Special Issue 1,
2024 pp 1–References

††footnotetext:  © The author(s). Distributed under a Creative Commons
Attribution 4.0 International License

This article is part of an OCNMP Special Issue
in Memory of Professor Decio Levi

Bäcklund transformations as integrable discretization. The geometric approach

Adam Doliwa

Faculty of Mathematics and Computer Science
University of Warmia and Mazury in Olsztyn
ul. Słoneczna 54, 10-710 Olsztyn, Poland

Received August 31, 2023; Accepted November 27, 2023

To memory of Decio Levi and Antoni Sym, distinguished researchers and friends

###### Abstract

We present interpretation of known results in the theory of discrete
asymptotic and discrete conjugate nets from the _discretization by Bäcklund
transformations_ point of view. We collect both classical formulas of XIXth
century differential geometry of surfaces and their transformations, and more
recent results from geometric theory of integrable discrete equations. We
first present transformations of hyperbolic surfaces within the context of the
Moutard equation and Weingarten congruences. The permutability property of the
transformations provides a way to construct integrable discrete analogs of the
asymptotic nets for such surfaces. Then after presenting the theory of
conjugate nets and their transformations we apply the principle that Bäcklund
transformations provide integrable discretization to obtain known results on
the discrete conjugate nets. The same approach gives, via the Ribaucour
transformations, discrete integrable analogs of orthogonal conjugate nets.

##  1 Introduction

Given integrable system of differential equations one is often interested in
finding the corresponding (in the sense of small lattice step size limit)
discrete system while preserving the integrability properties. It turns out
that usually the simple/naive replacement of derivatives by difference
operators spoils the integrability. The discretization has to be made on the
level where the integrability features are visible and transparent. Such
methods are, for example (i) discrete version of the linear problem [1], (ii)
the Hirota method via a bilinear form [52], (iii) extensions of the
Zakharov–Shabat dressing method [71], (iv) direct linearization using linear
integral equations [86].

Another technique, which is the subject of the present work, is based on
Bäcklund transformations, which are discrete symmetries of integrable
equations. It is the fundamental observation made by Decio Levi, which we
present quoting abstracts of two of his papers:

_It is shown that any Bäcklund transformation of a nonlinear differential
equation integrable by the multichannel Schrödinger eigenvalue problem can be
written in the form $V_{x}=U^{\prime}V-VU$. This allows us to interpret the
Bäcklund transformation formally as a nonlinear differential difference
equation for which we can immediately construct the soliton solutions._ [68]

_In this paper, one shows that the best known nonlinear differential
difference equations associated with the discrete Schrödinger spectral problem
and also with the discrete Zakharov-Shabat spectral problem can be interpreted
as Bäcklund transformations for some continuous nonlinear evolution
equations._ [69]

Bäcklund transformations arose in connection with the construction, by XIXth
century geometers [6, 5], of pseudospherical surfaces and corresponding
solutions of the sine-Gordon equation. It was shown by Bianchi [7] that such
transformations can be iterated leading to an algebraic superposition formula.
More recently, Wahlquist and Estabrook [110] demonstrated that also the
Korteweg–de Vries equation, which is a paradigmatic example of integrable
partial differential equation [48], admits invariance under a Bäcklund-type
transformation and possesses an associated permutability theorem. They have
used for that purpose the transformation introduced by Darboux [18] in the
context of Sturm–Liouville problems. The subject is generally known in the
soliton theory as Bäcklund or Darboux transformations [63, 97, 81, 98, 50,
46], but in the geometric theory of transformations of surfaces exhibiting
permutability property, also other names are relevant [24, 96, 56]. The
classical results of the old differential geometry of surfaces and their
transformations are summarized in [20, 19, 8, 45, 109, 65, 47].

The most general transformations of conjugate nets and their permutability
were introduced and studied by Jonas [56, 45]. The Darboux equations of
multidimensional conjugate nets [19] have been rediscovered by Zakharov and
Manakov [112, 113] as the most general systems solvable by the non-local
$\bar{\partial}$-dressing method. The discrete analogue of conjugate nets on a
surface was introduced first on the geometric level [100, 102], and connected
to integrability theory in [25]. The integrable discrete version of the
corresponding Darboux system was given in [14]. Geometric studies of
multidimensional discrete conjugate nets have been initiated in [34] and were
followed in [79, 36, 38]. The integrability of circular lattices, which form a
distinguished reduction of discrete conjugate nets and in the continuous limit
give orthogonal conjugate nets, was first studied geometrically in [17] and
then confirmed by other tools [36, 60, 4, 39]. Compelling reasons for such an
interpretation were given in [80, 91] on the basis of the computer graphics,
and in [11, 9] from the theory of discrete isothermic nets.

I met first time Decio Levi in late eighties when he came to Institute of
Theoretical Physics of Warsaw University to visit his friend Antoni Sym, who
supervised both my master and PhD theses. They worked together on integrable
generalization of pseudospherical surfaces, known now as Bianchi surfaces
[70]. In the present work we show a harmonious coexistence of two points of
view: _Bäcklund transformations provide integrable discretization_ by Decio
Levi, and _soliton theory is surface theory_ [108] by Antoni Sym. In the
theory of integrable systems it is quite common that the same equations can be
derived using different methods, which give different perspective and
emphasize different connections.

The paper is constructed as follows. In Section 2 we present the classical
theory of hyperbolic surfaces in asymptotic parametrization emphasizing their
connection with the Moutard equation [83]. The corresponding transformations
and their permutability property [89] give rise to discrete asymptotic nets,
which coincide with their natural geometric analogs [102]. The next Section 3
is devoted to conjugate nets and their transformations. We give a
discretization of the nets starting from their fundamental transformation and
exploiting its permutability properties. In Section 4 we present derivation of
integrable discrete version of orthogonal conjugate nets on the base of
geometric interpretation of the Ribaucour reduction [96, 24] of the
fundamental transformation. We conclude the paper with additional discussion
on geometry of Hirota’s discrete Kadomtsev–Petviashvili (KP) system and with
some remarks about dispersionless systems.

In the paper we tried to present old results, most of them more than one
hundred years old, from contemporary perspective and in unified notation. We
remark that interpretation of the fundamental transformation and its Ribaucour
reduction in terms of vertex operators within the free-fermion formalism of
the multicomponent KP hierarchy was the subject of [40, 39], where also the
aspects of integrable discretization of conjugate and orthogonal nets were
investigated.

##  2 Discretization of asymptotic nets, and the Moutard
transformation

The classical transformation of Bianchi and Bäcklund for pseudospherical
surfaces and the sine-Gordon equation can be considered as a reduction of
Weingarten transformation of hyperbolic surfaces in asymptotic
parametrization. We devote the present Section to such asymptotic nets and to
the Moutard equation [83], which governs the behaviour of their normal vector.
We use the standard notation and terminology of the classical theory of
surfaces [44], see also [98].

###  2.1 Hyperbolic surfaces, the Moutard equation, and the Lelieuvre
formulas

Let $(u,v)$ be local coordinate system on a surface $\Sigma$ in
${\mathbb{R}}^{3}$, and by $\boldsymbol{r}(u,v)$ denote the position vector of
a generic point. The coordinate lines are called asymptotic when in every
point their tangent planes coincide with the tangent plane to the surface.
Surfaces admitting the asymptotic coordinates are called hyperbolic. In such
case we have

$\displaystyle\boldsymbol{r}_{,uu}$
$\displaystyle=a_{1}\boldsymbol{r}_{,u}+b_{1}\boldsymbol{r}_{,v}\;,$ (2.1)
$\displaystyle\boldsymbol{r}_{,vv}$
$\displaystyle=a_{2}\boldsymbol{r}_{,u}+b_{2}\boldsymbol{r}_{,v}\;,$ (2.2)

where $a_{i}$, and $b_{i}$, $i=1,2$, are functions of the local coordinates;
here and in all the paper by a subscript after comma we denote the partial
derivative with respect to the corresponding variable. As a consequence of the
compatibility condition $\boldsymbol{r}_{,uuvv}=\boldsymbol{r}_{,vvuu}$ there
exists a function $\phi$, given up to an additive constant, such that

$a_{1}=\phi_{,u}\;,\qquad b_{2}=\phi_{,v}\;.$ (2.3)

By direct calculation one can check that the normal vector

$\boldsymbol{\nu}=\mathrm{e}^{-\phi}\boldsymbol{r}_{,u}\times\boldsymbol{r}_{,v}$
(2.4)

satisfies the Moutard equation

$\boldsymbol{\nu}_{,uv}=f\boldsymbol{\nu}\;,$ (2.5)

with the potential

$f=\phi_{,uv}-b_{1}a_{2}\;.$ (2.6)

Moreover, using eventually the allowed freedom in definition of $\phi$ and/or
changing the orientation, the position vector is given by the Lelieuvre
formulas [67]

$\boldsymbol{r}_{,u}=\boldsymbol{\nu}_{,u}\times\boldsymbol{\nu}\;,\qquad\boldsymbol{r}_{,v}=\boldsymbol{\nu}\times\boldsymbol{\nu}_{,v}\;.$
(2.7)

###  2.2 The Moutard transformation and its permutability property

The following result provides a way how to transform solutions of a Moutard
equation into new solution of equation of the same form but with different
potential. In consequence, given a hyperbolic surface it allows to construct
new surface of the same type.

######  Theorem 2.1.

[83] Given vector-valued solution $\boldsymbol{\nu}$ of the Moutard equation
corresponding to a given potential $f$, and given scalar solution $\theta$ of
the same equation, then the vector-valued function $\hat{\boldsymbol{\nu}}$
defined by the compatible equations

$\displaystyle(\theta\hat{\boldsymbol{\nu}})_{,u}$
$\displaystyle=\;\;\theta_{,u}\boldsymbol{\nu}-\theta\boldsymbol{\nu}_{,u}\;,$
(2.8) $\displaystyle(\theta\hat{\boldsymbol{\nu}})_{,v}$
$\displaystyle=-\theta_{,v}\boldsymbol{\nu}+\theta\boldsymbol{\nu}_{,v}\;,$
(2.9)

satisfies the Moutard equation with the potential

$\hat{f}=\frac{\hat{\theta}_{,uv}}{\hat{\theta}}\;,\qquad\text{where}\qquad\hat{\theta}=\frac{1}{\theta}\;.$

######  Corollary 2.2.

Notice that $\boldsymbol{\nu}$ is the Moutard transform of
$\hat{\boldsymbol{\nu}}$ with $\hat{\theta}$ taken as the transformation
function.

