Discrete Differential Geometry: Integrable Structure (Graduate Studies in Mathematics 98)

Alexander I. Bobenko Yuri B. Suris Discrete Differential Geometry Integrable Structure Alexander I. Bobenko Yuri B...

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Alexander I. Bobenko

Yuri B. Suris

Discrete Differential Geometry

Integrable Structure Alexander I. Bobenko

Yuri B. Suris

Graduate Studies in Mathematics Volume 98

American Mathematical Society Providence, Rhode Island

Editorial Board

David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlernann 2000 Mathematics Subject Classification. Primary 53-01, 53-02; Secondary 5lAxx, 5IBxx, 53Axx, 37Kxx, 39A12, 52C26.

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www.ams.org/bookpages/gsm-98

Library of Congress Cataloging-in-Publication Data Bobenko, Alexander 1. Discrete differential geometry: integrable structure / Alexander 1. Bobenko, Yuri B. Suris. p. cm. - (Graduate studies in mathematics; v. 98) Includes bibliographical references and index. ISBN 978-0-8218-4700-8 (alk. paper) 1. Integral geometry. 2. Geometry, Differential. 3. Discrete geometry. 1. Suris, Yuri B., 1963- II. Title. QA672.B63 2008 516.3'62---dc22

2008029305

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissionlCams. ~rg.

© 2008

by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America.

§

The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www . ams .org/ 10987654321

13 12 11 10 09 08

Contents

Preface

Xl

Introduction What is discrete differential geometry Integrability From discrete to smooth Structure of this book How to read this book Acknowledgements Chapter 1. Classical Differential Geometry 1.1.

Conjugate nets 1.1.1. Notion of conjugate nets 1.1.2. Alternative analytic description of conjugate nets 1.1.3. Transformations of conjugate nets 1.1.4. Classical formulation of F-transformation

1.2.

Koenigs and Moutard nets 1.2.1. Notion of Koenigs and Moutard nets 1.2.2. Transformations of Koenigs and Moutard nets 1.2.3. Classical formulation of the Moutard transformation

Xlll

xiii xv XVll

XXI

xxii xxiii

1

2 2 3 4 5 7 7 9 10

1.3. Asymptotic nets

11

1.4. Orthogonal nets 1.4.1. Notion of orthogonal nets 1.4.2. Analytic description of orthogonal nets 1.4.3. Spinor frames of orthogonal nets 1.4.4. Curvatures of surfaces and curvature line parametrized surfaces

12 12 14

15 16

-

v

Contents

vi

1.4.5.

Ribaucour transformations of orthogonal nets

17

1.5. Principally parametrized sphere congruences

19

1.6. Surfaces with constant negative Gaussian curvature

20

1.7. Isothermic surfaces

22

1.8. Surfaces with constant mean curvature

26

1.9. Bibliographical notes

28

Chapter 2. Discretization Principles. Multidimensional Nets

31

2.1.

Discrete conjugate nets (Q-nets) 2.1.1. Notion and consistency of Q-nets 2.1.2. Transformations of Q-nets 2.1.3. Alternative analytic description of Q-nets 2.1.4. Continuous limit

32 32 38 40 42

2.2.

Discrete line congruences

43

2.3.

Discrete Koenigs and Moutard nets 2.3.1. Notion of dual quadrilaterals 2.3.2. Notion of discrete Koenigs nets 2.3.3. Geometric characterization of two-dimensional discrete Koenigs nets 2.3.4. Geometric characterization of three-dimensional discrete Koenigs nets 2.3.5. Function v and Christoffel duality 2.3.6. Moutard representative of a discrete Koenigs net 2.3.7. Continuous limit 2.3.8. Notion and consistency of T-nets 2.3.9. Transformations of T-nets 2.3.10. Discrete M-nets

47 47 49

2.4.

Discrete asymptotic nets 2.4.1. Notion and consistency of discrete asymptotic nets 2.4.2. Discrete Lelieuvre representation 2.4.3. Transformations of discrete A-nets

66 66 70 72

2.5.

Exercises

73 82

2.6. Bibliographical notes Chapter 3. Discretization Principles. Nets in Quadrics 3.1.

Circular nets 3.1.1. Notion and consistency of circular nets 3.1.2. Transformations of circular nets 3.1.3. Analytic description of circular nets 3.1.4. Mobius-geometric description of circular nets

54 56 58 60 60 61 63 65

87 88 88 92 93 96

Contents

Vll

3.2.

Q-nets in quadrics

99

3.3.

Discrete line congruences in quadrics

101

3.4.

Conical nets

103

3.5.

Principal contact element nets

106

3.6.

Q-congruences of spheres

110

3.7.

Ribaucour congruences of spheres

113

3.8.

Discrete curvature line parametrization in Lie, Mobius and Laguerre geometries

115

3.9. Discrete asymptotic nets in Plucker line geometry

118

3.10. Exercises

120


123

Chapter 4. Special Classes of Discrete Surfaces

127

4.1.

Discrete Moutard nets in quadrics

127

4.2.

Discrete K-nets 4.2.1. Notion of a discrete K-net 4.2.2. Backlund transformation 4.2.3. Hirota equation 4.2.4. Discrete zero curvature representation 4.2.5. Discrete K-surfaces 4.2.6. Discrete sine-Gordon equation

130 130 133 133 139 139 142

4.3.

Discrete isothermic nets 4.3.1. Notion of a discrete isothermic net 4.3.2. Cross-ratio characterization of discrete isothermic nets 4.3.3. Darboux transformation of discrete isothermic nets 4.3.4. Metric of a discrete isothermic net 4.3.5. Moutard representatives of discrete isothermic nets 4.3.6. Christoffel duality for discrete isothermic nets 4.3.7. 3D consistency and zero curvature representation 4.3.8. Continuous limit

145 145 147 151 152 155 156 158 160

4.4.

S-isothermic nets

161

4.5.

Discrete surfaces with constant curvature 4.5.1. Parallel discrete surfaces and line congruences 4.5.2. Polygons with parallel edges and mixed area 4.5.3. Curvatures of a polyhedral surface with a parallel Gauss map 4.5.4. Q-nets with constant curvature 4.5.5. Curvature of principal contact element nets

170 170 170 173 175 177

viii

Contents

4.5.6.

Circular minimal nets and nets with constant mean curvature

178

4.6. Exercises

179

4.7.

183

Bibliographical notes

Chapter 5. Approximation

187

5.1. Discrete hyperbolic systems

187

5.2.

Approximation in discrete hyperbolic systems

190

5.3.

Convergence of Q-nets

196

5.4.

Convergence of discrete Moutard nets

197

5.5. Convergence of discrete asymptotic nets

199

5.6.

Convergence of circular nets

200

5.7.

Convergence of discrete K-surfaces

205

5.8. Exercises

206


207

Chapter 6. Consistency as Integrability

209

6.1.

Continuous integrable systems

210

6.2.

Discrete integrable systems

213

6.3.

Discrete 2D integrable systems on graphs

215

6.4.

Discrete Laplace type equations

217

6.5.

Quad-graphs

218

6.6.

Three-dimensional consistency

220

6.7.

From 3D consistency to zero curvature representations and Backlund transformations

222

Geometry of boundary value problems for integrable 2D equations 6.8.1. Initial value problem 6.8.2. Extension to a multidimensional lattice

227 228 231

3D consistent equations with noncommutative fields

235

6.10. Classification of discrete integrable 2D systems with fields on vertices. I

239

6.8.

6.9.

6.11. Proof of the classification theorem 6.11.1. 3D consistent systems, biquadratics and tetrahedron property 6.11.2. Analysis: descending from multi affine Q to quartic r 6.11.3. Synthesis: ascending from quartic r to biquadratic h

242 242 245

247

Contents

x

9.2.4.

Planar families of spheres; Dupin cyclides

340

9.3.

Mobius 9.3.1. 9.3.2. 9.3.3.

geometry Objects of Mobius geometry Projective model of Mobius geometry Mobius transformations

341 341 344 348

9.4.

Laguerre geometry

350

9.5.

Plucker line geometry

353

9.6. Incidence theorems 9.6.1. Menelaus' and Ceva's theorems 9.6.2. Generalized Menelaus' theorem 9.6.3. Desargues' theorem 9.6.4. Quadrangular sets 9.6.5. Carnot's and Pascal's theorems 9.6.6. Brianchon's theorem 9.6.7. Miquel's theorem Appendix. Solutions of Selected Exercises

357 357 360 361 362 364 366 367 369

A.1.

Solutions of exercises to Chapter 2

369

A.2.


376

A.3.


377

A.4.


381

Bibliography

385

Notation

399

Index

401

Preface

The intended audience of this book is threefold. We wrote it as a textbook on discrete differential geometry and integrable systems. A one semester graduate course in discrete differential geometry based on this book was held at TV Berlin and TV Miinchen several times. At the end of each chapter we included numerous exercises which we recommend for the classes. For some of them (marked with asterisks) solutions are supplied. The standard undergraduate background, i.e., calculus and linear algebra, is required. In particular, no knowledge of differential geometry is expected, although some familiarity with curves and surfaces can be helpful. On the other hand, this book is also written for specialists in geometry and mathematical physics. It is the first monograph on discrete differential geometry which reflects the progress in this field during the last decade, and it contains many original results. The bibliographical notes at the end of each chapter are intended to provide the reader with an overview of the relevant research literature. The third group at which this book is targeted are specialists in geometry processing, computer graphics, architectural design, numerical simulations and animation. There is a growing evidence of the importance of intelligent geometric discretizations in these fields. Talking with researchers in these fields, we were asked many questions regarding the discretization of differential geometry. We hope to have answered some of them in this book. AU the readers are encouraged to read or at least to skim the Introduction (some parts of it assume a broader knowledge than the minimum) to see the words and pictures and to get a sense of how the ideas fit together and what does the book cover.

-

xi

Introduction

What is discrete differential geometry. A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry; see, for instance, the recent book Bobenko-SchroderSullivan-Ziegler (2008). Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra), discrete differential geometry aims at the development of discrete equivalents of notions and methods of smooth surface theory. The latter appears as a limit of refinement of the discretization. Current interest in this field derives not only from its importance in pure mathematics but also from its relevance for other fields: see the lecture course on discrete differential geometry in computer graphics by Desbrun-Grinspun-Schroder (2005), the recent book on architectural geometry by Pottmann-Asperl-Hofer-Kilian (2007), and the mathematical video on polyhedral meshes and their role in geometry, numerics and computer graphics by Janzen-Polthier (2007). For a given smooth geometry one can suggest many different discretizations with the same continuous limit. Which is the best one? From the theoretical point of view, one would strive to preserve fundamental properties of the smooth theory. For applications the requirements of a good discretization are different: one aims at the best approximation of a smooth shape, on the one hand, and at on the other hand, its representation by a discrete shape with as few elements as possible. Although these criteria are different, it turns out that intelligent theoretical discretizations are distinguished also by their good performance in applications. We mention here as an example the discrete Laplace operator on simplicial surfaces ("cotan formula") introduced by Pinkall-Polthier (1993) in their investigation of discrete minimal

-

xiii

xiv

Introduction

surfaces, which turned out to be extremely important in geometry processing where it found numerous applications, e.g., Desbrun-Meyer-Alliez (2002), Botsch-Kobbelt (2004), to name but two. Another example is the theory of discrete minimal surfaces by Bobenko-Hoffmann-Springborn (2006), which turned out to have striking convergence properties: these discrete surfaces approximate their smooth analogs with all derivatives. A straightforward way to discretize differential geometry would be to take its analytic description in terms of differential equations and to apply standard methods of numerical analysis. Such a discretization makes smooth problems amenable to numerical methods. Discrete differential geometry does not proceed in this way. Its main message is:

Discretize the whole theory, not just the equations. This means that one should develop a discrete theory which respects fundamental aspects of the smooth one; which of the properties are to be taken into account is a nontrivial problem. The discrete geometric theory turns out to be as rich as its smooth counterpart, if not even richer. In particular, there are many famous existence theorems at the core of the classical theory. Proper discretizations open a way to make them constructive. For now, the statement about the richness of discrete differential geometry might seem exaggerated, as the number of supporting examples is restricted (although steadily growing). However, one should not forget that we are at the beginning of the development of this discipline, while classical differential geometry has been developed for centuries by the most outstanding mathematicians. As soon as one takes advantage of the apparatus of differential equations to describe geometry, one naturally deals with parametrizations. There is a part of classical differential geometry dealing with parametrized surfaces, coordinate systems and their transformations, which is the content of the fundamental treatises by DarboliX (1914-27) and Bianchi (1923). Nowadays one associates this part of differential geometry with the theory of integrable systems; see Fordy-Wood (1994), Rogers-Schief (2002). Recent progress in discrete differential geometry has led not only to the discretization of a large body of classical results, but also, somewhat unexpectedly, to a better understanding of some fundamental structures at the very basis of the classical differential geometry and of the theory of integrable systems. It is the aim of this book to provide a systematic presentation of current achievements in this field. Returning to the analytic description of geometric objects, it is not surprising that remarkable discretizations yield remarkable discrete equations.

Introduction

xv

The main message of discrete differential geometry, addressed to the integrable systems community, becomes:

Discretize equations by discretizing the geometry. The profundity and fruitfulness of this principle will be demonstrated throughout the book.

Integrability. We will now give a short overview of the historical development of the integrability aspects of discrete differential geometry. The classical period of surface theory resulted in the beginning of the 20th century in an enormous wealth of knowledge about numerous special classes of surfaces, coordinate systems and their transformations, which is summarized in extensive volumes by Darboux (1910, 1914-27), Bianchi (1923), etc. One can say that the local differential geometry of special classes of surfaces and coordinate systems has been completed during this period. Mathematicians of that era have found most (if not all) geometries of interest and knew nearly everything about their properties. It was observed that special geometries such as minimal surfaces, surfaces with constant curvature, isothermic surfaces, orthogonal and conjugate coordinate systems, Ribaucour sphere congruences, Weingarten line congruences etc. have many similar features. Among others we mention Backlund and Darboux type transformations with remarkable permutability properties investigated mainly by Bianchi, and the existence of special deformations within the class (associated family). Geometers realized that there should be a unifying fundamental structure behind all these common properties of quite different geometries; and they were definitely searching for this structure; see Jonas (1915) and Eisenhart (1923). Much later, after the advent of the theory of integrable systems in the the last quarter of the 20th century, these common features were recognized as being associated with the integrability of the underlying differential equations. The theory of integrable systems (called also the theory of solitons) is a vast field in mathematical physics with a huge literature. It has applications in fields ranging from algebraic and differential geometry, enumerative topology, statistical physics, quantum groups and knot theory to nonlinear optics, hydrodynamics and cosmology. The most famous models of this theory are the Korteweg-de Vries (KdV), the nonlinear Schrodinger and the sine-Gordon equations. The KdV equation played the most prominent role in the early stage of the theory. It was derived by Korteweg-de Vries (1895) to describe the propagation of waves in shallow water. Localized solutions of this equation called solitons gave the whole theory its name. The birth of the theory of solitons is associated with the famous paper by Gardner-Green-Kruskal-Miura (1967), where the inverse scattering method for the analytic treatment of the KdV equation

XVI

Introduction

was invented. The sine-Gordon equation is the oldest integrable equation and the most important one for geometry. It describes surfaces with constant negative Gaussian curvature and goes back at least to Bour (1862) and Bonnet (1867). Many properties of this equation which are nowadays associated with integrability were known in classical surface theory. One can read about the basic structures of the theory of integrable systems in numerous books. We mention just a few of them: Newell (1985), Faddeev-Takhtajan (1986), Hitchin-Segal-Ward (1999), Dubrovin-Krichever-Novikov (2001). The most commonly accepted features of integrable systems include: In the theory of solitons nonlinear integrable equations are usually represented as a compatibility condition of a linear system called the zero curvature representation (also known as Lax or ZakharovShabat representations). Various analytic methods of investigation of soliton equations (like the inverse scattering method, algebrogeometric integration, asymptotic analysis, etc.) are based on this representation. Another indispensable feature of integrable systems is that they possess Biicklund-Darboux transformations. These special transformations are often used to generate new solutions from the known ones. It is a characteristic feature of soliton (integrable) partial differential equations that they appear not separately but are always organized in hierarchies of commuting flows. It should be mentioned that there is no commonly accepted mathematical definition of integrability (as the title of the volume "What is integrability?", Zakharov (1991), clearly demonstrates). Different scientists suggest different properties as the defining ones. Usually, one refers to some additional structures, such as those mentioned above. In this book, we propose an algorithmic definition of integrability given in terms of the system itself.

In both areas, in differential geometry and in the theory of integrable systems, there were substantial efforts to discretize the fundamental structures. In the theory of solitons the problem is to discretize an integrable differential equation preserving its integrability. Various approaches to this problem began to be discussed in the soliton literature starting from the mid-1970s. The basic idea is to discretize the zero curvature representation of the smooth system, i.e., to find proper discrete analogues of the corresponding linear problems. This idea appeared first in Ablowitz-Ladik (1975).

Introduction

XVll

Its various realizations based on the bilinear method, algebro-geometric integration, integral equations, R-matrices, and Lagrangian mechanics were developed in Hirota (1977a,b), Krichever (1978), Date-Jimbo-Miwa (1982-3), Quispel-Nijhoff-Capel-Van der Linden (1984), Faddeev-Takhtajan (1986), Moser-Veselov (1991) (here we give just a few representative references). An encyclopedic presentation of the Hamiltonian approach to the problem of integrable discretization is given in Suris (2003). The development of this field led to a progress in various branches of mathematics. Pairs of commuting difference operators were classified in Krichever-Novikov (2003). Laplace transformations of difference operators on regular lattices were constructed in Dynnikov-Novikov (1997); see also Dynnikov-Novikov (2003) for a related development of a discrete complex analysis on triangulated manifolds. A characterization of Jacobians of algebraic curves based on algebro-geometric methods of integration of difference equations was given in Krichever (2006).

From discrete to smooth. In differential geometry the original idea of an intelligent discretization was to find a simple explanation of sophisticated properties of smooth geometric objects. This was the main motivation for the early work in this field documented in Sauer (1937, 1970) and Wunderlich (1951). The modern period began with the works by Bobenko-Pinkall (1996a,b) and by Doliwa-Santini (1997), where the relation to the theory of integrable systems was established. During the next decade this area witnessed a rapid development reflected in numerous publications. In particular, joint efforts of the main contributors to this field resulted in the books Bobenko-Seiler (1999) and Bobenko-Schroder-Sullivan-Ziegler (2008). The present book gives a comprehensive presentation of the results of discrete differential geometry of parametrized surfaces and coordinate systems along with its relation to integrable systems. We leave the detailed bibliographical remarks to the notes at the end of individual chapters of the book. Discrete differential geometry deals with multidimensional discrete nets (Le., maps from the regular cubic lattice zm into]RN or some other suitable space) specified by certain geometric properties. 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 considered on an equal footing with respect to the defining geometric properties. Due to this symmetry, discrete surfaces and their transformations become indistinguishable. We associate such a situation with the multidimensional consistency (of geometric properties, and of the equations which serve for their analytic description). In each case, multidimensional consistency, and therefore the existence and construction of multidimensional

Introduction

xviii

discrete nets, is seen to rely on some incidence theorems of elementary geometry. Conceptually, one can think of passing to a continuous limit by refining the mesh size in some of the lattice directions. In these directions the net converges to smooth surfaces whereas those directions that remain discrete correspond to transformations of the surfaces (see Figure 0.1). Differential geometric properties of special classes of surfaces and their transformations arise in this way from (and find their simple explanation in) the elementary geometric properties of the original multidimensional discrete nets. In particular, difficult classical theorems about the permutability of BiicklundDarboux type transformations (Bianchi permutability) for various geometries follow directly from the symmetry of the underlying discrete nets, and are therefore built in to the very core of the theory. Thus the transition from differential geometry to elementary geometry via discretization (or, in the opposite direction, the derivation of differential geometry from the discrete differential geometry) leads to enormous conceptual simplifications, and the true roots of the classical theory of special classes of surfaces are found in various incidence theorems of elementary geometry. In the classical differential geometry these elementary roots remain hidden. The limiting process taking the discrete master theory to the classical one is inevitably accompanied by a break of the symmetry among the lattice directions, which always leads to structural complications.

Figure 0.1. From the discrete master theory to the classical theory: surfaces and their transformations appear by refining two of three net directions.

Finding simple discrete explanations for complicated differential-geometric theories is not the only outcome of this development. It is well known that differential equations which analytically describe interesting special classes of surfaces are integrable (in the sense of the theory of integrable systems),

Introduction

xix

and conversely, many interesting integrable systems admit a differentialgeometric interpretation. Having identified the roots of integrable differential geometry in the multidimensional consistency of discrete nets, one is led to a new (geometric) understanding of integrability itself. First of all, we adopt the point of view that the central role in this theory is played by discrete integrable systems. In particular, a great variety of integrable differential equations can be derived from several fundamental discrete systems by performing different continuous limits. Further, and more importantly, we arrive at the idea that the multidimensional consistency of discrete equations may serve as a constructive and almost algorithmic definition of their integrability. This idea was introduced in Bobenko-Suris (2002a) (and independently in Nijhoff (2002)). This definition of integrability captures enough structure to guarantee such traditional attributes of integrable equations as zero curvature representations and Biicklund-Darboux transformations (which, in turn, serve as the basis for applying analytic methods such as inverse scattering, finite gap integration, Riemann-Hilbert problems, etc.). A continuous counterpart (and consequence) of multidimensional consistency is the well-known fact that integrable systems never appear alone but are organized into hierarchies of commuting flows. This conceptual view of discrete differential geometry as the basis of the theory of surfaces and their transformations as well as of the theory of integrable systems is schematically represented in Figure 0.2. This general picture looks very natural, and there is a common belief that the smooth theories can be obtained in a limit from the corresponding discrete ones. This belief is supported by formal similarities of the corresponding difference and differential equations. However one should not underestimate the difficulty of the convergence theorems required for a rigorous justification of this philosophy. Solutions to similar problems are substantial in various areas of differential geometry. Classical examples to be mentioned here are the fundamental results of Alexandrov and Pogorelov on the metric geometry of polyhedra and convex surfaces (see Alexandrov (2005) and Pogorelov (1973)). Alexandrov's theorem states that any abstract convex polyhedral metric is uniquely realized by a convex polyhedron in Euclidean 3-space. Pogorelov proved the corresponding existence and uniqueness result for convex Riemannian metrics by approximating smooth surfaces by polyhedra. Another example is Thurston's approximation of conformal mappings by circle packings (see Thurston (1985)). The theory of circle packings (see the book by Stephenson (2005)) is treated as discrete complex analysis. At the core of this theory is the Koebe-Andreev-Thurston theorem which states that any simplicial decomposition of a sphere can be uniquely (up to Mobius transformations)

Introduction

xx

Differential Geometry

surfaces

Discrete Differential Geometry

D

Integrability

===}-

integrable equations

===}-

zero-curvature representation BacklundDarboux transformations

===}-

hierarchies of commuting flows

discrete nets

/

J---- -

/

surfaces and their transformations

Bianchi permutability

CONSISTENCY

multidimensional consistency

Figure 0.2. The consistency principle of discrete differential geometry as conceptual basis of the differential geometry of special surfaces and of integrability.

realized by a circle packing. According to Rodin-Sullivan (1987) the conformal Riemann map can be approximated by such circle packings (even with all the derivatives as shown by He-Schramm (1998)). The first convergence results concerning the transition from the middle to the left column in Figure 0.2 (from discrete to smooth differential geometry) were proven in Bobenko-Matthes-Suris (2003, 2005). This turns the general philosophy of discrete differential geometry into a firmly established

Introduction

XXI

mathematical truth for several important classes of surfaces and coordinate systems, such as conjugate nets, orthogonal nets, including general curvature line parametrized surfaces, surfaces with constant negative Gaussian curvature, and general asymptotic line parametrized surfaces. For some other classes, such as isothermic surfaces, the convergence results are yet to be rigorously established. The geometric way of thinking about discrete integrability has also led to novel concepts in that theory. An immanent and important feature of various surface parametrizations is the existence of distinguished points, where the combinatorics of coordinate lines changes (like umbilic points, where the combinatorics of the curvature lines is special). In the discrete setup this can be modelled by quad-graphs, which are cell decompositions of topological two-manifolds with quadrilateral faces; see Bobenko-Pinkall (1999). Their elementary building blocks are still quadrilaterals, but they are attached to one another in a manner which can be more complicated than in 'Ii}. A systematic development of the theory of integrable systems on quad-graphs has been undertaken in Bobenko-Suris (2002a). In the framework of the multidimensional consistency, quad-graphs can be realized as quad-surfaces embedded in a higher-dimensional lattice 7l,d. This interpretation proves to be fruitful for the analytic treatment of integrable systems on quad-graphs, such as the integral representation of discrete holomorphic functions and the isomonodromic Green's function in Bobenko-Mercat-Suris (2005).