######  Corollary 2.3.

One can check that the surface

$\hat{\boldsymbol{r}}=\boldsymbol{r}+\hat{\boldsymbol{\nu}}\times\boldsymbol{\nu},$
(2.10)

can be obtained from $\hat{\boldsymbol{\nu}}$ via the Lelieuvre formulas
(2.7). In particular the local parameters $(u,v)$ form an asymptotic
coordinate system on the transformed surface.

######  Remark.

The two-parameter family of lines
$\langle\boldsymbol{r},\hat{\boldsymbol{r}}\rangle$ , which are tangent to
both surfaces $\Sigma$ and $\hat{\Sigma}$ at the corresponding points, forms
the so called Weingarten congruence. The corresponding transformation between
hyperbolic surfaces $\Sigma$ and $\hat{\Sigma}$ is called the Weingarten
transformation.

Let $\theta^{i}$, $i=1,2$, be two solutions of the Moutard equation satisfied
by the normal $\boldsymbol{\nu}$, and let $\boldsymbol{\nu}^{\\{i\\}}$,
$i=1,2$, be its corresponding two transforms. By $\theta^{2\\{1\\}}$ denote
also the Moutard transform of $\theta^{2}$ with respect to $\theta^{1}$

$\begin{pmatrix}\theta^{1}\begin{pmatrix}\boldsymbol{\nu}^{\\{1\\}}\\\
\theta^{2\\{1\\}}\end{pmatrix}\end{pmatrix}_{,u}=\theta^{1}_{,u}\begin{pmatrix}\boldsymbol{\nu}\\\
\theta^{2}\end{pmatrix}-\theta^{1}\begin{pmatrix}\boldsymbol{\nu}\\\
\theta^{2}\end{pmatrix}_{,u},\qquad\begin{pmatrix}\theta^{1}\begin{pmatrix}\boldsymbol{\nu}^{\\{1\\}}\\\
\theta^{2\\{1\\}}\end{pmatrix}\end{pmatrix}_{,v}=-\theta^{1}_{,v}\begin{pmatrix}\boldsymbol{\nu}\\\
\theta^{2}\end{pmatrix}+\theta^{1}\begin{pmatrix}\boldsymbol{\nu}\\\
\theta^{2}\end{pmatrix}_{,v},$ (2.11)

and by $\theta^{1\\{2\\}}$ denote the Moutard transform of $\theta^{1}$ with
respect to $\theta^{2}$

$\begin{pmatrix}\theta^{2}\begin{pmatrix}\boldsymbol{\nu}^{\\{2\\}}\\\
\theta^{1\\{2\\}}\end{pmatrix}\end{pmatrix}_{,u}=\theta^{2}_{,u}\begin{pmatrix}\boldsymbol{\nu}\\\
\theta^{1}\end{pmatrix}-\theta^{2}\begin{pmatrix}\boldsymbol{\nu}\\\
\theta^{1}\end{pmatrix}_{,u},\qquad\begin{pmatrix}\theta^{2}\begin{pmatrix}\boldsymbol{\nu}^{\\{2\\}}\\\
\theta^{1\\{2\\}}\end{pmatrix}\end{pmatrix}_{,v}=-\theta^{2}_{,v}\begin{pmatrix}\boldsymbol{\nu}\\\
\theta^{1}\end{pmatrix}+\theta^{2}\begin{pmatrix}\boldsymbol{\nu}\\\
\theta^{1}\end{pmatrix}_{,v}.$ (2.12)

In consequence, the transformation formulas (2.10) on the hyperbolic surfaces
level $\Sigma^{\\{1\\}}$ and $\Sigma^{\\{2\\}}$ take the form

$\boldsymbol{r}^{\\{1\\}}=\boldsymbol{r}+\boldsymbol{\nu}^{\\{1\\}}\times\boldsymbol{\nu},\qquad\boldsymbol{r}^{\\{2\\}}=\boldsymbol{r}+\boldsymbol{\nu}^{\\{2\\}}\times\boldsymbol{\nu}.$
(2.13)

Let us apply to $\boldsymbol{\nu}^{\\{1\\}}$ the Moutard transformation with
$\theta^{2\\{1\\}}$,

$\left(\theta^{2\\{1\\}}\boldsymbol{\nu}^{\\{1,2\\}}\right)_{,u}=\theta^{2\\{1\\}}_{,u}\boldsymbol{\nu}^{\\{1\\}}-\theta^{2\\{1\\}}\boldsymbol{\nu}^{\\{1\\}}_{,u},\qquad\left(\theta^{2\\{1\\}}\boldsymbol{\nu}^{\\{1,2\\}}\right)_{,v}=-\theta^{2\\{1\\}}_{,v}\boldsymbol{\nu}^{\\{1\\}}+\theta^{2\\{1\\}}\boldsymbol{\nu}^{\\{1\\}}_{,v},$
(2.14)

and to $\boldsymbol{\nu}^{\\{2\\}}$ the transformation with
$\theta^{1\\{2\\}}$

$\left(\theta^{1\\{2\\}}\boldsymbol{\nu}^{\\{2,1\\}}\right)_{,u}=\theta^{1\\{2\\}}_{,u}\boldsymbol{\nu}^{\\{2\\}}-\theta^{1\\{2\\}}\boldsymbol{\nu}^{\\{2\\}}_{,u},\qquad\left(\theta^{1\\{2\\}}\boldsymbol{\nu}^{\\{2,1\\}}\right)_{,v}=-\theta^{1\\{2\\}}_{,v}\boldsymbol{\nu}^{\\{2\\}}+\theta^{1\\{2\\}}\boldsymbol{\nu}^{\\{2\\}}_{,v}.$
(2.15)

We would like both transformations give the same result
$\boldsymbol{\nu}^{\\{2,1\\}}=\boldsymbol{\nu}^{\\{1,2\\}}$. Moreover since we
have

$(\theta^{1}\theta^{2\\{1\\}}+\theta^{2}\theta^{1\\{2\\}})_{,u}=0=(\theta^{1}\theta^{2\\{1\\}}+\theta^{2}\theta^{1\\{2\\}})_{,v},$

then the additive constants in definition of $\theta^{1}\theta^{2\\{1\\}}$ and
of $\theta^{2}\theta^{1\\{2\\}}$ can be fixed such that

$\theta^{2}\theta^{1\\{2\\}}=\theta^{12}=-\theta^{1}\theta^{2\\{1\\}}\;.$
(2.16)

By elimination of derivatives of the normal vectors from equations
(2.11)-(2.12) and (2.14)-(2.15), and using identity (2.16) we can arrive to
the the following important result.

######  Theorem 2.4 (Permutability of the Moutard transformations).

The vector-valued functions $\boldsymbol{\nu}^{\\{12\\}}$ given by algebraic
formula

$\boldsymbol{\nu}^{\\{12\\}}-\boldsymbol{\nu}=\frac{\theta^{1}\theta^{2}}{\theta^{12}}\left(\boldsymbol{\nu}^{\\{1\\}}-\boldsymbol{\nu}^{\\{2\\}}\right)$
(2.17)

is simultaneously Moutard transform of $\boldsymbol{\nu}^{\\{1\\}}$ with
respect to $\theta^{2\\{1\\}}$ and the Moutard transform of
$\boldsymbol{\nu}^{\\{2\\}}$ with respect to $\theta^{1\\{2\\}}$.

$\begin{CD}\boldsymbol{\nu}@>{\theta^{1}}>{}>\boldsymbol{\nu}^{\\{1\\}}\\\
@V{\theta^{2}}V{}V@V{}V{\theta^{2\\{1\\}}}V\\\
<EMAIL_ADDRESS>
######  Remark.

Actually, because of free additive parameter in definition of $\theta^{12}$ we
obtain this way one-parameter family of the transforms.

######  Corollary 2.5.

The one-parameter family of vector-valued functions
$\boldsymbol{r}^{\\{12\\}}$ given by algebraic formula

$\boldsymbol{r}^{\\{12\\}}=\boldsymbol{r}+\frac{\theta^{1}\theta^{2}}{\theta^{12}}\boldsymbol{\nu}^{\\{1\\}}\times\boldsymbol{\nu}^{\\{2\\}}$
(2.18)

provides simultaneously the Weingarten transforms of the hyperbolic surface
$\boldsymbol{r}^{\\{1\\}}$ with respect to $\theta^{2\\{1\\}}$ and the
Weingarten transforms of the hyperbolic surface $\boldsymbol{r}^{\\{2\\}}$
with respect to $\theta^{1\\{2\\}}$.

###  2.3 The Moutard transformation as integrable discretization

The successive application of the Weingarten-Moutard transforms, taking into
account their algebraic superposition formula, to a given hyperbolic surface
$\Sigma$ allows to build a two dimensional lattice $\Sigma^{m,n}$ of such
surfaces. Let us fix a point on $\Sigma$ and trace properties of a lattice in
${\mathbb{R}}^{3}$ of the corresponding points, represented by vectors
$\boldsymbol{r}^{m,n}$.

######  Proposition 2.6.

The five points $\boldsymbol{r}^{m,n}$, $\boldsymbol{r}^{m\pm 1,n}$,
$\boldsymbol{r}^{m,n\pm 1}$ belong to a common plane.

  * Proof.

The statement follows from equations (2.13), which imply that lines
$\langle\boldsymbol{r}^{m,n},\boldsymbol{r}^{m+1,n}\rangle$ are orthogonal to
both the vectors $\boldsymbol{\nu}^{m,n}$ and $\boldsymbol{\nu}^{m+1,n}$, and
the lines $\langle\boldsymbol{r}^{m,n},\boldsymbol{r}^{m,n+1}\rangle$ are
orthogonal to both the vectors $\boldsymbol{\nu}^{m,n}$ and
$\boldsymbol{\nu}^{m,n+1}$. The common plane of the five points is orthogonal
to the vector $\boldsymbol{\nu}^{m,n}$. ∎

######  Corollary 2.7.

Equations (2.13) are discrete analogs of the Lelieuvre formulas (2.7), see
also [59].