Structure of this book. The structure of this book follows the logic of this Introduction. We start in Chapter 1 with an overview of some classical results from surface theory, focusing on transformations of surfaces. The brief presentation in this chapter is oriented towards the specialists already familiar with the differential geometry of surfaces. The geometries considered include general conjugate and orthogonal nets in spaces of arbitrary dimension, Koenigs nets, asymptotic nets on general surfaces, as well as special classes of surfaces, such as isothermic ones and surfaces with constant negative Gaussian curvature. There are no proofs in this chapter. The analytic proofs are usually tedious and can be found in the original literature. The discrete approach which we develop in the subsequent chapters will lead to conceptually transparent and technically much simpler proofs. In Chapter 2 we define and investigate discrete analogs of the most fundamental objects of projective differential geometry: conjugate, Koenigs and asymptotic nets and line congruences. For instance, discrete conjugate nets are just multidimensional nets consisting of planar quadrilaterals. Our focus is on the idea of multidimensional consistency of discrete nets and discrete line congruences.

xxii

Introduction

According to Klein's Erlangen Program, the classical geometries (Euclidean, spherical, hyperbolic, Mobius, Plucker, Lie etc.) can be obtained by restricting the projective geometry to a quadric. In Chapter 3 we follow this approach and show that the nets and congruences defined in Chapter 2 can be restricted to quadrics. In this way we define and investigate discrete analogs of curvature line parametrized surfaces and orthogonal nets, and give a description of discrete asymptotic nets within the framework of Plucker line geometry. Imposing simultaneously several constraints on (discrete) conjugate nets, one comes to special classes of surfaces. This is the subject of Chapter 4. The main examples are discrete isothennic surfaces and discrete surfaces with constant curvature. From the analytic point of view, these are represented by 2-dimensional difference equations (as opposed to the 3-dimensional equations in Chapters 2, 3). Then in Chapter 5 we develop an approximation theory for hyperbolic difference systems, which is applied to derive the classical theory of smooth surfaces as a continuum limit of the discrete theory. We prove that the discrete nets of Chapters 2, 3, and 4 approximate the corresponding smooth geometries of Chapter 1 and simultaneously their transformations. In this setup, Bianchi's permutability theorems appear as simple corollaries. In Chapter 6 we formulate the concept of multidimensional consistency as a defining principle of integrability. We derive basic features of integrable systems such as the zero curvature representation and Bii,cklund-Darboux transformations from the consistency principle. Moreover, we obtain a complete list of 2-dimensional integrable systems. This classification is a striking application of the consistency principle. In Chapters 7 and 8 these ideas are applied to discrete complex analysis. We study Laplace operators on graphs and discrete harmonic and holomorphic functions. Linear discrete complex analysis appears as a linearization of the theory of circle patterns. The consistency principle allows us to single out distinguished cases where we obtain more detailed analytic results (like Green's function and isomonodromic special functions). Finally, in Chapter 9 we give for the reader's convenience a brief introduction to projective geometry and the geometries of Lie, Mobius, Laguerre and Plucker. We also include a number of classical incidence theorems relevant to discrete differential geometry. How to read this book. Different audiences (see the Preface) should read this book differently, as suggested in Figure 0.3. Namely, Chapter 1 on classical differential geometry is addressed to specialists working in this field. It is thought to be used as a short guide in the theory of surfaces and their

Introduction

XXlll

transformations. This is the reason why Chapter 1 does not contain proofs and exercises. Students who use this book for a graduate course and have less or no experience in differential geometry should not read this chapter and should start directly with Chapter 2 (and consult Chapter 1 at the end of the course, after mastering the discrete theory). This was the way how this course was taught in Berlin and Munchen, with no knowledge of differential geometry required. Those interested primarily in applications of discrete differential geometry are advised to browse through Chapters 2-4 and perhaps also Chapter 5 and to pick up the problems they are particularly interested in. Almost all results are supplied with elementary geometric formulations accessible for nonspecialists. Finally, researchers with interest in the theory of integrable systems could start reading with Chapter 6 and consult the previous chapters for better understanding of the geometric origin of the consistency approach to integrability. graduate course for those interested in integrable systems

for differential geometers

2,3,4,9

7,8 Figure 0.3. A suggestion for the focus on chapters, depending on the readers background.

Acknowledgements. Essential parts of this book are based on results obtained jointly with Vsevolod Adler, Tim Hoffmann, Daniel Matthes, Christian Mercat, Ulrich Pinkall, Helmut Pottmann, and Johannes Wallner. We warmly thank them for inspiring collaboration. We are very grateful to Adam Doliwa, Udo Hertrich-Jeromin, Nicolai Reshetikhin, Wolfgang Schief, Peter Schroder, Boris Springborn, Sergey Tsarev, Alexander Veselov, Gunter Ziegler for enjoyable and insightful discussions on discrete differential geometry which influenced the presentation in this book. Special thanks go to Emanuel Huhnen-Venedey and Stefan Sechelmann for their help with the preparation of the manuscript and with the figures. The support of the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged. During the work on this book the authors were partially

xxiv

Introduction

supported by the DFG Research Unit "Polyhedral Surfaces" and the DFG Research Center MATHEON "Mathematics for key technologies" in Berlin.

Chapter 1

Classical Differential Geometry

In this chapter we discuss some classical results of the differential geometry of nets (parametrized surfaces and coordinate systems) in ]RN, mainly concentrated around the topics of transformations of nets and of their permutability properties. This classical area was very popular in the differential geometry of the 19th and of the first quarter of the 20th century, and is well documented in the fundamental treatises by Bianchi, Darboux, Eisenhart and others. Our presentation mainly follows these classical treatments, of course with modifications which reflect our present points of view. We do not trace back the exact origin of the concrete classical results: often enough this turns out to be a complicated task in the history of mathematics, which still waits for its competent investigation. For the classes of nets described by essentially two-dimensional systems (special classes of surfaces such as surfaces with a constant negative Gaussian curvature or isothermic surfaces), the permutability theorems, mainly due to Bianchi, are dealing with a quadruple of surfaces (depicted as vertices of a so-called Bianchi quadrilateral). Given three surfaces of such a quadruple, the fourth one is uniquely defined; see Theorems 1.27 and 1.31. For the classes of nets described by essentially three-dimensional systems (conjugate nets; Moutard nets; asymptotic line parametrized surfaces; orthogonal nets, including curvature line parametrized surfaces), the situation is somewhat different. The corresponding permutability theorems (Theorems 1.3, 1.10, 1.15, and 1.20) consist of two parts. The first part of each theorem presents the traditional view and deals with Bianchi quadrilaterals. In our opinion, this is not the proper setting in the three-dimensional

-

1

1. Classical Differential Geometry

2

context, and the non uniqueness of the fourth net in these theorems reflects this. The natural setting for permutability is given in the second part, where the permutability is associated with an octuple of nets, depicted as vertices of a combinatorial cube, so that the eighth net is uniquely determined by the other seven (Eisenhart hexahedron). Our discrete philosophy makes the origin of such permutability theorems quite transparent. A few remarks on notation. We denote independent variables of a net ~ ]RN by u = (Ul, ... , um) E ]Rm, and we set 8i = 8/8ui. All nets are supposed to be sufficiently smooth, so that all the required partial derivatives exist. We write

f : ]Rm

13 i1 ... is ={UE]Rm:

Ui=Q

for i#-il, ... ,i s }

for s-dimensional coordinate planes (coordinate axes, if s = 1). 1.1. Conjugate nets 1.1.1. Notion of conjugate nets. We always suppose that the dimension of the ambient space N 2': 3.

Definition 1.1. (Conjugate net) A map f : ]Rm ~ ]RN is called an m-dimensional conjugate net in ]RN if at every U E ]Rm and for all pairs 1::; i #-j::; m we have 8i 8 j f E span(8d,8j J). Two-dimensional nets (m = 2) are nothing but parametrized surfaces. A parametrization of a surface in the three-space (m = 2, N = 3) is a conjugate net if its second fundamental form is diagonal. For a generic surface in the three-space, infinitely many such parametrizations can be found. A generic surface in the four-space carries an essentially unique conjugate net (uniqueness is understood here up to reparametrizations of coordinate lines). In higher-dimensional spaces such a parametrization does not need to exist at all (that is, only special surfaces of co dimension 2': 2 support conjugate nets). From Definition 1.1 it follows that the conjugate nets are described by the (linear) differential equations (1.1 ) with some functions Cij : ]Rm ~ R Compatibility of these equations, i.e. the requirement 8i(8j 8k!) = 8j(8i8kJ), is expressed by the following system of (nonlinear) differential equations:

(1.2)

8 i cjk

=

CijCjk

+ CjiCik -

CjkCik.

i #- j #- k #- i.

Note that the latter equations for the coefficients Cij do not contain f anymore. The system (1.1), (1.2) is hyperbolic (see Chapter 5); the following

1.1. Conjugate nets

3

data define a well-posed Goursat problem for this system and determine a conjugate net 1 uniquely: (Ql) the values of 1 on the coordinate axes 'Bi for 1 ~ i ~ m, i.e., m smooth curves Jr 13i with a common intersection point 1(0);

on the coordinate planes 'Bij for alII ~ i < j ~ m, 1) smooth real-valued functions cijf13 ij of two variables.

(Q2) the values of Cij, i.e.,

m(m -

Cji

It is important to note that Definition 1.1, as well as Definition 1.2 below, may be reformulated so as to deal with projectively invariant notions only, and thus they belong to projective differential geometry. In this setting the ambient space ~N of a conjugate net should be interpreted as an affine part of the projective space ~JIDN = J1D(~N+l), with ~N+l being the space of homogeneous coordinates. Equations (1.1) hold then for the standard lift U,I) E ~N+l of the conjugate net f rv [J : 1] E ~JIDN, while an arbitrary lift! = AU, 1) E ~N+l is characterized by a more general linear system

(1.3)

8/)j!

c

= Cji8J + ij 8j! + Pij!,

i -1= j

(with the corresponding compatibility conditions for the coefficients ~j, Pij, which generalize equations (1.2)). We will not pursue this description further. 1.1.2. Alternative analytic description of conjugate nets. A classical description of conjugate nets makes use of the following construction. Given the functions Cij, define functions 9i : ~m ---t ~* as solutions of the system of differential equations

(1.4)

8 i 9j

= Cij9j ,

i -1= j.

Compatibility of this system is expressed as 8 i cjk = 8jCik and is a consequence of equations (1.2) (whose right-hand sides are symmetric with respect to the flip i +-+ j). Solutions 9i can be specified by prescribing their values arbitrarily on the corresponding coordinate axes 'B i . Define vectors Wi = 9;18d. It follows from (1.1) and (1.4) that these vectors satisfy the following differential equations: (1.5)

8 i wj

=

9"

~CjiWi

,

i -1= j.

9j

Thus, defining the rotation coefficients as

(1.6)

"(ji

9i = -eji, 9j

we end up with the following system: (1.7)

8d

(1.8)

8 i wj

"(jiWi,

(1.9)

8 i 9j

9nij,

9i W i,

i -1= j, i -1= j.


4

Rotation coefficients satisfy a closed system of differential equations, which follow from (1.2) upon substitution (1.6): (1.10)

Oi'Ykj

= Iki'Yij ,

i

i= j i= k i= i.

Eqs. (1.10), known as the Darboux system, can be regarded as compatibility conditions of the linear differential equations (1.8). Observe an important difference between the two descriptions of conjugate nets: while the functions Cij describe the local geometry of a net, this is not the case for the rotation coefficients lij. Indeed, to define the latter, one needs first to find gi as solutions of differential equations (1.4). 1.1.3. Transformations of conjugate nets. The most general class of transformations of conjugate nets was introduced by Jonas and Eisenhart. Definition 1.2. (Fundamental transformation) Two m-dimensional conjugate nets f, f+ : ]Rm ~ ]RN are said to be related by a fundamental transformation (F-transformation) if at every point u E ]Rm of the domain and for each 1 SiS m the three vectors od, od+ and t5 f = f+ - fare coplanar. The net f+ is called an F-transform of the net f. This definition yields that F-transformations are described by the following (linear) differential equations: (1.11) Of course, the functions ai, bi : ]Rm ~ ]R must satisfy (nonlinear) differential equations, which express the compatibility of (1.11) with (1.1): (1.12)

Oiaj

(ai - aj)cij

(1.13)

Oibj

ct;bj

(1.14)

aJ·c-:r ~J

+ bi(aj -

1),

+ Ctbi - bjbi , aiCij + bi(aj - 1).

The following data determine an F-transform f+ of a given conjugate net f uniquely: (F I ) a point f+(O); (F 2 ) the values of ai, bi on the coordinate axes 2)i for 1 SiS m, i.e., 2m smooth real-valued functions ai I~i' bi I~i of one variable. Observe a remarkable conceptual similarity between Definitions 1.1 and 1.2. Indeed, one can interpret the condition of Definition 1.1 as planarity of infinitesimal quadrilaterals (J( u), f( u + Eied, f( u + Eiei + Ejej), f( u + Ejej)), while the condition of Definition 1.2 can be interpreted as planarity of infinitesimally narrow quadrilaterals (J(u), f(u + Eied, f+(u + Eiei), f+(u)).

1.1. Conjugate nets

5

One can iterate F-transformations and obtain a sequence f, f+, (f+)+, etc., of conjugate nets. We will see that this can be interpreted as generating a conjugate net of dimension M = m + 1, with m continuous directions and one discrete direction. The most remarkable property of F-transformations is the following permutability theorem.

Theorem 1.3. (Permutability of F-transformations)

1) Let f be an m-dimensional conjugate net, and let f(1) and f(2) be two of its F-transforms. Then there exists a two-parameter family of conjugate nets f(12) that are F-transforms of both f(1) and f(2). Corresponding points of the four conjugate nets f, f(1), f(2) and f(12) are coplanar.

2) Let f be an m-dimensional conjugate net. Let f(l), f(2) and f(3) be three of its F-transforms, and let three further conjugate nets f(12), f(23) and f(13) be given such that f(ij) is a simultaneous F-transform of f(i) and f(j). Then there e.'Eists generically a unique conjugate net f(123) that is an F-transform of f(12), f(23) and f(13). The net f(123) is uniquely defined by the condition that for every permutation (ijk) of (123) the corresponding points of f(i), f(ij), f(ik) and f(123) are coplanar. The situations described in this theorem can be interpreted as conjugate nets of dimension M = m + 2, resp. "'1 = m + 3, with m continuous and two (resp. three) discrete directions. The theory of discrete conjugate nets allows one to put all directions on an equal footing and to unify the theories of smooth nets and of their transformations. Moreover, we will see that both these theories may be seen as a continuum limit (in some precise sense) of the fully discrete theory, if the mesh sizes of all or some of the directions become infinitely small (see Figure 0.1). This way of thinking is the guiding idea and the philosophy of the discrete differential geometry. The following special F-transformation is important in the surface theory.

Definition 1.4. (Combescure transformation) We will say that two mdimensional conjugate nets f, f+ : ]Rm -; ]RN are related by a Combescure transformation if at every point u E ]Rm and for each 1 :S i :S m the vectors ad, ad+ are parallel. The net f+ is called parallel to f, or a Combescure transform of the net f.

1.1.4. Classical formulation of F-transformation. Our formulation of F-transformations is rather different from the classical one, due to Jonas and Eisenhart, based on the formula (1.15)


6

whose data are: an additional solution 1 : JRm -+ JR of (1.1), a Combescure transform p : JRm -+ JRN of f, and the function 'lj; : JRm -+ JR, associated to 1 in the same way as p is related to f. We now demonstrate how to identify these ingredients within our approach and how they are specified by the initial data (F 1,2). It follows from (1.12)-(1.14) that

(1.16) The symmetry of the right-hand sides of (1.16), (1.13) yields the existence of the functions 1,1+ : JRm -+ JR such that

1:::; i:::; m.

(1.17)

These equations define 1,1+ uniquely up to respective constant factors, which can be fixed by requiring 1(0) = 1+(0) = 1. An easy computation based on (1.16), (1.13) shows that the functions 1,1+ satisfy the following equations:

(1.18) (1.19)

CijOj1 + CjiOi1,

ct oj1+ + ct oi1+,

for all 1 :::; i =I- j :::; m. Thus, an F-transformation yields some additional scalar solutions 1 and 1+ of the equations describing the nets f and f+, respectively. Of these two, the solution 1 is directly specified by the original net f and the initial data (F2). Indeed, the data (F 2) yield the values of 1 along the coordinate axes, through integrating the first equations in (1.17); these values determine the solution of (1.18) with the known coefficients Cij uniquely. Further, introduce the quantities

(1.20) Then a direct computation based on (1.11), (1.12)-(1.14), and (1.17) shows that the following equations hold:

(1.21 ) (1.22) where

(1.23)

G.ioil, G.i Oi1,

1.2. Koenigs and Moutard nets

7

Thus, p is a Combescure transform of f, and 'ljJ is a function associated to ¢, in Eisenhart's terminology. Another computation leads to the relation (1.24) The same argument as above shows that the data (F2) yield the values of ¢+, and thus the values of ai, on the coordinate axes 'B i . This uniquely specifies the functions ai everywhere on ]Rm as solutions of the compatible linear system (1.24) with the known coefficients Cij. This, in turn, allows for a unique determination of the solutions p, 'ljJ of equations (1.21), (1.22) with the initial data p(O) = f+(O) - f(O) and 'ljJ(0) = 1 (here the data (F 1 ) enter into the construction). Thus, the classical formula (1.15) is recovered.

1.2. Koenigs and Moutard nets 1.2.1. Notion of Koenigs and Moutard nets. A geometrically important subclass of two-dimensional conjugate nets, very popular in the classical differential geometry, can be most directly defined as follows. Definition 1.5. (Koenigs net) A map f : ]R2 net if it satisfies a differential equation

-+ ]RN

is called a Koenigs

(1.25)

with some scalar function

1/ : ]R2 -+ ]R+.

In other words, a Koenigs net is a two-dimensional conjugate net with the coefficients C21, C12 satisfying a1C21 = a2C12. Classically, this property has been interpreted as equality of the so-called Laplace invariants of the net (for this reason the Koenigs nets are also known as nets with equal invariants). Remarkably, this property is invariant under projective transformations, so that the notion of Koenigs nets actually belongs to projective geometry. The following data determine a Koenigs net f uniquely: (KI) the values of f on the coordinate axes 131,132, i.e., two smooth curves with a common intersection point f(O);

n13;

(K 2 ) a smooth function

1/ : ]R2 -+ ]R+.

Leaving aside numerous geometric properties of Koenigs nets, discovered by the classics, we formulate here only the following characterization.

Theorem 1.6. (Christoffel dual for a Koenigs net) A conjugate net f : ]R2 -+ ]RN is a Koenigs net if and only if there exists a scalar function 1/ : ]R2 -+ ]R+ such that the differential one-form df* defined by ( 1.26)

a

1

1* =

ad

1/2 '

ad* = _ ad 1/ 2

8

1. Classical DiHerential Geometry

is closed. In this case the map !* : lR 2 --t lR N , defined (up to a translation) by the integration of this one-form, is also a Koenigs net, called Christoffel dual to f.

This follows immediately by cross-differentiating (1.26). A different way to formulate the latter equations is:

al!* II ad, a2!* II a2f, (1.27)

(al

+ a2)!* I

(al - a2)f,

(al - a2)!*

II

(al

+ a2)f.

If one considers the ambient space lRN of a Koenigs net as an affine part of lRpN, then there is an important choice of representatives for f '"" (f, 1) in the space lR N + l of homogeneous coordinates, namely

(1.28) Indeed, a straightforward computation shows that the representatives (1.28) satisfy the following simple differential equation:

ala2Y = q12Y

(1.29)

with the scalar function q12 = vala2(v-l). Differential equation (1.29) is known as the Moutard equation and Y is called a Moutard representative of the Koenigs net f.

Definition 1. 7. (Moutard net) A map y : lR 2 --t lR N+1 is called a Moutard net if it satisfies the Moutard differential equation (1.29) with some q12 : lR 2 --t R Thus, we see that Moutard nets appear as special lifts of Koenigs nets to the space of homogeneous coordinates. Conversely, if y is a Moutard net in lR N +l , then it is not difficult to figure out the condition for a scalar function v: lR 2 --t lR, under which = vy satisfies an equation of the type (1.1): v-I has to be a solution of the same Moutard equation (1.29), and then

J

ala2J = (a2log v)8d + (allog v)fhf. For instance, one can choose v-~ to be any component of the vector y; in this case the N components of f = vy which are different from 1 build a Koenigs net in lR N . Of course, Moutard nets can be considered also in their own right, i.e., one does not have to regard the ambient space lRN +1 of a Moutard net as the space of homogeneous coordinates for JlUDN. Nevertheless, such an interpretation is useful in most cases. The following data determine a Moutard net y uniquely: (Mr) the values of yon the coordinate axes '13 1, '13 2, i.e., two smooth curves yf 13i with a common intersection point y(O);

1.2. Koenigs and Moutard nets

(M 2) a smooth function qI2 : IR2 of the Moutard equation.

9

---t

IR that has the meaning of the coefficient

1.2.2. Transformations of Koenigs and Moutard nets. Moutard invented a remarkable analytic device for transforming Moutard nets. Definition 1.8. (Moutard transformation) Two Moutard nets y, y+ : IR2 ---t IRN are called Moutard transforms of one another if they satisfy (linear) differential equations

+ 8I y

(1.30)

fhy+

(1.31 )

82y+ - 82y

PI(Y+-y), P2(Y+

+ y),

with some functions PI, P2 : IR2 ---t IR (or similar equations with all plus and minus signs interchanged, which is also equivalent to renaming the coordinate axes 1 f-+ 2). The functions PI, P2, specifying the Moutard transform, must satisfy (nonlinear) differential equations that express compatibility of (1.30), (1.31) with (1.29):

+ PIP2, -q12 + 2pIP2·

(1.32)

-q12

(1.33)

The following data determine a Moutard transform y+ of a given Moutard net y: (MTt) a point y+(O) E IRN; (MT2) the values of the functions Pi on the coordinate axes 13 i for i = 1,2, i.e., two smooth functions Pi l:Bi of one variable.

If the Moutard nets y, y+ in IR N +I are considered as lifts of Koenigs nets f = [y], f+ = [y+] in IR N , then a geometric content of the Moutard transformation can be easily revealed. Introduce two surfaces F(l), p(2) IR2 ---t IRN with the homogeneous coordinates F(l)

= [y+ + y],

F(2)

= [y+ - y].

Then for every u E IR2 the points F(1), F(2) lie on the line (J f+), and equations (1.30), (1.31) show that this line is tangent to both surfaces F(1), F(2). One says that these surfaces are focal surfaces of the line congruence (J f+)· Now an easy computation shows that on each such line the four points f, F(I), f+, p(2) build a harmonic set, that is,

(1.34) where q is the cross-ratio of four collinear points; see (9.54).


10

Definition 1.9. (Koenigs transformation) Two Koenigs nets f, f+ : ]R2 ~ ]RN are said to be related by a Koenigs transformation if the focal

points F(l), F(2) of the line congruence (f f+) separate the points f, f+ harmonically.

It can be shown that any Koenigs transformation is analytically represented as the Moutard transformation (1.30), (1.31) by a suitable choice of Moutard lifts y, y+.

Theorem 1.10. (Permutability of Moutard transformations)

1) Let y be a Moutard net, and let y(1) and y(2) be two of its Moutard transforms. Then there exists a one-parameter family of Moutard nets y(12) that are Moutard transforms of both y(1) and y(2).

2) Let y be a Moutard net. Let y(1), y(2) and y(3) be three of its Moutard transforms, and let three further Moutard nets y(12), y(23) and y(13) be given such that y(ij) is a simultaneous Moutard transform of y(i) and y(j). Then generically there exists a unique Moutard net y(123) that is a Moutard transform of y(12), y(23) and y(13) .

1.2.3. Classical formulation of the Moutard transformation. Due to the first equation in (1.32), for any Moutard transformation there exists a function () : ]R2 ~ ]R, unique up to a constant factor, such that (1.35)

al()

PI =

-0'

P2 =

a2 () -0·

The last equation in (1.32) implies that () satisfies (1.29). This scalar solution of (1.29) can be specified by its values on the coordinate axes 'Bi (i = 1,2), which are readily obtained from the data (MT 2 ) by integrating the corresponding equations (1.35). This establishes a bridge to the classical formulation of the Moutard transformation, according to which a Moutard transform y+ of the solution y of the Moutard equation (1.29) is specified by an additional scalar solution () of this equation, via (1.30), (1.31) with (1.35). Note that these equations can be equivalently rewritten as (1.36) From these equations one can conclude that y+ solves the Moutard equation (1.29) with the transformed potential (1.37) In our formulation, the origin of the function () becomes clear: it comes from PI, P2 by integrating the system (1.35). Equation (1.37) is then nothing but

an equivalent form of (1.33).

1.3. Asymptotic nets

11

1.3. Asymptotic nets Definition 1.11. (A-surface) A map f : ]R2 -) IR3 is called an A-surface (an asymptotic line parametrized surface) if at every point the vectors 8i f lie in the tangent plane to the surface f spanned by 8d, 82 f.

8U,

Thus, the second fundamental form of an A-surface in IR3 is off-diagonal. Such a parametrization exists for a general surface with a negative Gaussian curvature. Definition 1.11, like the definition of conjugate nets, can be reformulated so as to contain projectively invariant notions only. Therefore, A-surfaces actually belong to the geometry of the three-dimensional projective space. In our presentation, however, we will use for convenien~e additional structures on IR3 (the Euclidean structure and the cross-product). A convenient description of A-surfaces is provided by the Lelieuvre representation which states: there exists a unique (up to sign) normal field n : IR2 -) IR3 to the surface f such that (1.38)

8d

=

8 1 n x n,

82 f

= n x

82 n.