######  Remark.

The three points $\boldsymbol{r}^{m,n}$, $\boldsymbol{r}^{m\pm 1,n}$, define
the tangent plane of the first discrete coordinate curve at point
$\boldsymbol{r}^{m,n}$, and the three points $\boldsymbol{r}^{m,n}$,
$\boldsymbol{r}^{m,n\pm 1}$, define the tangent plane of the second curve at
this point. Both planes coincide with the common plane of Proposition 2.6,
what allows to call the curves the discrete asymptotic lines.

######  Definition 2.1 ([102, 100]).

The discrete asymptotic net is a map
$\boldsymbol{r}\colon{\mathbb{Z}}^{2}\to{\mathbb{R}}^{3}$ such that for
arbitrary $(m,n)\in{\mathbb{Z}}^{2}$ the five points $\boldsymbol{r}^{m,n}$,
$\boldsymbol{r}^{m\pm 1,n}$, $\boldsymbol{r}^{m,n\pm 1}$ are coplanar.

Figure 1: Discrete asymptotic nets

######  Corollary 2.8.

The algebraic superposition formula (2.17) of the Moutard transformations on
the level of the normal vector $\boldsymbol{\nu}^{m,n}$ can be interpreted as
the discrete Moutard equation [89]

$\boldsymbol{\nu}^{m+1,n+1}-\boldsymbol{\nu}^{m,n}=f^{m,n}\left(\boldsymbol{\nu}^{m+1,n}-\boldsymbol{\nu}^{m,n+1}\right).$
(2.19)

######  Remark.

To obtain the Moutard equation (2.5) and the Lelieuvre formulas (2.7) from
(2.19) and (2.13) one first has to change the orientation of the normal vector
according to $\boldsymbol{\nu}^{m,n}\to(-1)^{n}\boldsymbol{\nu}^{m,n}$.

Integrability of the hyperbolic surfaces can be understood as existence of
transformations satisfying algebraic permutability principle. It turns out
that one can construct analogous transformations for integrable discrete
hyperbolic surfaces (discrete asymptotic nets). The corresponding
transformation formulas, their geometric interpretation, and their
permutability property have been considered in [89, 84, 27]. See also [55] for
development of their theory in direction of application in computer graphic.

Hyperbolic surfaces with constant Gauss curvature, i.e. pseudospherical
surfaces, are described by the celebrated sine-Gordon equation for the angle
between the asymptotic coordinates on such surfaces. Restricting the
Weingarten–Moutard transformation to such surfaces we obtain discrete
asymptotic nets whose elementary quadrilaterals have opposite sides of equal
length, and which provide integrable discrete analogs of pseudospherical
surfaces [101, 111, 10]. The corresponding Bianchi’s permutability theorem for
the Bäcklund transformation of the sine-Gordon equation provides its
integrable difference analog [53], and describes the angle between asymptotic
coordinates on discrete pseudospherical surface. The Bianchi surfaces [8],
whose integrability was discussed in [70], are hyperbolic surfaces
characterized by the property that their Weingarten transformation preserves
Gauss curvature in the corresponding points; see [12, 103, 37, 105] for
discussion of discrete analogs and this and other integrable reductions of
asymptotic nets.

###  2.4 Discrete BKP equation

Integrable non-linear equations appear in the context of discrete Moutard
equation, prior to reductions of discrete hyperbolic surfaces, for more then
two discrete variables [89, 28]. Consider a map
$\boldsymbol{\nu}\colon{\mathbb{Z}}^{N}\to{\mathbb{R}}^{M}$, $N,M\geq 3$,
which satisfies the system of discrete Moutard equations in each pair of
variables

$\boldsymbol{\nu}^{\boldsymbol{n}+\boldsymbol{e}_{i}+\boldsymbol{e}_{j}}-\boldsymbol{\nu}^{\boldsymbol{n}}=f_{ij}^{\boldsymbol{n}}\left(\boldsymbol{\nu}^{\boldsymbol{n}+\boldsymbol{e}_{i}}-\boldsymbol{\nu}^{\boldsymbol{n}+\boldsymbol{e}_{j}}\right),\qquad
1\leq i\neq j\leq N;$ (2.20)

here $\boldsymbol{n}\in{\mathbb{Z}}^{N}$, and $\boldsymbol{e}_{i}$ is the unit
vector in $i$th direction, $i=1,\dots,N$. Compatibility of the system leads to
the following set of nonlinear equations

$1+f_{jk}^{\boldsymbol{n}+\boldsymbol{e}_{i}}(f_{ij}^{\boldsymbol{n}}-f_{ik}^{\boldsymbol{n}})=f_{ik}^{\boldsymbol{n}+\boldsymbol{e}_{j}}f_{ij}^{\boldsymbol{n}}=f_{ij}^{\boldsymbol{n}+\boldsymbol{e}_{k}}f_{ik}^{\boldsymbol{n}},\qquad
i,j,k\quad\text{distinct},\qquad
f_{ji}^{\boldsymbol{n}}=-f_{ij}^{\boldsymbol{n}}.$ (2.21)

The second equality implies existence of the potential
$\tau:{\mathbb{Z}}^{N}\to{\mathbb{R}}$, in terms of which the functions
$f_{ij}$ can be written as

$f_{ij}^{\boldsymbol{n}}=\frac{\tau^{\boldsymbol{n}+\boldsymbol{e}_{i}}\tau^{\boldsymbol{n}+\boldsymbol{e}_{j}}}{\tau^{\boldsymbol{n}}\,\tau^{\boldsymbol{n}+\boldsymbol{e}_{i}+\boldsymbol{e}_{j}}},\qquad
i<j.$ (2.22)

The first equality can be then rewritten in the form of the system of Miwa’s
discrete BKP equations [82]

$\tau^{\boldsymbol{n}}\tau^{\boldsymbol{n}+\boldsymbol{e}_{i}+\boldsymbol{e}_{j}+\boldsymbol{e}_{k}}=\tau^{\boldsymbol{n}+\boldsymbol{e}_{i}+\boldsymbol{e}_{j}}\tau^{\boldsymbol{n}+\boldsymbol{e}_{k}}-\tau^{\boldsymbol{n}+\boldsymbol{e}_{i}+\boldsymbol{e}_{k}}\tau^{\boldsymbol{n}+\boldsymbol{e}_{j}}+\tau^{\boldsymbol{n}+\boldsymbol{e}_{j}+\boldsymbol{e}_{k}}\tau^{\boldsymbol{n}+\boldsymbol{e}_{i}},\quad
1\leq i<j<k\leq N.$ (2.23)

Geometric meaning of the above equations goes beyond the theory of discrete
asymptotic nets, and can be incorporated [28] into the theory of discrete
conjugate nets [34, 38].

##  3 Multidimensional conjugate nets, their fundamental
transformation, and multidimensional lattices
of planar quadrilaterals

Conjugate nets on a surface are second, after asymptotic nets, distinguished
coordinate systems studied in depth by geometers of XIXth century. They
include, as a special subcase, curvature coordinates, whose theory will be
studied in the next Section. For classical results on the subject see works of
Gabriel Lamé [64], Luigi Bianchi [8] or Gaston Darboux [19]. The relation of
conjugate nets on a surface to linear partial differential equations of the
second order allows to transfer special theorems on such equations [66, 72,
83, 49] to the geometric level. The theory of transformations within special
classes of such equations/conjugate nets has obtained mature form in works of
Hans Jonas [56] and Luther P. Eisenhart [45], see also [109, 65, 47]. The
Darboux equations, which describe multidimensional conjugate nets, were
rediscovered in [112, 113] in the context of soliton theory as the most
general partial differential equations integrable by the non-local
$\bar{\partial}$-dressing method. Moreover, in [57] they were isolated as the
simplest equations within multicomponent KP hierarchy. We start this Section
with presenting basic elements of the theory of conjugate nets on a surface
and their transformations. Then we move to the multidimensional nets.

###  3.1 Conjugate coordinates on a surface, and the Lévy transformation

Local coordinates $(u,v)$, on a surface in the space ${\mathbb{R}}^{N}$ of
arbitrary dimension $N\geq 3$, are called conjugate if the second mixed
derivative of the position vector $\boldsymbol{r}(u,v)$ are tangent to the
surface. The defining equation takes then the form of the _Laplace equation_

$\boldsymbol{r}_{,uv}=a\boldsymbol{r}_{,u}+b\boldsymbol{r}_{,v},$ (3.1)

where $a(u,v)$ and $b(u,v)$ are corresponding functions of the conjugate
parameters. The following considerations allow to introduce concepts relevant
in the general theory of transformations of multidimensional conjugate nets.

Let $\theta$ be a scalar solution of equation (3.1), linearly independent of
the components of $\boldsymbol{r}$, then by direct calculation one can check
that the so called _Lévy transforms_ of $\boldsymbol{r}$, given by

$\boldsymbol{r}^{(u)}=\boldsymbol{r}-\frac{\theta}{\theta_{,u}}\boldsymbol{r}_{,u},\qquad\boldsymbol{r}^{(v)}=\boldsymbol{r}-\frac{\theta}{\theta_{,v}}\boldsymbol{r}_{,v},$
(3.2)

are new surfaces with $(u,v)$ being local conjugate coordinates. The
corresponding Laplace equations (3.1) of the new nets have coefficients

$\displaystyle
a^{(u)}=a+\left(\log\frac{\theta}{\theta_{,u}}\right)_{,v},\qquad$
$\displaystyle b^{(u)}=b^{(v)}+\left(\log a^{(u)}\right)_{,u},$ (3.3)
$\displaystyle a^{(v)}=a^{(u)}+\left(\log b^{(v)}\right)_{,v},\qquad$
$\displaystyle b^{(v)}=b+\left(\log\frac{\theta}{\theta_{,v}}\right)_{,u},$
(3.4)

what can be verified by direct calculation.

Figure 2: The fundamental transform $\hat{\boldsymbol{r}}$, the Lévy
transforms $\boldsymbol{r}^{(u)}$, $\boldsymbol{r}^{(v)}$, the adjoint Lévy
transforms $\boldsymbol{r}_{(u)}$, $\boldsymbol{r}_{(v)}$, and the Laplace
transforms $\boldsymbol{r}^{(u)}_{(v)}$, $\boldsymbol{r}^{(v)}_{(u)}$ of two-
dimensional conjugate net $\boldsymbol{r}$

In the proof it is convenient to show first that

$\boldsymbol{r}^{(u)}_{,v}=a^{(u)}\left(\boldsymbol{r}^{(u)}-\boldsymbol{r}^{(v)}\right),\qquad\boldsymbol{r}^{(v)}_{,u}=b^{(v)}\left(\boldsymbol{r}^{(v)}-\boldsymbol{r}^{(u)}\right),$

what means that the lines joining the corresponding points of both Lévy
transforms of the conjugate net $\boldsymbol{r}$ are simultaneously tangent to
$v$-coordinate lines of $\boldsymbol{r}^{(u)}$ and to $u$-coordinate lines of
$\boldsymbol{r}^{(v)}$, see Figure 2.

Few comments are in order:

  * •

Lines through $\boldsymbol{r}$ in direction of $\boldsymbol{r}_{,u}$ form the
_$u$ -tangent congruence_ of the conjugate net. Equivalently, $\boldsymbol{r}$
is _$u$ -focal net_ of the congruence. Similarly one defines the $v$-tangent
congruence of $\boldsymbol{r}$, which is its $v$-focal net. Using such a
terminology one can state that $\boldsymbol{r}^{(v)}$ is $u$-focal net of the
$v$-tangent congruence of $\boldsymbol{r}^{(u)}$, and _vice versa_.

  * •

Notice that we use the notion of congruence of lines in the narrow sense, i.e.
the parameters $(u,v)$ of the family define its focal nets. This means that
one-dimensional family of lines parametrized by $u$ forms a developable
surface and is made of tangents to $u$-coordinate on the $u$-focal net;
similar condition folds for $v$-parameter family of the congruence. We say
that the congruence is referred to its developables.

  * •

The unique $v$-focal net of the $u$-tangent congruence to $\boldsymbol{r}$,
given by

$\boldsymbol{r}^{(u)}_{(v)}=\boldsymbol{r}-\frac{1}{b}\boldsymbol{r}_{,u},$
(3.5)

is called the _$uv$ -Laplace transform of $\boldsymbol{r}$_. Analogously one
defines the $vu$-Laplace transform of $\boldsymbol{r}$

$\boldsymbol{r}^{(v)}_{(u)}=\boldsymbol{r}-\frac{1}{a}\boldsymbol{r}_{,v}.$
(3.6)

In particular the conjugate net $\boldsymbol{r}^{(u)}$ is the $uv$-Laplace
transform of $\boldsymbol{r}^{(v)}$.

  * •

The net $\boldsymbol{r}^{(u)}$ is called _conjugate_ to the $u$-tangent
congruence of $\boldsymbol{r}$, similarly the net $\boldsymbol{r}^{(v)}$ is
conjugate to the $v$-tangent congruence of $\boldsymbol{r}$. In such relation
the conjugate coordinates on a surface are the parameters which define the
focal nets of the congruence. One can reverse the situation and try to find
focal nets of a congruence conjugate to a given conjugate net. Such focal
nets, denoted by $\boldsymbol{r}_{(u)}$ and $\boldsymbol{r}_{(v)}$, are called
_adjoint Lévy transforms_ of the net.

  * •

The conjugate net $\hat{\boldsymbol{r}}$ is called a _fundamental transform_
of $\boldsymbol{r}$ when the two-dimensional family of lines joining their
corresponding points forms a congruence (in the narrow sense explained above)
whose developables cut both nets along the conjugate coordinate lines, i.e.
both nets are conjugate to the same congruence.

###  3.2 Multidimensional conjugate nets and the Darboux equations

Consider $N$-dimensional submanifold in ${\mathbb{R}}^{M}$ with local
parameters $\boldsymbol{u}=(u_{1},\dots,u_{N})$ satisfying in each pair
$(u_{i},u_{j})$, $i\neq j$, the conjugate net condition

$\boldsymbol{r}_{,ij}=a_{ij}\boldsymbol{r}_{,i}+a_{ji}\boldsymbol{r}_{,j}.$
(3.7)

The functions $a_{ij}$ of the local conjugate parameters cannot be arbitrary,
because for $N>2$ they should satisfy the following compatibility conditions
of the above system (3.7) of Laplace equations

$a_{ij,k}=a_{ij}a_{jk}+a_{ik}a_{kj}-a_{ij}a_{ik},\qquad
i,j,k\quad\text{distinct}.$ (3.8)

The nonlinear _Darboux equations_ (3.8) imply in particular that
$a_{ij,k}=a_{ik,j}$, what allows to introduce potentials $h_{i}$, called _Lamé
coefficients_ , such that

$a_{ij}=\frac{h_{i,j}}{h_{i}},\qquad i\neq j,$ (3.9)

and correspondingly the Laplace system (3.7) takes the form

$\boldsymbol{r}_{,ij}=(\log h_{i})_{,j}\boldsymbol{r}_{,i}+(\log
h_{j})_{,i}\boldsymbol{r}_{,j}\qquad i\neq j.$ (3.10)

The remaining part of the Darboux equations in Laplace coefficients $a_{ij}$
can be written in terms of the Lamé coefficients as

$h_{i,jk}=\frac{h_{j,k}h_{i,j}}{h_{j}}+\frac{h_{k,j}h_{i,k}}{h_{k}},\qquad
i,j,k\quad\text{distinct}.$ (3.11)

Following Darboux, let us introduce the suitably scaled tangent vectors
$\boldsymbol{X}_{i}$, $i=1,\dots,N$, from equations

$\boldsymbol{r}_{,i}=h_{i}\boldsymbol{X}_{i}.$ (3.12)

Then the Laplace equations (3.7) take the particularly simple form

$\boldsymbol{X}_{i,j}=\beta_{ij}\boldsymbol{X}_{j},\qquad i\neq j,$ (3.13)

where the _rotation coefficients_ $\beta_{ij}$ are defined by the linear
system adjoint to (3.13)

$h_{j,i}=\beta_{ij}h_{i},\qquad i\neq j.$ (3.14)

The corresponding version of the Darboux equation reads

$\beta_{ij,k}=\beta_{ik}\beta_{kj},\qquad i,j,k\quad\text{distinct}.$ (3.15)

The tangent lines to $i$-th coordinate on the conjugate net $\boldsymbol{r}$
form its $i$-th tangent congruence. Its $j$-th focal net, called $ij$-Laplace
transform of $\boldsymbol{r}$, is given by

$\boldsymbol{r}^{(i)}_{(j)}=\boldsymbol{r}-\frac{1}{a_{ji}}\boldsymbol{r}_{,i}.$
(3.16)

The Laplace transforms satisfy the following identities

$(\boldsymbol{r}^{(i)}_{(j)})^{(j)}_{(i)}=\boldsymbol{r},\qquad(\boldsymbol{r}^{(i)}_{(j)})^{(j)}_{(k)}=\boldsymbol{r}^{(i)}_{(k)},\qquad(\boldsymbol{r}^{(i)}_{(j)})^{(k)}_{(i)}=\boldsymbol{r}^{(k)}_{(j)}.$
(3.17)

This means that each generic $N$-dimensional conjugate net comes together with
whole system of conjugate nets enumerated by points of the $Q(A_{N-1})$ root
lattice.

###  3.3 The vectorial fundamental (binary Darboux) transformation
of conjugate nets

The contemporary theory of transformations of conjugate nets and of the
Darboux equations is based on the following Lemma, given first in the discrete
setting in [79]. The present version is its direct limit.

######  Lemma 3.1.

Given solution $\beta_{ij}$ of the Darboux equations (3.15), and given
solution ${\boldsymbol{Y}}_{i}$ of the linear system (3.13) taking values in
the (column vector) space ${\mathbb{R}}^{K}$, and given solution
${\boldsymbol{Y}}^{*}_{i}$ of the adjoint linear system (3.14) taking values
in the (row vector) space ${\mathbb{R}}^{L}$.
1\. There exists the $K\times L$ matrix-valued potential
$\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]$ defined by the
following compatible system

$\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]_{,i}={\boldsymbol{Y}}_{i}\otimes{\boldsymbol{Y}}_{i}^{*},\qquad
i=1,\dots,N.$ (3.18)

2\. If $K=L$ and the potential
$\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]$ is invertible,
then the functions

$\hat{\beta}_{ij}=\beta_{ij}-{\boldsymbol{Y}}^{*}_{j}\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]^{-1}{\boldsymbol{Y}}_{i},$
(3.19)

are new solutions of the Darboux equations.
3\. The vector-valued functions

$\hat{{\boldsymbol{Y}}}_{i}=\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]^{-1}{\boldsymbol{Y}}_{i},\qquad\hat{{\boldsymbol{Y}}}^{*}_{i}={\boldsymbol{Y}}^{*}_{i}\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]^{-1},$
(3.20)

are the corresponding new solutions of the linear and the adjoint linear
problem equations (3.13)-(3.14). In addition, the new matrix-valued potential
is of the form

$\boldsymbol{\Theta}[\hat{{\boldsymbol{Y}}},\hat{{\boldsymbol{Y}}}^{*}]=C-\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]^{-1},$
(3.21)

where $C$ is a constant operator.

######  Remark.

Notice that equations (3.12) mean that one can write
$\boldsymbol{r}=\boldsymbol{\Theta}[\boldsymbol{X},h]$.

######  Remark.

In the theory of Darboux transformations of the KP hierarchy [92] the function
$\boldsymbol{\Theta}$ is called the squared-eigenfunction potential.

By suitable arrangement of the transformation data [38] we arrive at the
following version of the above Lemma.