Cross-differentiation of (1.38) reveals that 8 1 82 n x n = 0, that is, the Lelieuvre normal field satisfies the M outard equation (1.39) with some q12 : ]R2 -) R This reasoning can be reversed: integration of eqs. (1.38) with any solution n : ]R2 -) ]R3 of the Moutard equation generates an A-surface f : ]R2 -) IR3. Theorem 1.12. (Lelieuvre normals of A-surfaces are Moutard nets) A-surfaces f : ]R2 -) ]R3 are in a one-to-one correspondence (up to translations of f) with Moutard nets n : IR2 -) IR3 , via the Lelieuvre representation (1.38). An A-surface f is reconstructed uniquely (up to a translation) from its Lelieuvre normal field n. In turn, a Moutard net n is uniquely determined by the initial data (M l ,2), which we denote in this context by (A 1,2): (Ad the values of the Lelieuvre normal field on the coordinate axes 'B l , 'B 2 , i.e., two smooth curves nf'Bi with a common intersection point nCO); (A 2 ) a smooth function q12 : IR2 -) ]R that has the meaning of the coefficient of the Moutard equation for n. Definition 1.13. (Weingarten transformation) A pair of A-surfaces f, f+ : IR2 -) IR3 is related by a Weingarten transformation if, for every u E IR2, the line (J (11,) f+ (11,») is tangent to both surfaces f and f+ at the corresponding points. The surface f+ is called a Weingarten transform of the surface f. The lines (J (u) f+ (11,») are said to build a W-congruence.


12

It can be demonstrated that the Lelieuvre normal fields of a Weingarten pair f, f+ of A-surfaces satisfy (with the suitable choice of their signs) the following relation: f+ - f = n+ x n.

(1.40)

Differentiating the last equation and using the Lelieuvre formulas (1.38) for f and for f+, one easily sees that the normal fields of a Weingarten pair are related by (linear) differential equations:

+ Ehn

(1.41)

Ehn+

(1.42)

8 2n+ - 8 2n

with some functions P1,P2 : ]R2

-t

P1(n+ - n), pdn+

+ n),

R Thus:

Theorem 1.14. (Weingarten transformation = Moutard transformation for Lelieuvre normals) The Lelieuvre normal fields n, n+ of a Weingarten pair f, f+ of A-surfaces are Moutard transforms of one another.

A Weingarten transform f+ of a given A-surface f is reconstructed from a Moutard transform n+ of the Lelieuvre normal field n. The data necessary for this are the data (MT 1,2) for n: (WI) a point n+(O) E

]R3;

(W 2 ) the values of the functions Pi on the coordinate axes Ei for i = 1,2, i.e., two smooth functions Pi l13i of one variable. The following statement is a direct consequence of Theorem 1.10.

Theorem 1.15. (Permutability of Weingarten transformations) 1) Let f be an A-surface, and let f(1) and f(2) be two of its Weingarten transforms. Then there exists a one-parameter family of A-surfaces f(12) that are Weingarten transforms of both f(1) and f(2).

2) Let f be an A-surface. Let f(l), f(2) and f(3) be three of its Weingarten transforms, and let three further A-surfaces f(12), f(23) and f(13) be given such that f(i j ) is a simultaneous Weingarten transform of f(i) and f{j). Then generically there exists a unique A-surface f(123) that is a Weingarten transform of f(12), f(23) and f(13). The net f(123) is uniquely defined by the condition that its every point lies in the tangent planes to f(12), f(23) and f(13) at the corresponding points.

1.4. Orthogonal nets 1.4.1. Notion of orthogonal nets. An important subclass of conjugate nets is fixed in the following definition.

1.4. Orthogonal nets

13

Definition 1.16. (Orthogonal net) A conjugate net f : ~m ---4 ~N is called an m-dimensional orthogonal net in ~N if at every u E ~m and for all pairs 1 S; i i- j S; m we have ad 1- OJ f. Such a net is called an orthogonal coordinate system if m = N. The class of orthogonal nets (as well as their Ribaucour transformations; see Definition 1.19 below) are invariant under Mobius transformations and therefore belong to Mobius differential geometry. To demonstrate this, it is enough to show the invariance with respect to the inversion f t--+ 1= f Ilfl2. A direct computation shows that the inversion maps a conjugate net with the coefficients Cij and with the orthogonality property to a conjugate net with the coefficients Cij = Cij - 2(od,f)/lfI2, which is orthogonal again. Since orthogonal nets belong to Mobius differential geometry, it is useful to describe them with the help of the corresponding apparatus (a sketch of which is given in Section 9.3). In this formalism, the points of ~N (or, better, of the conformal N-sphere §N, which is a compactification of ~N) are represented by elements of the projectivized light cone lP'(ILN+l,l) in the projectivized Minkowski space lP'(~N+l,l). The light cone

ILN+l,l = {~E ~N+l,l : (~,~) =

o}

is of central importance in Mobius geometry (the absolute quadric). Let {el,"" eN +2} denote the standard basis of the Minkowski space ~N+l,l. We denote also eo = !(eN+2 - eN+d and e oo = !(eN+2 + eN+l)' The Euclidean space ~N is identified, via (1.43)

7T"o: ~

N

3

f

t--+

'

f = f

+ eo + If I2 e oo

E

N

Ql o ,

with the section Ql~ of the cone ILN+l,l by the affine hyperplane {~o = I}, where ~o is the eo-component of ~ E ~N+l,l in the basis {el,"" eN, eo, e oo }. An elegant characterization of orthogonal nets is due to Darboux:

Theorem 1.17. (Mobius-geometric characterization of orthogonal nets) A conjugate net f : ~m ---4 ~N is orthogonal if and only if the scalar function Ifl2 satisfies the same equation (1.1) as f does, or, equivalently, if the lift 1 = 7T"o 0 f : ~m ---4 Ql~ is a conjugate net in ~N+l,l. In other words, the image of an orthogonal net in the projectivized light cone lP'(ILN+l,l) is a conjugate net in lP'(~N+1,l). In particular, any lift 1= )..1 of fin ILN+l,l, not necessarily normalized as in (1.43), satisfies linear differential equations (1.3). This criterion makes the invariance of orthogonal nets under Mobius transformations self-evident. It will be important to preserve this symmetry group under discretization. This deep result by Darboux is an instance of a very general phenomenon which will be used many times within this book. It turns out that conjugate

1. Classical DiHerential Geometry

14

nets can be consistently restricted to any quadric in a projective space. As we will see in Chapter 3, discrete differential geometry gives a clear insight into the origin of this nontrivial statement (and a simple proof). The quadric responsible for Mobius geometry is the light cone JPl(JLN+l,l). Choosing various quadrics, we come to the classical geometries of Klein's Erlangen program including the hyperbolic, spherical, Lie, Plucker, Laguerre, etc. geometry.

1.4.2. Analytic description of orthogonal nets. For an analytic description of an orthogonal net f : ]Rm ---t ]RN, introduce metric coefficients hi = ladl and (pairwise orthogonal) unit vectors Vi = h-;lad. Then the following equations hold: (1.44)

ad

hiVi,

(1.45)

aiVj

!3jiVi,

(1.46)

aihj

hdJij ,

(1.47)

ai!3kj

!3ki!3ij,

i= j, i i= j, i i= j i= k i= i,

i

which are analogous to (1.7)-(1.10). Indeed, equation (1.45) holds since f is a conjugate net and the Vj are orthonormal, and it serves as a definition of rotation coefficients !3ji. Equation (1.46) is a direct consequence of (1.44), (1.45), while the Darboux system (1.47) expresses the compatibility of the linear system (1.45). So, one of distinctive features of orthogonal nets among general conjugate nets is that the system (1.4) admits a solution given by the locally defined metric coefficients hi. In the same spirit, the rotation coefficients !3ji reflect the local geometry of the net. The Darboux system (1.47) has to be supplemented by the orthogonality constraint

(1.48) To derive (1.48), one considers the identity aiaj(Vi, Vj) = 0 and makes use of (1.45). Equation (1.48) is an admissible constraint for the system (1.44)(1.47). This is understood as follows: (1.48) involves two independent variables i, j only, and therefore it makes sense to require that it be fulfilled on the coordinate plane 'B ij . One can easily check that if a solution to the system (1.44)-(1.47) satisfies (1.48) on all coordinate planes 'Bij for 1 :S i < j :S m, then it is fulfilled everywhere on ]Rm. The meaning of the orthogonality condition (1.48) is that the coordinate surfaces 'Bij are parametrized along curvature lines. The fact that this condition propagates is a sort of inversion of the classical Dupin theorem, which says that the coordinate surfaces of a triply orthogonal coordinate system intersect along their curvature lines. The coordinate surfaces Jr 'B'j can be characterized by the functions TJij = (ai !3ij - aj !3jd on 'B ij .

n

!


15

Thus, the following data can be used to determine an orthogonal net uniquely:

f

(0 1 ) the values of f on the coordinate axes 'Bi for 1 :S i :S m, i.e., m smooth curves f r:Bi' intersecting pairwise orthogonally at f (0); (0 2 ) m(m - 1)/2 smooth functions T/ij : 'Bij

---t

]R

for all 1 :S i

< j :S m.

1.4.3. Spinor frames of orthogonal nets. The Mobius-geometric description of orthogonal nets has major conceptual and technical advantages. First, this description linearizes the invariance group of orthogonal nets, i.e., the Mobius group of the sphere §N (which can be considered as a compactification of ]RN by a point at infinity). Orientation preserving Euclidean motions of ]RN are represented as conjugations by elements of 9{oo, the isotropy subgroup of e oo in Spin+(N + 1,1). Further, using the Clifford algebra model of Mobius differential geometry enables us to give a frame description of orthogonal nets, which turns out to be a key technical device. As is easily seen, the metric coefficients hi = lodl satisfy also hi = lodl, where j = f +eo + Ifl 2 e oo . Hence, the vectors Vi = h-;lod = Vi + 2(1, vi)e oo have the (Lorentz) length 1. Since (j, j) = 0, one readily finds that (j, Vi) = o and hi = -(OiVi, /). Theorem 1.18. (Spinor frame of an orthogonal net) For an orthogonal net f : ]Rm ---t ]RN, i. e., for the corresponding conjugate net j : ]Rm ---t Q-s', there exists a function 'l/J : ]Rm ---t 9{oo (called a frame of j), such that

(1.49)

j

(1.50)

Vi

'l/J-1 eo 'l/J, 'l/J-1 ei 'l/J,

1:S i :S m,

and satisfying the system of differential equations:

(1.51) Note that for an orthogonal coordinate system (m = N) the frame 'l/J is uniquely determined at any point by the requirements (1.49) and (1.50). It is readily seen that the unit tangent vectors Vi satisfy eq. (1.45) with the same rotation coefficients {3ji = (OiVj, Vi) = -(OiVi, Vj). With the help of the frame 'l/J we extend the set of vectors {Vi : 1 :S i :S m} to an orthonormal basis {Vk : 1 :S k :S N} of TjQ-s':

(1.52)

1 :S k :S N.

Respectively, we extend the set of rotation coefficients according to the formula


16

Recall that we also have:

hi

=

-(tM!i' j)

=

-(8(Vi' 1/J- 1 eo1/J),

1 SiS m.

Thus, introducing vectors Si = ~)Si1/J-l, we have the following expansion with respect to the vectors ek:

(1.53)

Si = 1/J si1/J-l = !1/J(8(vd1/J-l =

-! L (3ki ek + hieOCJ' k~i

It is easy to see that (1.47) still holds, if the range of the indices is extended to all pairwise distinct i,j, k with 1 S i,j S m and 1 S k S N, and that the orthogonality constraint (1.48) can be now put as

(1.54)

8d3ij

+ 8j(3ji = -

L (3ki(3kj . k~i,j

The system consisting of (1.47), (1.54) carries the name of the Lame system.

1.4.4. Curvatures of surfaces and curvature line parametrized surfaces. Two-dimensional (m = 2) orthogonal nets in]R3 are nothing but surfaces parametrized along curvature lines, or, otherwise said, parametrized so that both the first and the second fundamental forms are diagonal. Such a parametrization exists and is essentially unique for a general surface in ]R3 in the neighborhood of a nonumbilic point. In dimensions N > 3 only special surfaces support such a parametrization.

n

Figure 1.1. Principal directions through touching spheres.

Curvature lines are subject of Lie geometry, i.e., are invariant with respect to Mobius transformations and normal shifts. To see this, consider an infinitesimal neighborhood U of a point f of an oriented smooth surface in ]R3, and the pencil of spheres S (I'\,) with the curvatures 1'\" touching the surface at f; see Figure 1.1. The curvature 1'\" as well as the signed radius r = 1/1'\" is assumed positive if S (I'\,) lies on the same side of the tangent plane as the normal n, and negative otherwise; the tangent plane itself is S(O). For big 1'\,0 > 0 the spheres S(l'\,o) and S( -1'\,0) intersect U in f only.


17

The set of the touching spheres with this property (intersecting U in f only) has two connected components: M+ containing S(K:o) and !vL containing S( -K:o) for big K:o > o. The boundary values K:l

= inf {K: : S(K:)

E

M+},

= sup {K: : S(K:) E !vL} surface in J. The directions K:2

are the principal curvatures of the in which S(K:d and S(K:2) touch U are the principal directions. Curvature lines are integral curves of the principal directions fields. The symmetric functions K

=

K:IK:2,

H __

K:l

+ K:2

2 are called the Gaussian curvature and the mean curvature, respectively.