######  Theorem 3.2.

Consider conjugate net $\boldsymbol{r}$ with Lamé coefficients $h_{i}$,
normalized tangent vectors $\boldsymbol{X}_{i}$ and rotation coefficients
$\beta_{ij}$. Given transformation data ${\boldsymbol{Y}}_{i}$,
${\boldsymbol{Y}}^{*}_{i}$ which satisfy point 2. of Lemma 3.1 then

$\hat{\boldsymbol{r}}=\boldsymbol{r}-\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}]\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]^{-1}\boldsymbol{\Theta}[{\boldsymbol{Y}},h],$
(3.22)

is new conjugate net, called the fundamental transform of $\boldsymbol{r}$.
The new rotation coefficients are given by equation (3.19), and the Lamé
coefficients and normalized tangent vectors read

$\hat{h}_{i}=h_{i}-{\boldsymbol{Y}}^{*}_{i}\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]^{-1}\boldsymbol{\Theta}[{\boldsymbol{Y}},h],\qquad\hat{\boldsymbol{X}}_{i}=\boldsymbol{X}_{i}-\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}]\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]^{-1}{\boldsymbol{Y}}_{i}.$
(3.23)

When the transformation data are scalar functions $Y_{i}$ and $Y^{*}_{i}$ then
$\theta=\boldsymbol{\Theta}[Y,h]$ is a scalar additional solution of the
Laplace equation of $\boldsymbol{r}$. The $N$-dimensional vector-valued
function $\boldsymbol{X}^{\prime}=\boldsymbol{\Theta}[\boldsymbol{X},Y^{*}]$
defines new conjugate net with Lamé coefficients $Y_{i}^{*}$ and the same
normalized tangent vectors $\boldsymbol{X}_{i}$. The function
$\theta^{\prime}=\boldsymbol{\Theta}[Y,Y^{*}]$ is a scalar solution of the
Laplace equation of $\boldsymbol{X}^{\prime}$ and in such a notation the
scalar fundamental transformation reads

$\hat{\boldsymbol{r}}=\boldsymbol{r}-\frac{\theta}{\theta^{\prime}}\boldsymbol{X}^{\prime}.$
(3.24)

The vector $\boldsymbol{X}^{\prime}$ points in the direction of the congruence
of the transformation, and is called the _Combescure transform_ of
$\boldsymbol{r}$. The corresponding $i$-th Lévy transform
$\boldsymbol{r}^{(i)}$ of $\boldsymbol{r}$, the intersection of $i$-tangents
of $\boldsymbol{r}$ and $\hat{\boldsymbol{r}}$ reads

$\boldsymbol{r}^{(i)}=\boldsymbol{r}-\frac{\theta}{\theta_{,i}}\boldsymbol{r}_{,i}=\boldsymbol{r}-\frac{\theta}{Y_{i}}\boldsymbol{X}_{i},$
(3.25)

while the $i$-th adjoint Lévy transform $\boldsymbol{r}_{(i)}$ of
$\boldsymbol{r}$, whose $i$-th tangent congruence is the congruence of the
transformation is given by

$\boldsymbol{r}_{(i)}=\boldsymbol{r}-\frac{h_{i}}{Y^{*}_{i}}\boldsymbol{X}^{\prime}.$
(3.26)

See Figure 2 to compare with the two-dimensional case.

###  3.4 Superpositions of fundamental transformations, and
multidimensional lattices of planar quadrilaterals

The vectorial form of the fundamental transformation given in Lemma 3.1
contains already their permutability theorem.

######  Theorem 3.3.

Assume the following splitting of the data of the vectorial fundamental
transformation

${\boldsymbol{Y}}_{i}=\begin{pmatrix}{\boldsymbol{Y}}^{a}_{i}\\\
{\boldsymbol{Y}}^{b}_{i}\end{pmatrix},\qquad{\boldsymbol{Y}}^{*}_{i}=\left({\boldsymbol{Y}}^{*}_{ai}\;{\boldsymbol{Y}}^{*}_{bi}\right),$
(3.27)

associated with the partition
${\mathbb{R}}^{K}={\mathbb{R}}^{K_{a}}\oplus{\mathbb{R}}^{K_{b}}$, which
implies the following splitting of the potentials

$\displaystyle\boldsymbol{\Theta}[{\boldsymbol{Y}},h]=\begin{pmatrix}\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},h]\\\
\boldsymbol{\Theta}[{\boldsymbol{Y}}^{b},h]\end{pmatrix},\qquad\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}]=\left(\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}_{a}]\;\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}_{b}]\right),$
$\displaystyle\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]=\begin{pmatrix}\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},{\boldsymbol{Y}}^{*}_{a}]&\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},{\boldsymbol{Y}}^{*}_{b}]\\\
\boldsymbol{\Theta}[{\boldsymbol{Y}}^{b},{\boldsymbol{Y}}^{*}_{a}]&\boldsymbol{\Theta}[{\boldsymbol{Y}}^{b},{\boldsymbol{Y}}^{*}_{b}]\end{pmatrix}.$

Then the vectorial fundamental transformation is equivalent to the following
superposition of vectorial fundamental transformations:
1\. Transformation $\boldsymbol{r}\to\boldsymbol{r}^{\\{a\\}}$ with the data
${\boldsymbol{Y}}^{a}_{i}$, ${\boldsymbol{Y}}^{*}_{ai}$ and the corresponding
potentials $\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},h]$,
$\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},{\boldsymbol{Y}}^{*}_{a}]$,
$\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}_{a}]$

$\displaystyle\boldsymbol{r}^{\\{a\\}}$
$\displaystyle=\boldsymbol{r}-\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}_{a}]\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},{\boldsymbol{Y}}^{*}_{a}]^{-1}\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},h],$
(3.28) $\displaystyle h^{\\{a\\}}_{i}$
$\displaystyle=h_{i}-{\boldsymbol{Y}}^{*}_{ai}\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},{\boldsymbol{Y}}^{*}_{a}]^{-1}\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},h],$
(3.29) $\displaystyle{\boldsymbol{X}}^{\\{a\\}}_{i}$
$\displaystyle=\boldsymbol{X}_{i}-\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}_{a}]\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},{\boldsymbol{Y}}^{*}_{a}]^{-1}{\boldsymbol{Y}}_{i}^{a}.$
(3.30)

2\. Application on the result the vectorial fundamental transformation with
the transformed data

$\displaystyle{\boldsymbol{Y}}^{*\\{a\\}}_{bi}$
$\displaystyle={\boldsymbol{Y}}^{*}_{bi}-{\boldsymbol{Y}}^{*}_{ai}\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},{\boldsymbol{Y}}^{*}_{a}]^{-1}\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},{\boldsymbol{Y}}^{*}_{b}],$
(3.31) $\displaystyle{{\boldsymbol{Y}}}^{b\\{a\\}}_{i}$
$\displaystyle={\boldsymbol{Y}}^{b}_{i}-\boldsymbol{\Theta}[{\boldsymbol{Y}}^{b},{\boldsymbol{Y}}^{*}_{a}]\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},{\boldsymbol{Y}}^{*}_{a}]^{-1}{\boldsymbol{Y}}_{i}^{a},$
(3.32)

and potentials

$\displaystyle\boldsymbol{\Theta}[{\boldsymbol{Y}}^{b},h]^{\\{a\\}}$
$\displaystyle=\boldsymbol{\Theta}[{\boldsymbol{Y}}^{b},h]-\boldsymbol{\Theta}[{\boldsymbol{Y}}^{b},{\boldsymbol{Y}}^{*}_{a}]\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},{\boldsymbol{Y}}^{*}_{a}]^{-1}\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},h]=\boldsymbol{\Theta}[{\boldsymbol{Y}}^{b\\{a\\}},h^{\\{a\\}}],$
$\displaystyle\boldsymbol{\Theta}[{\boldsymbol{Y}}^{b},{\boldsymbol{Y}}^{*}_{b}]^{\\{a\\}}$
$\displaystyle=\boldsymbol{\Theta}[{\boldsymbol{Y}}^{b},{\boldsymbol{Y}}^{*}_{b}]-\boldsymbol{\Theta}[{\boldsymbol{Y}}^{b},{\boldsymbol{Y}}^{*}_{a}]\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},{\boldsymbol{Y}}^{*}_{a}]^{-1}\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},{\boldsymbol{Y}}^{*}_{b}]=\boldsymbol{\Theta}[{\boldsymbol{Y}}^{b\\{a\\}},{\boldsymbol{Y}}_{b}^{*\\{a\\}}],$
$\displaystyle\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}_{b}]^{\\{a\\}}$
$\displaystyle=\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}_{b}]-\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}_{a}]\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},{\boldsymbol{Y}}^{*}_{a}]^{-1}\boldsymbol{\Theta}[{\boldsymbol{Y}}^{a},{\boldsymbol{Y}}^{*}_{b}]=\boldsymbol{\Theta}[\boldsymbol{X}^{\\{a\\}},{\boldsymbol{Y}}_{b}^{*\\{a\\}}],$

i.e.

$\hat{\boldsymbol{r}}=\boldsymbol{r}^{\\{a,b\\}}=\boldsymbol{r}^{\\{a\\}}-\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}_{b}]^{\\{a\\}}\left(\boldsymbol{\Theta}[{\boldsymbol{Y}}^{b},{\boldsymbol{Y}}^{*}_{b}]^{\\{a\\}}\right)^{-1}\boldsymbol{\Theta}[{\boldsymbol{Y}}^{b},h]^{\\{a\\}}.$
(3.33)

######  Remark.

The final result (3.33) is independent of the order of making the partial
transformations.

######  Remark.

The above procedure fixes already the integration constants when integrating
equations (3.18) in constructions of the potentials.

Lest us consider the simplest case of $K=2$ when the vectorial fundamental
transformation is obtained as superposition of two scalar transformations

$\displaystyle\boldsymbol{r}^{\\{a\\}}$
$\displaystyle=\boldsymbol{r}-\frac{\theta^{a}}{\theta^{a}_{a}}\boldsymbol{X}_{a},\qquad\boldsymbol{X}_{a}=\Theta[\boldsymbol{X},Y^{*}_{a}],\quad\theta^{a}=\Theta[Y^{a},h],\quad\theta^{a}_{a}=\Theta[Y^{a},Y^{*}_{a}],$
(3.34) $\displaystyle\boldsymbol{r}^{\\{b\\}}$
$\displaystyle=\boldsymbol{r}-\frac{\theta^{b}}{\theta^{b}_{b}}\boldsymbol{X}_{b},\qquad\boldsymbol{X}_{b}=\Theta[\boldsymbol{X},Y^{*}_{b}],\quad\theta^{b}=\Theta[Y^{b},h],\quad\theta^{b}_{b}=\Theta[Y^{b},Y^{*}_{b}],$
(3.35)

and the final result reads

$\boldsymbol{r}^{\\{a,b\\}}=\boldsymbol{r}-\left(\boldsymbol{X}_{a},\boldsymbol{X}_{b}\right)\begin{pmatrix}\theta^{a}_{a}&\theta^{a}_{b}\\\
\theta^{b}_{a}&\theta^{b}_{b}\end{pmatrix}^{-1}\begin{pmatrix}\theta^{a}\\\
\theta^{b}\end{pmatrix},\qquad\theta^{a}_{b}=\Theta[Y^{a},Y^{*}_{b}],\quad\theta^{b}_{a}=\Theta[Y^{b},Y^{*}_{a}].$
(3.36)

The point $\boldsymbol{r}^{\\{a,b\\}}$ belongs to the plane passing through
$\boldsymbol{r}$, $\boldsymbol{r}^{\\{a\\}}$, $\boldsymbol{r}^{\\{b\\}}$ (the
plane containing $\boldsymbol{r}$ and spanned by $\boldsymbol{X}_{a}$ and
$\boldsymbol{X}_{b}$), see Figure 3, and the superposition formula (3.36) can
be rewritten in the form of a discrete analogue of the Laplace equation (3.7)

$\boldsymbol{r}^{\\{a,b\\}}-\boldsymbol{r}=\frac{\theta^{a\\{b\\}}\theta^{a}_{a}}{\theta^{a\\{b\\}}_{a}\theta^{a}}(\boldsymbol{r}^{\\{a\\}}-\boldsymbol{r})+\frac{\theta^{b\\{a\\}}\theta^{b}_{b}}{\theta^{b\\{a\\}}_{b}\theta^{b}}(\boldsymbol{r}^{\\{b\\}}-\boldsymbol{r}).$
(3.37)

If $\boldsymbol{r}$, $\boldsymbol{r}^{\\{a\\}}$, $\boldsymbol{r}^{\\{b\\}}$
are given then the position of $\boldsymbol{r}^{\\{a,b\\}}$ on the plane is
arbitrary due to the integration constants in definitions of $\theta^{a}_{b}$
and $\theta^{b}_{a}$.

Figure 3: Elementary quadrilateral of superposition of two fundamental
transformations

The superposition of three scalar fundamental transformations

$\boldsymbol{r}^{\\{a,b,c\\}}=\boldsymbol{r}-\left(\boldsymbol{X}_{a},\boldsymbol{X}_{b},\boldsymbol{X}_{c}\right)\begin{pmatrix}\theta^{a}_{a}&\theta^{a}_{b}&\theta^{a}_{c}\\\
\theta^{b}_{a}&\theta^{b}_{b}&\theta^{b}_{c}\\\
\theta^{c}_{a}&\theta^{c}_{b}&\theta^{c}_{c}\end{pmatrix}^{-1}\begin{pmatrix}\theta^{a}\\\
\theta^{b}\\\ \theta^{c}\end{pmatrix},$ (3.38)

does not generate new integration constants. This result can be found in [45]
as _the extended theorem of permutability_. It implies, in particular, that
the point $\boldsymbol{r}^{\\{a,b,c\\}}$ is uniquely given as intersection of
three planes
$\langle\boldsymbol{r}^{\\{a\\}},\boldsymbol{r}^{\\{a,b\\}},\boldsymbol{r}^{\\{a,c\\}}\rangle$,
$\langle\boldsymbol{r}^{\\{b\\}},\boldsymbol{r}^{\\{a,b\\}},\boldsymbol{r}^{\\{b,c\\}}\rangle$
and
$\langle\boldsymbol{r}^{\\{c\\}},\boldsymbol{r}^{\\{a,c\\}},\boldsymbol{r}^{\\{b,c\\}}\rangle$;
see also [34].

Due to equation (3.34) the vectors $\boldsymbol{X}_{a}$ can be interpreted as
normalized tangent vectors to the lattice direction, and
$-\frac{\theta^{a}}{\theta^{a}_{a}}$ playing the role of the Lamé
coefficients. The transformation formulas

$\boldsymbol{X}_{b}^{\\{a\\}}=\boldsymbol{X}_{b}-\frac{\theta^{a}_{b}}{\theta^{a}_{a}}\boldsymbol{X}_{a}$
(3.39)

give the discrete analog of the linear problem (3.13) with
$-\frac{\theta^{a}_{b}}{\theta^{a}_{a}}$ playing the role of the rotation
coefficients. The corresponding transformation rules provide nonlinear
relations, which can be interpreted as discrete Darboux equations. To close
this Section we briefly recapitulate the basic theory of discrete conjugate
nets [34], which we have just obtained from the theory of transformations of
conjugate nets in the spirit of works [68, 69].

######  Definition 3.1.

The discrete conjugate net is a map
$\boldsymbol{r}\colon{\mathbb{Z}}^{N}\to{\mathbb{R}}^{M}$ of $N$-dimensional
integer lattice such that for arbitrary $\boldsymbol{n}\in{\mathbb{Z}}^{N}$
and any two directions $a\neq b$ of the lattice, the vertices
$\boldsymbol{r}^{\boldsymbol{n}}$,
$\boldsymbol{r}^{\boldsymbol{n}+\boldsymbol{e}_{a}}$,
$\boldsymbol{r}^{\boldsymbol{n}+\boldsymbol{e}_{b}}$, and
$\boldsymbol{r}^{\boldsymbol{n}+\boldsymbol{e}_{a}+\boldsymbol{e}_{b}}$ of
elementary quadrilaterals are coplanar.

The coplanarity condition can be written in terms of the system of discrete
Laplace equations

$\boldsymbol{r}^{\boldsymbol{n}+\boldsymbol{e}_{a}+\boldsymbol{e}_{b}}-\boldsymbol{r}^{\boldsymbol{n}}=A_{ab}^{\boldsymbol{n}}\left(\boldsymbol{r}^{\boldsymbol{n}+\boldsymbol{e}_{a}}-\boldsymbol{r}^{\boldsymbol{n}}\right)+A_{ba}^{\boldsymbol{n}}\left(\boldsymbol{r}^{\boldsymbol{n}+\boldsymbol{e}_{b}}-\boldsymbol{r}^{\boldsymbol{n}}\right),\qquad
a\not=b.$ (3.40)

Due to compatibility of the system (3.40) the functions
$A_{ab}^{\boldsymbol{n}}$ can be expressed in terms of the discrete Lamé
coefficients

$A_{ab}^{\boldsymbol{n}}=\frac{H_{a}^{\boldsymbol{n}+\boldsymbol{e}_{b}}}{H_{a}^{\boldsymbol{n}}}\;,\qquad
a\neq b\;,$ (3.41)

which satisfy equations

$H_{c}^{\boldsymbol{n}+\boldsymbol{e}_{a}+\boldsymbol{e}_{b}}-H_{c}^{\boldsymbol{n}}=\frac{H_{a}^{\boldsymbol{n}+\boldsymbol{e}_{b}+\boldsymbol{e}_{c}}}{H_{a}^{\boldsymbol{n}+\boldsymbol{e}_{c}}}\left(H_{c}^{\boldsymbol{n}+\boldsymbol{e}_{a}}-H_{c}^{\boldsymbol{n}}\right)+\frac{H_{b}^{\boldsymbol{n}+\boldsymbol{e}_{a}+\boldsymbol{e}_{c}}}{H_{b}^{\boldsymbol{n}+\boldsymbol{e}_{c}}}\left(H_{c}^{\boldsymbol{n}+\boldsymbol{e}_{b}}-H_{c}^{\boldsymbol{n}}\right),$
(3.42)

for distinct $a$, $b$ and $c$. If we introduce the suitably scaled tangent
vectors $\boldsymbol{X}_{a}$, $a=1,...,N$,

$\boldsymbol{r}^{\boldsymbol{n}+\boldsymbol{e}_{a}}-\boldsymbol{r}^{\boldsymbol{n}}=H_{a}^{\boldsymbol{n}}\boldsymbol{X}_{a}^{\boldsymbol{n}},$
(3.43)

then equations (3.40) can be rewritten as a first order system

$\boldsymbol{X}_{a}^{\boldsymbol{n}+\boldsymbol{e}_{b}}-\boldsymbol{X}_{a}^{\boldsymbol{n}}=Q_{ab}^{\boldsymbol{n}}\boldsymbol{X}_{b}^{\boldsymbol{n}},\qquad
a\neq b\;.$ (3.44)

The proportionality factors $Q_{ab}$, called the discrete rotation
coefficients, can be found from the linear equations

$H_{b}^{\boldsymbol{n}+\boldsymbol{e}_{a}}-H_{b}^{\boldsymbol{n}}=H_{a}^{\boldsymbol{n}+\boldsymbol{e}_{b}}Q_{ab}^{\boldsymbol{n}},\qquad
a\neq b\;,$ (3.45)

adjoint to (3.13). The compatibility condition for the system (3.13) (or its
adjoint) gives the following form of the discrete Darboux equations

$Q_{ab}^{\boldsymbol{n}+\boldsymbol{e}_{c}}-Q_{ab}^{\boldsymbol{n}}=Q_{ac}^{\boldsymbol{n}+\boldsymbol{e}_{b}}Q_{cb}^{\boldsymbol{n}},\qquad
a,b,c\quad\text{distinct}.$ (3.46)

The integrable discretization of the Darboux system was achieved first in [14]
within the $\bar{\partial}$ technique. The discrete analog of a conjugate net
on a surface was introduced on a purely geometric basis in [100, 102]. Its
connection with integrability theory was made first in [25] by connecting the
Laplace sequence of two-dimensional discrete conjugate nets to Hirota’s
discrete generalized Toda lattice [54]. Soon after that the discrete analogs
of multidimensional conjugate nets were introduced in [34]. In particular it
was shown there that the number of discrete variables in the discrete Darboux
equations can be arbitrary large, and this augmentation does not restrict the
solution space of the basic three-dimensional system — this property is known
nowadays as multidimensional consistency [3, 85] and is considered as the
basic concept of the theory of discrete integrable systems [51].