Clearly, all ingredients of this description are Mobius-invariant. Under a normal shift by the distance d the centers of the principal curvature spheres are preserved and their radii are shifted by d. This implies that the principal directions and thus the curvature lines are preserved under normal shifts, as well. A Lie-geometric nature of the curvature line parametrization yields that it has a Lie-invariant description. A surface in Lie geometry is considered as consisting of contact elements. A contact element can be identified with a pencil of spheres through a common point with a common (directed) normal in that point. Two infinitesimally close contact elements (sphere pencils) belong to the same curvature line if and only if they have a sphere in common, which is the principal curvature sphere. Let us consider an infinitesimal neighborhood of a surface Gauss map n. For sufficiently small t the formula

f

with the

it = f + tn defines smooth surfaces parallel to f. The infinitesimal area of the parallel surface it turns out to be a quadratic polynomial of t and is described by the classical Steiner formula

(1.55)

dA(Jt} = (1 - 2Ht + Kt 2 )dA(J),

Here dA is the infinitesimal area of the corresponding surface and Hand K are the mean and the Gaussian curvatures of the surface f, respectively. 1.4.5. Ribaucour transformations of orthogonal nets. An important class of transformations between orthogonal nets is specified in the following definition.

Definition 1.19. (Ribaucour transformation) A pair ofm-dimensional orthogonal nets f, f+ : ]Rm -+ ]RN is related by a Ribaucour transformation if the corresponding coordinate curves of f and f+ envelope one-parameter families of circles, i. e., if at every u E ]Rm and for every 1 :S i :S m the


18

straight lines spanned by the vectors ad, ad+ at the respective points f, f+ are interchanged by the reflection in the orthogonal bisecting hyperplane of the segment [f, f+]· The net f+ is called a Ribaucour transform of f.

The nets f, f+ serve as two envelopes of a Ribaucour sphere congruence

Sm : ~m

->

{m-spheres in ~N}.

In other words, f(u), f+(u) E sm(u), and the tangent m-spaces to sm(u) at f(u), resp. f+(u), are spanned by ad(u), resp. by ad+(u), i = 1, . .. ,m. To describe a Ribaucour transformation analytically, we write: (1.56)

ad

+

,81)) = ri ( ad - 2 (a (8f,81) 8f , d

with some functions ri : ~m -> ~*. It is convenient to define the metric coefficients of the transformed net as ht = rihi = sign(ri)/ad+/, with the corresponding unit vectors vt = (hn- 1ad+. Further, denote £ = /8f/ and introduce the unit vector y = £-18f, so that f+ = f + £y. Then we find: (1.57)

vi

= Vi

- 2(Vi, y)y,

aiy

=

!Oi(vt

+ Vi)'

with the functions Oi : ~m -> ~ defined as Oi = (ht - h i )/£ = (ri - 1)hd£. Equations (1.57) imply equations for the metric coefficients: (1.58) Compatibility of the system (1.57) yields that Oi have to satisfy certain differential equations: (1.59)

(31; = (3ij -

2(Vi,y)Oj,

aiOj

= !Oi((31; + (3ij).

The following data determine a Ribaucour transform f+ of a given orthogonal net f uniquely: (R 1 ) the point f+(O); (R2) the values of Oi on the coordinate axes 13 i for 1 :::; i :::; m, i.e., m smooth functions Oi I'Bi of one variable. According to the general philosophy, iterating Ribaucour transformations can be interpreted as adding an additional (discrete) dimension to an orthogonal net. The situation arising by adding two or three discrete dimensions is described in the following fundamental theorem.

Theorem 1.20. (Permutability of Ribaucour transformations)

1) Let f be an m-dimensional orthogonal net, and let f(l) and f(2) be two of its Ribaucour transforms. Then there exists a one-parameter family of orthogonal nets f(12) that are Ribaucour transforms of both f(1) and f(2) . The corresponding points of the four orthogonal nets f, f(l), f(2) and f(12) are con circular.

19

1.5. Principally parametrized sphere congruences

2) Let f be an m-dimensional orthogonal net. Let f(1), f(2) and f(3) be three of its Ribaucour transforms, and let three further orthogonal nets f(12), f(23) and f(13) be given such that f(ij) is a simultaneous Ribaucour transform of f(i) and f(j). Then generically there exists a unique orthogonal net f(123) that is a Ribaucour transform of f(12) , f(23) and f(13). The net f(123) is uniquely defined by the condition that for every permutation (ij k) of (123) the corresponding points of f(i), f Ui ), f(ik) and f(123) are concircular.

The theory of discrete orthogonal nets will unify the theories of smooth orthogonal nets and of their transformations.