The Darboux–Bäcklund transformations of the discrete Darboux equations were
formulated first on the algebraic level in [79], and then in [38] the full
geometric flavour of the theory was presented together with the interpretation
of the transformations on the nonlocal $\bar{\partial}$-dressing method level,
see also [73, 75, 40, 77, 78]. In particular it was pointed out in [38],
referring to [68] and other similar works, that for discrete conjugate nets
there is no essential difference between transformations and generation of new
dimensions of the lattice. On the other hand, the conjugate nets are natural
continuous limits of the lattices of planar quadrilaterals, what can be easily
seen on the level of their Laplace equations (3.7) and (3.40). Therefore, the
principle of getting the integrable discretization via Bäcklund transformation
approach in this particular case finds natural explanation. This shows that,
from the point of view of the theory of integrable systems, the discrete ones
seem to be more basic.

The transformation theory of discrete conjugate nets can be constructed [38]
following the geometric principles of the continuous case. Also here the
notion of congruence of lines (any two neighbouring lines of the family are
coplanar) turns out to be crucial. In particular, it implies natural
definition of focal lattices of such congruences. The simplest congruences are
given by tangent lines in a fixed direction of a discrete conjugate net.

######  Remark.

The discrete Weingerten congruences, which are relevant in the theory of
transformations of discrete asymptotic nets described in Section 2, do not
satisfy the definition of a discrete congruences of the theory of
transformations of discrete conjugate nets, as it was pointed to me by Maciej
Nieszporski. In fact, as it was explained in [27], the discrete Weingarten
congruences provide two-dimensional lattices of planar quadrilaterals in the
Plücker quadric. Their theory fits therefore into the general scheme of
quadratic reductions of discrete conjugate nets [26].

##  4 Curvature coordinates, the Ribaucour transformation,
and circular lattices

The notion of conjugate nets is invariant with respect to projective
transformations of the ambient space. By imposing an additional geometric
structure one can consider the corresponding reduction of the general theory.
In the Euclidean space (we work in the standard orthonormal basis where the
scalar product of two vectors is given with the help of transposition
$\boldsymbol{u}\cdot\boldsymbol{v}=\boldsymbol{u}^{t}\,\boldsymbol{v}$) one
can consider orthogonal conjugate nets, which turn out to be curvature
coordinates on the given submanifold. The corresponding reduction of the
fundamental transformation is provided by the Ribaucour transformation.

The classical Ribaucour transformation [96] concerns surfaces in three
dimensional Euclidean space ${\mathbb{E}}^{3}$ such that lines of curvature
(which are both conjugate and orthogonal) correspond and such that the normals
to both surfaces in corresponding points in a point (the center of the
Ribaucour sphere) equidistant to both of them. In general one can consider
$N$-dimensional submanifold of ${\mathbb{E}}^{M}$, $N\leq M$, parametrized by
conjugate and orthogonal coordinates.

The orthogonality constraint

$\boldsymbol{r}_{,i}\cdot\boldsymbol{r}_{,j}=0,\qquad i\neq j,$ (4.1)

implies that the function

$\rho=\frac{1}{2}\,\boldsymbol{r}\cdot\boldsymbol{r},$

satisfies the Laplace equations of the orthogonal conjugate net
$\boldsymbol{r}$, and the functions

$X^{\circ}_{i}=\boldsymbol{r}\cdot\boldsymbol{X}_{i},$

give the corresponding solution to the linear system (3.13), i.e.
$\rho=\boldsymbol{\Theta}[X^{\circ},h]$. Equivalently, the above facts imply
orthogonality of the conjugate net.

The corresponding Ribaucour reduction of the vectorial fundamental
transformation, compatible with the orthogonality of a given conjugate net,
can be constructed using only half of the transformation data, for analogous
but different description see [74]. The following result can be checked
directly by calculating derivatives of both sides of the formulas.

######  Lemma 4.1.

Given solution ${\boldsymbol{Y}}^{*}_{i}$ of the adjoint linear problem (3.14)
of the orthogonal conjugate net $\boldsymbol{r}$ then

${\boldsymbol{Y}}_{i}=\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}]^{t}\boldsymbol{X}_{i},$
(4.2)

give a solution to the linear problem (3.13) of the net. Moreover, the
integration constants in construction of the corresponding potentials
$\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]$,
$\boldsymbol{\Theta}[{\boldsymbol{Y}},h]$ and
$\boldsymbol{\Theta}[X^{\circ},{\boldsymbol{Y}}^{*}]$ can be chosen such that

$\displaystyle\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]^{t}+\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]=\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}]^{t}\>\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}],$
(4.3)
$\displaystyle\boldsymbol{\Theta}[{\boldsymbol{Y}},h]=\boldsymbol{\Theta}[\boldsymbol{X},{\boldsymbol{Y}}^{*}]^{t}\boldsymbol{r}-\boldsymbol{\Theta}[X^{\circ},{\boldsymbol{Y}}^{*}]^{t}.$
(4.4)

######  Theorem 4.2.

The vectorial fundamental transformation of orthogonal conjugate net
$\boldsymbol{r}$ calculated by (3.22) with application of the above
constraints (4.3)- (4.4) is again conjugate orthogonal net.

The proof is based on the observation that orthogonality of the conjugate net
$\hat{\boldsymbol{r}}$ is equivalent to the fact that the function
$\frac{1}{2}\hat{\boldsymbol{r}}\cdot\hat{\boldsymbol{r}}$ satisfies the
Laplace system of the net. The statement follows then from direct
verification, using the constraints (4.3)-(4.4), and showing that

$\frac{1}{2}\hat{\boldsymbol{r}}\cdot\hat{\boldsymbol{r}}=\frac{1}{2}\boldsymbol{r}\cdot\boldsymbol{r}-\boldsymbol{\Theta}[X^{\circ},{\boldsymbol{Y}}^{*}]\boldsymbol{\Theta}[{\boldsymbol{Y}},{\boldsymbol{Y}}^{*}]^{-1}\boldsymbol{\Theta}[{\boldsymbol{Y}},h]=\hat{\rho},$
(4.5)

is therefore the corresponding transform of the solution $\rho$ of the Laplace
system of $\boldsymbol{r}$.

The formulation of the theorem on permutability of superpositions of vectorial
Ribaucour transformations reads as in Theorem 3.3. One has to check, what can
be done by direct calculation, that on the intermediate level the reduction
conditions (4.2)-(4.4) are satisfied by the transformed data.

Geometry of integrable discrete analog of orthogonal conjugate nets follows
from the observation made by Demoulin, who showed [24] that the vertices
$\boldsymbol{r}$, $\boldsymbol{r}^{\\{a\\}}$, $\boldsymbol{r}^{\\{b\\}}$, and
$\boldsymbol{r}^{\\{a,b\\}}$ of the Ribaucour transform are concircular, see
Figure 4. In proving that we will follow [45], where this fact was shown as an
implication of the orthogonality constraint imposed on the fundamental
transformation. Given three points $\boldsymbol{r}$,
$\boldsymbol{r}^{\\{a\\}}$ and $\boldsymbol{r}^{\\{b\\}}$, the coordinates of
the center of the circle passing through them are of the form

$\boldsymbol{r}+\lambda\boldsymbol{X}_{a}+\mu\boldsymbol{X}_{b},$

where $\lambda$ and $\mu$ are determined by the condition that the lines
joining the center to the mid-points of the segments
$[\boldsymbol{r},\boldsymbol{r}^{\\{a\\}}]$ and
$[\boldsymbol{r},\boldsymbol{r}^{\\{b\\}}]$, are perpendicular to these
segments. These conditions, due to the transformation formulas (3.34) and the
diagonal elements of the reduction condition (4.3)

$2\theta^{a}_{a}=\boldsymbol{X}_{a}\cdot\boldsymbol{X}_{a},$ (4.6)

are reducible to

$\theta^{a}+(\lambda\boldsymbol{X}_{a}+\mu\boldsymbol{X}_{b})\cdot\boldsymbol{X}_{a}=0,\qquad\theta^{b}+(\lambda\boldsymbol{X}_{a}+\mu\boldsymbol{X}_{b})\cdot\boldsymbol{X}_{b}=0.$
(4.7)

Analogously, the condition that the line joining the center to the mid-point
of the segments $[\boldsymbol{r}^{\\{a\\}},\boldsymbol{r}^{\\{a,b\\}}]$ is
perpendicular to the segment is

$\left(\boldsymbol{r}^{\\{a\\}}+\frac{1}{2}\left(\boldsymbol{r}^{\\{a,b\\}}-\boldsymbol{r}^{\\{a\\}}\right)-\boldsymbol{r}-\lambda\boldsymbol{X}_{a}-\mu\boldsymbol{X}_{b}\right)\cdot\boldsymbol{X}_{b}^{\\{a\\}}=0,$
(4.8)

what can be verified using the transformation formulas, equations (4.7), and
off-diagonal elements of the reduction condition (4.3)

$\theta^{a}_{b}+\theta^{b}_{a}=\boldsymbol{X}_{a}\cdot\boldsymbol{X}_{b}.$
(4.9)

######  Remark.

In [36] we proved the circularity of the quadrilaterals by showing equivalence
of the constraint with equation

$\boldsymbol{X}_{a}\cdot\boldsymbol{X}_{b}^{\\{a\\}}+\boldsymbol{X}_{b}\cdot\boldsymbol{X}_{a}^{\\{b\\}}=0,$
(4.10)

an immediate consequence of the transformation formulas (3.39) and of the
condition (4.3).

Figure 4: The superposition of two scalar Ribaucour transforms of an
orthogonal conjugate net. The vertices $\boldsymbol{r}$,
$\boldsymbol{r}^{\\{a\\}}$, $\boldsymbol{r}^{\\{b\\}}$, and
$\boldsymbol{r}^{\\{a,b\\}}$ of the elementary quadrilateral are concircular.
For circular quadrilaterals opposite angles sum up to $\pi$. The center of the
circle is the intersection point of bisectors of all the four sides of the
quadrilateral

In the spirit of works [68, 69] one can conclude that the integrable discrete
analogs of orthogonal conjugate nets is provided by circular lattices.

######  Definition 4.1.

The integrable discrete analogue of orthogonal conjugate net is a map
$\boldsymbol{r}\colon{\mathbb{Z}}^{N}\to{\mathbb{E}}^{M}$ of $N$-dimensional
integer lattice such that for arbitrary $\boldsymbol{n}\in{\mathbb{Z}}^{N}$
and any two directions $a\neq b$ of the lattice, the vertices
$\boldsymbol{r}^{\boldsymbol{n}}$,
$\boldsymbol{r}^{\boldsymbol{n}+\boldsymbol{e}_{a}}$,
$\boldsymbol{r}^{\boldsymbol{n}+\boldsymbol{e}_{b}}$, and
$\boldsymbol{r}^{\boldsymbol{n}+\boldsymbol{e}_{a}+\boldsymbol{e}_{b}}$ of
elementary quadrilaterals are concircular.

Integrability of the circular reduction of lattices of planar quadrilaterals
follows from the fact that the circularity constraint is preserved [17] by
evolution of the lattices expressed by the extended permutability theorem,
i.e. if the three quadrilaterals with vertices

$\displaystyle\\{\boldsymbol{r},\boldsymbol{r}^{\\{a\\}},\boldsymbol{r}^{\\{b\\}},\boldsymbol{r}^{\\{a,b\\}}\\},\qquad\\{\boldsymbol{r},\boldsymbol{r}^{\\{a\\}},\boldsymbol{r}^{\\{c\\}},\boldsymbol{r}^{\\{a,c\\}}\\},\qquad\\{\boldsymbol{r},\boldsymbol{r}^{\\{b\\}},\boldsymbol{r}^{\\{c\\}},\boldsymbol{r}^{\\{b,c\\}}\\},\qquad$

are circular, then the three circles through the triplets

$\\{\boldsymbol{r}^{\\{a\\}},\boldsymbol{r}^{\\{a,b\\}},\boldsymbol{r}^{\\{a,c\\}}\\},\qquad\\{\boldsymbol{r}^{\\{a\\}},\boldsymbol{r}^{\\{a,b\\}},\boldsymbol{r}^{\\{a,c\\}}\\},\qquad\\{\boldsymbol{r}^{\\{a\\}},\boldsymbol{r}^{\\{a,b\\}},\boldsymbol{r}^{\\{a,c\\}}\\},$

intersect in the point $\boldsymbol{r}^{\\{a,b,c\\}}$, what is equivalent to
the Miquel theorem of elementary geometry. In view of multidimensional
consistency of lattices of planar quadrilaterals this implies the
multidimensional consistency of circular lattices. Soon after geometric proof
of their integrability, it was confirmed by the non-local $\bar{\partial}$
dressing technique in [36], the algebro-geometric construction [4], by
construction of the corresponding reduction of the fundamental transformation
[26, 75], and within the free-fermion description of the KP hierarchy [39].

Integrability of other basic reductions of the multidimensional lattice of
planar quadrilaterals was investigated in [35], see also [13]. We remark that
transformations [89] of the system of discrete Moutard equations (2.20) and of
the corresponding Miwa’s discrete BKP system (2.23) can be obtained as an
integrable reduction of the fundamental transformation [28]. For review of the
theory and many other geometric aspects of integrable discrete systems see
[13]. Interesting applications of the theory to architectural design and
computer graphics are discussed in [94, 95].

##  5 Conclusion and open problems

We presented interpretation of known results in the theory of discrete
asymptotic and discrete conjugate nets from the _discretization by Bäcklund
transformations_ point of view. We collected both classical formulas of XIXth
century differential geometry of surfaces and their transformations, and more
recent results from geometric theory of integrable discrete equations.
Darboux–Bäcklund transformations of difference operators are reviewed in [33].
Old ideas of differential geometry of surfaces relevant to integrability are
still sources of inspiration for contemporary research, see for example [15,
23].

The theory of multidimensional discrete conjugate nets is based on the simple
geometric principle of planarity of elementary quadrilaterals. Their
integrable reductions often come from geometric understanding, in the spirit
of Klein, of various reductions of projective geometry by introducing absolute
objects and corresponding restrictions of the group of projective
transformations. Moreover, basic analytic tools of the theory of integrable
systems, like the non-local $\bar{\partial}$-dressing method and the algebro-
geometric techniques, could be applied to construct such lattices and
corresponding solutions to discrete Darboux equations in a rather pure form.
It turns out however, surprisingly, that the principle of _coplanarity of four
points_ can be replaced, without loss of generality from the integrability
viewpoint, by the condition of _collinearity of three points_ [30]. The
multidimensional consistency of such lattices is equivalent to the Desargues
theorem of projective geometry. The collinearity condition for such
$N$-dimensional Desargues lattice in natural homogeneous coordinates of the
projective space can be rewritten in the form

$\boldsymbol{\psi}^{\boldsymbol{n}+\boldsymbol{e}_{i}}-\boldsymbol{\psi}^{\boldsymbol{n}+\boldsymbol{e}_{j}}=u^{\boldsymbol{n}}_{ij}\boldsymbol{\psi}^{\boldsymbol{n}},\qquad
1\leq i\neq j\leq N.$ (5.1)

The compatibility condition

$u^{\boldsymbol{n}}_{ji}+u^{\boldsymbol{n}}_{ij}=0,\qquad
u^{\boldsymbol{n}}_{ij}+u^{\boldsymbol{n}}_{jk}+u^{\boldsymbol{n}}_{ki}=0,\qquad
u^{\boldsymbol{n}}_{ij}u^{\boldsymbol{n}+\boldsymbol{e}_{j}}_{ik}=u^{\boldsymbol{n}}_{ik}u^{\boldsymbol{n}+\boldsymbol{e}_{k}}_{ij},\qquad
i,j,k\quad\text{disctinct}$ (5.2)

can be simplified by introducing the single potential $\tau$ such that

$u^{\boldsymbol{n}}_{ij}=\frac{\tau^{\boldsymbol{n}+\boldsymbol{e}_{i}+\boldsymbol{e}_{j}}\tau^{\boldsymbol{n}}}{\tau^{\boldsymbol{n}+\boldsymbol{e}_{i}}\tau^{\boldsymbol{n}+\boldsymbol{e}_{j}}}=-u^{\boldsymbol{n}}_{ji},\qquad
i<j,$ (5.3)

which satisfies the system of Hirota equations [54, 82]

$\tau^{\boldsymbol{n}+\boldsymbol{e}_{i}}\tau^{\boldsymbol{n}+\boldsymbol{e}_{j}+\boldsymbol{e}_{k}}-\tau^{\boldsymbol{n}+\boldsymbol{e}_{j}}\tau^{\boldsymbol{n}+\boldsymbol{e}_{i}+\boldsymbol{e}_{k}}+\tau^{\boldsymbol{n}+\boldsymbol{e}_{k}}\tau^{\boldsymbol{n}+\boldsymbol{e}_{i}+\boldsymbol{e}_{j}}=0,\qquad
i<j<k.$ (5.4)

As it was shown in [30] the $2N-1$-dimensional Hirota system is equivalent to
discrete Darboux equations of $N$-dimensional discrete conjugate net
supplemented by $(N-1)$ dimensional lattice of its Laplace transformations.

The Hirota system (5.4) is probably the most important discrete integrable
system both from theoretical [82] and practical [62] viewpoint. In particular,
its multidimensional consistency gives rise to the full hierarchy of commuting
symmetries of the KP equation [22]. Originally it was called the discrete Toda
system, which is another sign of a general rule that single integrable
discrete equation can lead, via different limits, to many differential
equations. In fact, the original name is related to the theory of two-
dimensional lattices of planar quadrilaterals and their Laplace transforms
[25], while the other name is related to the Desargues lattices [30].
Darboux–Bäcklund transformations of the Hirota system were studied in [87,
88], where the fundamental transformation of the theory of discrete conjugate
nets appears under the name of binary Darboux transformation (the Lévy
transformation is called there the elementary Darboux transformation).

Among distinguished integrable reductions of the KP hierarchy of equations
there is the BKP hierarchy, which is encompassed by the single Miwa equation
(2.23). The CKP hierarchy [21] leads, via the Darboux–Bäcklund transformations
to the corresponding reduction [58, 104] of the Hirota system, see also [35,
29] for geometry of the corresponding reduction of the discrete conjugate
nets. In parallel to the reductions of the KP hierarchy, which is based on
restrictions of the Lie algebra $\mathfrak{gl}(\infty)$ to its orthogonal and
symplectic subalgebras $\mathfrak{so}(\infty)$ and $\mathfrak{sp}(\infty)$, on
the discrete level one has the corresponding root lattices and their affine
Weyl group interpretations [31, 32] of the Hirota system, the Miwa (discrete
BKP) and the Kashaev (discrete CKP) equations. Many of known integrable
systems, both discrete and continuous, can be obtained as its further
reductions, see [90, 99] for applications of the root lattices and affine Weyl
groups to understand Painlevé equations. It may seem therefore that integrable
discrete systems are enough to fully understand integrability.

There is however a distinguished class of integrable differential equations,
called dispersionless systems [76, 16, 42, 43, 61, 107], which escapes the
above interpretation. The distinguished examples of such systems are the so
called generalized heavenly equations [106, 41], which describe self-dual
Einstein spaces [93], or the self-dual Yang–Mills equations [2], and both of
them are genuine four dimensional.

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