1.5. Principally parametrized sphere congruences The Mobius-geometric formalism is very convenient in description of hypersphere congruences. The classical case is, of course, that of two-parametric families of spheres in ~3. Nonoriented spheres in ~3 can be represented as elements of ]P>(~~~t), where (1.60)

~~~t =

{.9 E ~4,1:

(s,s) >

o}

is the space-like part of ~4,1 .

Definition 1.21. (Principally parametrized sphere congruence) A map

(1.61 )

S : ~2 ~ {nonoriented spheres in ~3}

is called a principally parametrized sphere congruence if the corresponding map s : ~2 ~ ]P>(~~~t) is a conjugate net, i.e., if for any lift of it to ~~'~tI

(1.62) A principal parametrization exists and is unique for a generic congruence. The classical description of this is as follows. In an arbitrary parametrization of a congruence, consider two neighbors S(u + dud, S(u + dU2) of a sphere S(u), obtained by infinitesimal shifts along both coordinate lines; they intersect the original sphere along two circles C 1 (u) and C 2 (u). Thus, in the projective model of Mobius geometry of ~3, based on the Minkowski space ~4,1 of pentaspherical coordinates, these circles are described as lL 4 ,1 n (span(s, 8 1 s)) 1., resp. lL 4 ,1 n (span(s, 82 s)) 1.; i.e. their points are represented by elements :i; E lL 4 ,1 satisfying (1.63) resp. (1.64)

20


These two circles intersect in two points. Such pairs of points comprise the two enveloping surfaces of the congruence, described in the M6biusgeometric formalism as lL 4 ,1 n (span(s, 81 s, 8 2s)).l. In other words, the envelopes are represented by the elements x E lL 4,1 satisfying (1.65)

(s, x) = 0,

(8 1 s, x) = 0,

(82 s, x) = 0.

Now, the principal parametrization is characterized by the following condition: when an infinitesimal displacement is made along one of the coordinate lines, say along the u2-line, the four points of contact of the two infinitely close spheres S(u), S(u + dU2) with the envelopes lie on a circle, namely on C 1 (u). Indeed, differentiating the first two equations in (1.65) with respect to U2 and making use of the third and of equation (1.62), we come to (s,82 x) = 0, (8 1 s,82 x) = 0, which, compared with (1.63), demonstrates the claim. A convenient choice of representatives s of hyperspheres S is the Euclidean one, in terms of the centers c and radii r: (1.66) The condition for this to be a conjugate net in classical statement.

]R4,1

leads to the following

Theorem 1.22. (Principally parametrized sphere congruences; centers and radii) A map (1.61) is a principally parametrized sphere congruence if and only if the centers c : ]R2 -+ ]R3 of the spheres S form a conjugate net in ]R3, and the radii r : ]R2 -+ ]R+ are such that the function Icl 2 - r2 satisfies the same equation (1.1) as the centers c. 1.6. Surfaces with constant negative Gaussian curvature Up to now, we discussed special classes of coordinate systems in space, or special parametrizations of a general surface. Now, we turn to the discussion of several special classes of surfaces. The distinctive feature of these classes is the existence of transformations with certain permutability properties. One of the most prominent examples of integrability in differential geometry is given by the K-surfaces.

Definition 1.23. (K-surface) An asymptotic line parametrized surface f : ]R2 -+ ]R3 is called a K-surface (or a pseudospherical surface) if its Gaussian curvature K is constant, i.e., does not depend on u E ]R2. The following is their equivalent characterization as Chebyshev nets, i.e. nets with infinitesimal coordinate strips of constant width.

1.6. Surfaces with constant negative Gaussian curvature

21

Theorem 1.24. (K-surfaces = A-surfaces with Chebyshev property) An asymptotic line parametrized surface f : ]R2 ----+ ]R3 is a K-surface if and only if the functions (3i = lodl Ci = 1,2) depend on Ui only: (3i = (3i(ud· One of the approaches to the analytical study of K-surfaces is based on the investigation of the angle cP( u], U2) between asymptotic lines which is governed by the equation O]02cP = -K(3] (Ul)(32(U2) sincP. After a reparametrization of asymptotic lines one arrives at the famous sine-Gordon equation

(1.67)

O]02cP

= sin cPo

Another description is based on the Gauss maps. Theorem 1.25. (Gauss map of a K-surface is a Moutard net) The Lelieuvre normal field n : ]R2 ----+ ]R3 of a K-surface with K = -1 takes values in the sphere §2 C ]R3, thus coinciding with the Gauss map. Conversely, any Moutard net 'in the unit sphere §2 is the Gauss map and the Lelieuvre normal field of a K-surface with K = -1. Moreover, IOinl = (3i (i = 1,2), with the same functions (3i = (3i (Ui) as in Theorem 1.24.

Thus, the K-surfaces with K = -1 are in a one-to-one correspondence with the Moutard nets in §2. Functions n : ]R2 ----+ §2 satisfying a Moutard equation (1.39) are sometimes called Lorentz-harmonic maps to §2 (one means hereby harmonicity with respect to the Lorentz metric on the plane ]R2 with coordinates (U], U2)). It is important to observe that the coefficient q]2 of the Moutard equation (1.39) satisfied by a Lorentz-harmonic map n is completely determined by its first order derivatives:

(1.68) Therefore, the following data determine the Gauss map n of a K-surface f: (K) the values of the Gauss map on the coordinate axes '13], '13 2 , i.e., two smooth curves nl13i in §2 intersecting at a point n(O). The K-surface f is reconstructed from n uniquely, up to a translation, via formulas (1.38). Definition 1.26. (Backlund transformation) A Weingarten transform f+ of a K-surface f : ]R2 ----+ ]R3 is called a Backlund transform if the distance

If+

- f

I is

constant, i. e., does not depend on u

E ]R2.

As for a general Weingarten pair, the Lelieuvre normal fields (Gauss maps) 11,,11,+ of a Backlund pair of K-surfaces f, f+ can be chosen so that eq. (1.40) holds, and hence n, 11,+ are related by the Moutard transformation (1.41), (1.42). From these equations it easily follows that for a Backlund pair the quantity (11" n+) is constant; thus, the intersection angle of the


22

tangent planes at the corresponding points of a Backlund pair is constant. Moreover, (1.40) yields that this constant angle is related to the constant distance between f and f+ via

The fact that n, n+ E §2 allows one to express the coefficients PI, P2 in eqs. (1.41), (1.42) in terms of the solutions themselves: (1.69)

PI

(1. 70)

P2

=

(8 1n, n+) - (n,8 1n+) 2 - 2(n, n+) (n,8 2n+) - (82n, n+) 2 + 2(n, n+)

(8 1n, n+) 1 - (n, n+)' -(82n, n+) 1 + (n, n+)'

With these expressions, (1.41), (1.42) become a compatible system of first order differential equations for n +; therefore the following data determine a Backlund transform f+ of the given K-surface f uniquely: (B) a point n+(O) E

§2.

Theorem 1.27. (Permutability of Backlund transformations) Let f be a K-surface, and let f(1) and f(2) be two of its Backlund transforms. Then there exists a unique K-surface f(12) which is simultaneously a Backlund transform of f(1) and of f(2). The points of the fourth surface f(12) lie in the intersection of the tangent planes to f(l) and to f(2) at the corresponding points, and are uniquely defined by the properties If(12) - f(1) I = If(2) - fl and If(12) - f(2) I = If(1) - fl, or, in terms of the Gauss maps, (n(l), n(12)) = (n, n(2)) and (n(2), n(12)) = (n, n(I)). Equivalently, the Gauss map n(12) of f(12) can be characterized by the condition that n(12) - n is parallel to n(1) - n(2). We will see how the theory of discrete K-surfaces unifies the theories of smooth K-surfaces and of their Backlund transformations.

1. 7. Isothermic surfaces Classically, theory of isothermic surfaces and their transformations was considered as one of the highest achievements of the local differential geometry.

Definition 1.28. (Isothermic surface) A curvature line parametrized surface f : ]R2 -t]RN is called an isothermic surface if its first fundamental form is conformal, possibly upon a reparametrization of independent variables Ui 1--+ (lL N +l,l) conjugate net s in ]R~~t Moutard net n in §:t Moutard net s in lLlV+l,l Isothermic net n in §:t

All these notions and relations will be discretized in the main text of the book. The actual list of examples treated in this book is even longer. We discretize some other classical examples including line congruences and constant mean curvature surfaces. In the context of Lie and Plucker geometry, isotropic line congruences are interpreted as curvature and asymptotic line parametrized surfaces, respectively. A discrete version of this theory is also developed in the main text of the book.

28


1.9. Bibliographical notes Achievements of the classical period of the differential geometry of surfaces and their transformations are documented in the treatises by Darboux (1910, 1914-27), Bianchi (1923) and Eisenhart (1909, 1923). These books cover huge material and are indispensable sources for a detailed treatment of the special geometries of this chapter. Section 1.1: Conjugate nets and their transformations. The classical description of multidimensional conjugate nets, given in Section 1.1.2, can be found in Darboux (1914-27). The fundamental transformations of conjugate nets, given in Section 1.1.4, as well as the permutability theorem for F-transformations (part one of Theorem 1.3) are due to Eisenhart (1923) and Jonas (1915). The first instance of the Eisenhart hexahedron (part two of Theorem 1.3) we were able to localize is the "extended theorem of permutability" for conjugate nets in Eisenhart (1923, §24). In the modern literature on integrable systems, the Darboux system (1.10) is known as the n-wave equation; see Novikov-Manakov-PitaevskiiZakharov (1984). Section 1.2: Koenigs and Moutard nets and their transformations. Classically, Koenigs nets were known as nets with equal Laplace invariants. Their geometry was studied, among others, by Koenigs (1891, 1892a,b), Darboux (1914-27), Tzitzeica (1924). An exhaustive treatment of nets with dependent Laplace invariants is in Bol (1967). For the classical formulation of the Moutard transformation see, e.g., Moutard (1878), and for its geometric interpretation as Koenigs transformation see Koenigs (1891). The two- and three-dimensional permutability theorems for Koenigs transformations are due to Eisenhart (1923). In terms of Moutard transformations this was formulated in Ganzha-Tsarev (1996) and Nimmo-Schief (1997). Section 1.3: Asymptotic nets and their transformations. For the description of asymptotic nets, W-congruences and their Weingarten transformations in terms of Lelieuvre normals see the classical books by Darboux (1914-27), Bianchi (1923) and Eisenhart (1923) or, for example, Lane (1942). The two-dimensional permutability of Weingarten transformations (part one of Theorem 1.15) is due to Bianchi (1923). For the projective interpretation of the Lelieuvre normals, see Konopelchenko-Pinkall (2000). A survey on integrable systems in projective differential geometry based on asymptotic line parametrization is in Ferapontov (2000a). Section 1.4: Orthogonal nets and their transformations. A fundamental monograph on orthogonal coordinate systems is Darboux (1910). A


29

detailed study of conjugate nets in arbitrary quadric is in Tzitzeica (1924). The surface theory in the framework of Mobius, Laguerre and Lie geometry was developed by Blaschke (1929). Permutability theorem for Ribaucour transformations (part one of Theorem 1.20) is due to Bianchi (1923). Eisenhart type permutability theorem for Ribaucour transformations (part two of Theorem 1.20) was found in Ganzha-Tsarev (1996). Orthogonal coordinate systems from the viewpoint of the theory of integrable systems were investigated in Zakharov (1998). Algebro-geometric orthogonal coordinate systems were constructed by Krichever (1997). A survey of integrable systems in Lie geometry is given in Ferapontov (2000b). Spinor frames for orthogonal nets were introduced in Bobenko-Hertrich-Jeromin (2001). A modern textbook on the Mobius surface theory including the theory of orthogonal nets is Hertrich-Jeromin (2003). A Lie-geometric description of Ribaucour transformations is given in Burstall-Hertrich-Jeromin (2006). Section 1.5: Principally parametrized sphere congruences. This topic was rather popular in the classical literature; see, e.g., DarboliX (191427), Coolidge (1916), Eisenhart (1923), but is not well presented in the modern literature. Section 1.6: K-surfaces and their transformations. Surfaces with constant negative Gaussian curvature (pseudospherical surfaces) played an important role in the theory of surfaces and their transformations. The sineGordon equation is the oldest integrable equation. It goes back at least to Bour (1862) and Bonnet (1867). The Backlund transformation was found by Backlund (1884); the permutability theorem is due to Bianchi (1892). For a modern presentation, generalizations and description in terms of loop groups see Rogers-Schief (2002) and Terng-Uhlenbeck (2000). Section 1.7: Isothermic surfaces and their transformations. The classical period of the theory of isothermic surfaces is summarized in DarboliX (1914-27) and Bianchi (1923). In particular, the DarboliX transformations for isothermic surfaces as well as the characterization of isothermic surfaces as Moutard nets in the light cone from Theorem 1.32 are due to Darboux. The permutability of DarboliX transformations was established by Bianchi. The constant cross-ratio property in the permutability theorem was shown by Demoulin (1910). Moutard nets in general quadrics were investigated in Tzitzeica (1924). Isothermic surfaces played an important role in the development of the modern integrable differential geometry. A relation of the classical theory of isothermic surfaces to the theory of integrable systems was found in Cieslinski-Goldstein-Sym (1995). A spinor zero-curvature representation for isothermic surfaces was found in Bobenko-Pinkall (1996b). A relation of

30


isothermic surfaces to curved flats (see Ferus-Pedit (1996)) was established in Burstall-Hertrich-Jeromin-Pedit-Pinkall (1997). A description of Bonnet pairs (pairs of isometric surfaces with the same mean curvature) in terms of isothermic surfaces was given in Kamberov-Pedit-Pinkall (1998). Isothermic surfaces in multidimensional spaces were studied by Schief (2001) and Burstall (2006). A construction of Darboux transformations for isothermic surfaces using a quaternionic Riccati equation was suggested in HertrichJ eromin-Pedit (1997). A systematic presentation of the theory of isothermic surfaces and their transformations is given in Hertrich-Jeromin (2003). Section 1.8: Surfaces with constant mean curvature. The theory of surfaces with constant positive Gaussian curvature and their parallel surfaces (including surfaces with constant mean curvature) goes back to Bonnet and can be found, e.g., in Darboux (1914-27) and Bianchi (1923).

Chapter 2

Discretization Principles. Multidimensional Nets

In this chapter we start to develop discrete analogues of the classes of nets (parametrized surfaces and coordinate systems) and their transformations considered in Chapter 1. We will see that on the discrete level there is essentially no difference between nets and their transformations, which can be regarded just as various parts of multidimensional discrete nets characterized by certain elementary geometric properties. The very possibility to impose these properties on a multidimensional net, which usually relies on certain incidence theorems of elementary geometry, is the ultimate reason for the existence of transformations and their remarkable permutation properties. Since the existence of Backlund-like transformations with permutability properties is associated with integrability of the underlying differential equations, one is led to regard the multidimensional consistency of their discretizations as the core of integrability itself. About notation: in this chapter, not yet dealing with the approximation questions, we regard discrete nets as functions on zm. We define translation and difference operators in a standard manner:

where ei is the i-th coordinate vector of zm. Often we write Ii, !ij for Td, TiTj!, etc. The notation for (discrete) s-dimensional coordinate planes is the same as in the continuous case:

-

31

32

2. Discretization Principles. Multidimensional Nets

We will denote by

e· . - e· . (u) '1.1···'l.8

-

2,1···'1,5

the elementary s-dimensional cube with the 28 vertices U+Eil eil + Ei E

... +Eiseis'

{O,l}.

2.1. Discrete conjugate nets (Q-nets) 2.1.1. Notion and consistency of Q-nets. Recall that we always assume the dimension N of the ambient space ]RN of our nets to be 2: 3. Definition 2.1. (Q-net) A map f : zm ~]RN is called an m-dimensional Q-net (discrete conjugate net) in]RN if all its elementary quadrilaterals are planar, i. e., if at every u E zm and for every pair 1 :S i of. j :S m the four points f, Td, Tj f, and TiTj f are coplanar: oiojf

(2.1)

= cjiod + cijojf,

i

of. j,

or, equivalently, (2.2)

TiTjf - f = (1

+

cji)od

+

(1

+

cij)ojf,

i

of. j.

Here it is convenient to think that the real numbers Cij, Cji, as well as equation (2.1) itself, are assigned to elementary squares ij of zm parallel to the coordinate plane 13 ij .

e

Figure 2.1. A planar quadrilateral.

Definition 2.1, like its continuous counterpart Definition 1.1, actually belongs to projective geometry, since the coplanarity of four points is a property manifestly invariant under projective transformations. If the ambient space ]RN is interpreted as an affine part of ]RIP'N = 1P'(]RN+l) , then

2.1. Discrete conjugate nets (Q-nets)

33

an arbitrary lift] = p(j,1) E ]RN+l of a Q-net f to the space of homogeneous coordinates ]RN+l is characterized by the following condition: for every u E zm and for every pair i i= j, the four elements ], Td, Tj], and TiTj] are linearly dependent (span a three-dimensional vector subspace): (2.3)

-

-

TiTjf = D:jiTi!

+ D:ijTjf + f3ijf·

To analyze the existence and construction of discrete conjugate nets, consider various values of m.

m = 2 : discrete surfaces parametrized by conjugate lines. Consider a Q-surface f : Z2 -----) ]RN. Suppose its two coordinate lines, n"131 and n"132 , are given. To extend the surface into the quadrant Z~, say, one proceeds by induction whose step consists in choosing !I2 in the plane through f, !I and 12, provided the latter three points are known (and are in general position). The planarity condition is expressed as

(2.4) So, on each such step one has two free real parameters C2l, C12, attached to the elementary square e12 (u) of the lattice Z2. Thus, one can define a Q-surface f by prescribing its two coordinate lines f 1"131' f 1"132' and two real-valued functions C12, C2l defined on all elementary squares of Z2. The planes (j,!I, !I2, h) of a Q-surface can be assigned to elementary quadrilaterals of Z2, that is, to the vertices of the dual lattice (Z2)*. This corresponds to thinking about a surface as an envelope of its tangent planes rather than a set of its points. In the case of the dimension of the ambient space N = 3, this view of Q-surfaces makes them an example of the following notion. Definition 2.2. (Q*-net) A map

(2.5)

P : 'I!} -----) {planes in ]R3}

is called a Q* -net if at each u E Z2 the four planes P, PI, P2, P 12 have a common point. Clearly, this definition is projectively dual to Definition 2.1. Therefore it is more natural to consider Q* -nets in the framework of projective geometry in ]Rp3 rather than in ]R3. Each plane P in ]Rp3 is described as an element p E (]Rp3)*, with some representatives P E ]R4 in the space of homogeneous coordinates. The condition for P, PI, P2, P 12 to have a common point is equivalent to the condition for p, PI, P2, P12 to span a three-dimensional vector subspace. In other words, a Q* -net is nothing but a Q-net in the dual space p : Z2 -----) (]Rp3) * , while the intersection points of the planes of a Q* -net constitute a Q-net in lRJP>3.

34


It will be important to remark that the combinatorics of Q-surfaces may well be more complicated than that of Z2. Definition 2.1 can be literally extended to maps f : VeD) ----+ IR N , where V(D) is the set of vertices of an arbitrary quad-graph D. A quad-graph is a cell decomposition of a surface with all quadrilateral faces; see Section 6.7 for a precise definition. Thus, planar quadrilaterals of a generalized Q-surface may be attached to one another not necessarily in a regular manner, with the only condition that a nonempty intersection of two different quadrilaterals is either a common edge or a common vertex.

m = 3 : the basic 3D system. Suppose that three coordinate surfaces of a three-dimensional Q-net f are given, that is, f f1312' f f1323 and f f1313' Of course, each one of them is a Q-surface. To extend the net into the octant zt, one proceeds by induction whose step consists in determining h23, provided f, fi and fij are known. We now show that the planarity condition determines the point h23 uniquely. Remark. (General position assumption) In the spirit of local differential geometry, we will always assume in our statements and reasonings that all the data are in general position, without specifying this explicitly. In particular, in the following theorem it will be silently assumed that the four points f, fi span a three-dimensional space, and that no three points /i, fij, fik are collinear. Theorem 2.3. (Elementary hexahedron of a Q-net) Given seven points f, fi and fij (1::; i < j::; 3) in IR N , such that each of the three quadrilaterals (j, ii, /ij, fj) is planar, define three planes TiIIjk as those passing through the point triples (h /ij, /ik), respectively. Then these three planes intersect at one point:

(2.6) Proof. Planarity of the quadrilaterals (j, fi, /ij, fj) assures that all seven initial points f, fi and /ij belong to the three-dimensional affine space II 123 through the four points f, k Hence, the planes TiIIjk lie in this threedimensional space, and therefore generically they intersect at exactly one point. 0 Thus, an elementary construction step of a three-dimensional Q-net out of its three coordinate surfaces consists in finding the eighth vertex of an elementary hexahedron from the known seven vertices. This is symbolically represented in Figure 2.2, which is the picture we have in mind when thinking and speaking about discrete 3D systems with dependent variables (fields) attached to the vertices of a regular cubic lattice.


35

h3 __-------{

I

12.------/

/

/

1-----11 Figure 2.2. 3D system on an elementary cube; fields on vertices.

An analytic description of Q-nets leads to the following picture. The characteristic property of a Q-net is encoded in equation (2.1). Six such equations are attached to six faces of an elementary cube of Z3 depicted in Figure 2.2. These equations yield that 8i (8j 8k f) can be expressed as linear combinations of 8d (1 :s: f!. :s: 3). Equating coefficients of 8d in expressions for 8i (8 j 8k I) for the three cyclic permutations (i, j, k) of the indices (1,2,3), we see that the equations for Cjk split off from the equations for I: (2.7)

8i cjk =

(TkCij )Cjk

+ (TkCji)Cik -

hCjk)Cik,

i

i- j i- k i- i.

More precisely, equations (2.7) are sufficient for the consistency of the system of six equations (2.1), and are also necessary if the three vectors 8d (1 :s: f!. :s: 3) are linearly independent. Consider the numbers {cjd on the three faces adjacent to I as known, and the numbers {Ticjd on the three faces adjacent to 1123 as yet unknown. Then equations (2.7) can be seen as a system of six (linear) equations for six unknown variables TiCjk in terms of the known Cjk. For geometric reasons (existence and uniqueness of 1123 in the general position case), this system generically admits a unique solution. The resulting map

(2.8) is birational. Explicit formulas for this map are too complicated to be of any use. Nevertheless, this map, encoded in Figure 2.3, should be considered as another fundamental 3D system associated with Q-nets.

m ~ 4 : consistency. Turning to the case m ~ 4, we see that one can prescribe all two-dimensional coordinate surfaces of a Q-net, i.e., I r'Bij for all 1 :s: i < j :s: m. Indeed, these data are clearly independent, and one can construct the whole net from them. In doing so, one proceeds by induction, again. The induction step is essentially three-dimensional and consists in determining fijk, provided I, Ii and Iij are known. However, this inductive process works only if one does not encounter contradictions. To see the


36

}----/ / /

Figure 2.3. 3D system on an elementary cube; fields on faces.

possible source of contradictions, consider first the case of m = 4 in detail. From f, Ii and fij (1:S i < j :S 4) one determines all fijk uniquely. After that, one has, in principle, four different ways to determine iI234 from four 3D cubic faces adjacent to this point; see Figure 2.4. A remarkable property of Q-nets is that these four values for iI234 automatically coincide. We call this property the 4D consistency.

Definition 2.4. (4D consistency) A 3D system is called 4D consistent if it can be imposed on all three-dimensional faces of an elementary hypercube of 1:;4.

iI234

I', I'

h34

I',

~~

h4 f~

I

13

[2. __ _ h2

f -

.' ,,"" / / 124 ',

iI24

'/

.; /

Figure 2.4. 4D consistency of 3D systems.

Of course, this definition can be applied not only to discrete 3D systems with fields on vertices, such as the geometric construction of Q-nets, but also to other types of systems, such as the map (2.8) which is a discrete 3D system with fields on faces. For such a map {cjd !--t {TiCjd, the 4D


37

consistency means that the two values Ti(TjCke) and any permutation (i,j, k, £) of the indices (1,2,3,4).

Tjhcke)

coincide for

Theorem 2.5. (4D consistency of Q-nets) The 3D system governing Q-nets is 4D consistent. Proof. In the construction above, the four values in question are h234

= T1 T2II34 n T1 T3II24 n T1 T4II23 ,

and the three others are obtained by cyclic shifts of indices. Thus, we have to prove that the six planes TiTjIIkl' intersect in one point. First, assume that the ambient space jRN has dimension N ~ 4. Then, in general position, the affine space II 12 34 through the five points f, fi (1 ::; i ::; 4) is four-dimensional. It is easy to understand that the plane TiTjIIkl' is the intersection of two three-dimensional subspaces TiIIjkl' and TjIIikc. Indeed, the subspace TiIIjkC through the four points k lij, fik' fie contains also fijk, fije, and likC· Therefore, both TiIIjke and TjIIikc contain the three points lij, fijk and lijc, which determine the plane TiTjIIkl'. Now the intersection in question can be alternatively described as the intersection of the four three-dimensional subspaces T1II234, T2II134, T3II124 and T4II123 of one and the same four-dimensional space II 1234 . This intersection consists in the general case of exactly one point. This argument yields also the 4D consistency of the map {Cjk} f-----t {TiCjk}. This map, however, does not depend on the dimension N of the ambient space of f. The 4D consistency of the map {cjd f-----t {Ticjd yields, in turn, the 4D consistency of the construction of Q-nets also in the case N = 3. 0 The m-dimensional consistency of a 3D system for m > 4 is defined analogously to the case m = 4. Remarkably and quite generally, the 4dimensional consistency already implies m-dimensional consistency for all m >4.

Theorem 2.6. (4D consistency yields consistency in all higher dimensions) Any 4D consistent discrete 3D system is also m-dimensionally consistent for all m > 4. Proof. For concreteness, we give the proof only for discrete systems with fields on vertices (the case of fields on two-dimensional faces is analogous). The proof goes by induction from the m-dimensional consistency to the (m + I)-dimensional consistency, but, for the sake of notational simplicity, we present the details only for the case m = 5, the general case being absolutely similar.

e

Initial data for a 3D system on the 5D cube 12 345 with the fields on vertices consist of the fields f, fi and fij for all 1 ::; i < j ::; 5. From these

38


data one first gets ten fields fijk for 1 ~ i < j < k ~ 5, and then five fields fijk£ for 1 ~ i < j < k < £ ~ 5 (the fact that the latter are well defined is nothing but the assumed 4D consistency for the 4D cubes e ijkl ). Now, one has ten possibly different values for h2345, coming from ten 3D cubes TiTjek£m. To prove that these ten values coincide, consider five 4D cubes Tiejk£m. For instance, for the 4D cube T1 e 2 345 the assumed consistency assures that the four values for h2345 coming from four 3D cubes T1 T2 e 345,

T1 T3 e 245 ,

T1T4e235,

T1T5e234

are all the same. Similarly, for the 4D cube T2e1345 the 4D consistency leads to the conclusion that the four values for h2345 coming from T1 T 2 e 345,

T2 T 3 e 145,

T2 T4 e 135,

T2 T 5 e 134

coincide. Note that the 3D cube T1 T2e345, the intersection of T1 e 2345 and T2e1345, is present in both lists, so that we now have seven coinciding values for h2345. Adding similar conclusions for other 4D cubes Tiejk£m, we arrive at the desired result. D Theorems 2.5, 2.6 yield that Q-nets are m-dimensionally consistent for any m. On the level of formulas we have for m ::: 4 the system (2.1), (2.7), where now all indices i, j, k vary between 1 and m. This system consists of interrelated three-dimensional building blocks: for any triple of pairwise different indices (i, j, k) the equations involving these indices only, form a closed subset. The m-dimensional consistency of this system means that all three-dimensional building blocks can be imposed without contradictions. A set of initial data which determines a solution of the system (2.1), (2.7) consists of

(Qf) the values of f on the coordinate axes 13 i for 1 ~ i ~ m; (Q~) the values of Cij, Cji on all elementary squares of the coordinate planes 13 ij , for 1 ~ i < j ~ m.

2.1.2. Transformations of Q-nets. A natural generalization of Definition 1.2 would be the following. Definition 2.7. (Discrete fundamental transformation) Two m-dimensional Q-nets f, f+ : zm ~ ]RN are said to be related by a fundamental transformation (F-transformation) if at every u E zm and for every 1 ~ i ~ m the four points f, Td, f+ and Td+ are co-planar. The net f+ is called an F-transform of the net f.

However, this relation can be rephrased as follows: set F(u,O) = f(u) and F(u,1) = f+(u); then F : zm X {O, 1} ~ ]RN constitutes (two layers of) an M-dimensional Q-net, where M = m + 1. Thus, in the discrete case

39


there is no difference between conjugate nets and their F-transformations. The situation of Definition 2.7 is governed by the equation (2.9) w here the coefficients ai, bi are nothing but ai = 1 + cMi, bi = CiM. These coefficients are naturally attached to the elementary squares of tl m x {O, I} parallel to the coordinate plane 13 iM . It is also convenient to think of them as attached to the edges of tl m parallel to 13 i (to which the corresponding "vertical" squares are adjacent). Equations of the system (2.7) with one of the indices equal to M give: (2.10)

Oiaj

(Tjbi)(aj - 1) + (Tjai - Tiaj)Cij,

(2.11)

Oibj

c0 bj + ctbi -

0

(2.12)

aj c

(Tjai)Cij

(Tibj )bi'

+ (Tjbi)(aj

- 1).

The following data are needed to specify an F-transform f+ of a given m-dimensional Q-net f:

(Ff) the value of f+(O); (F~) the values of ai, bi on all edges of the respective coordinate axis 13 i ,

for 1

~

i ~ m.

The (m + 2)-, resp. (m + 3)-dimensional consistency of Q-nets may be reformulated as the following permutability properties of the discrete F -transformations.

Theorem 2.8. (Permutability of discrete F-transformations)

1) Let f be an m-dimensional Q-net, and let f(1) and f(2) be two of its discrete F-transforms. Then there exists a two-parameter family of Q-nets f(12) that are discrete F-transforms of both f(l) and f(2). The corresponding points of the four Q-nets f, f(l), f(2) and f(12) are coplanar.

2) Let f be an m-dimensional Q-net. Let f(1), f(2) and f(3) be three of its discrete F-transforms, and let three further Q-nets f(12), f(23) and f(13) be given such that f(ij) is a simultaneous discrete F-transform of f(i) and f(j). Then generically there exists a unique Q-net f(123) that is a discrete F-transform of f(12), f(23) and f(13). The net f(123) is uniquely determined by the condition that for every permutation (ijk) of (123) the corresponding points of f(i), f(ij), f(ik) and f(123) are coplanar. Proof. In the first part of the theorem, we define an (m Q-net F : tl m x {O, 1}2 -+ IRN such that

F(u, 0, 0)

= f(u),

F(u, 1,0)

= f(l)(u),

F(u, 0,1)

+ 2)-dimensional = f(2)(u).

40


The only additional initial datum which is required for the construction of F is the value of F(u, 1, 1) = j(12)(u) at one point u E zm, say at u = o. The point j(12)(0) E ]RN can be chosen arbitrarily in the plane through the points j(O), j(1)(O), j(2)(0), which leads to the two-parameter family of Q-nets j(12). The left part of Figure 2.5 illustrates the extension of j(12) along the edges of zm, which is of course governed by the 3D system for the construction of Q-nets; consistency of this procedure for different coordinate directions is assured by the multidimensional consistency of Q-nets. Similarly, the second part of the theorem is actually dealing with the (m + 3)-dimensional Q-net F: zm X {O, 1}3 ---+]RN such that F(u, 0, 0, 0) = j(u), and the shift of F in the (m + i)-th coordinate direction is denoted by the superscript i of j, so that, e.g., F(u, 1, 1, 1) = j(123)(u). For each u E zm the point j(123) is uniquely determined by the other seven points j, j{i), and j(ij), which is illustrated in the right part of Figure 2.5. 0 Td(12)

Td(2)

j(23)

j(12)

j(123) j(13)

j(2)

j(3) I I

Td.----- -

-

(2) I

Td(l)

-

/

j

/

j(1)

-

j(12)

j / / .-----

/

j

/

Figure 2.5. To the permutability theorem for discrete F-transformations.

2.1.3. Alternative analytic description of Q-nets. In a complete analogy with the smooth case, one can give a (nonlocal) description of discrete conjugate nets, with somewhat simpler equations. It should be mentioned that this description is of an affine (rather than projective) nature. One inquires about the existence of a normalization for vectors allel to the edges 8d of a Q-net j such that

Wi

par-

(2.13) with some real coefficients "(ji, called discrete rotation coefficients of a Q-net. Clearly, in general 8i wj is a linear combination of Wi and Wj, and the requirement in (2.13) is that the component along Wj vanishes. Geometrically, the tip of the vector TiWj is obtained as the intersection of the line (fdij) with the line through the tip of the vector Wj parallel to the vector Wi; see Figure 2.6.


Figure 2.6. Geometric construction of the vectors

41

TiWj.

This construction is well defined also in the multidimensional situation. To show this, one has to prove that Ti(TjWk) = Tj(TiWk). But the tips of both vectors can be defined as the intersection of the line (Jijfijk) with the plane passing through the tip of the vector Wk parallel to the plane IIij of the face (J, fi, fij, fj); see Figure 2.7.

Figure 2.7. Vectors of a Q-net.

Wk

are well defined in an elementary hexahedron

Turning to the analytic description, we introduce the real-valued quantities 9i attached to the edges parallel to the i-th coordinate axes according to (2.14)


42

Substituting (2.14) into (2.1), we see that requirement (2.13) is equivalent to (2.15)

i

i= j,

and then the discrete rotation coefficients are given by (2.16)

9i

9i

Cji

,ji = - C j i = Ti9j 9j

1 + Cij

Difference equations (2.15) explicitly define the evolution of the edge functions 9i within any elementary quadrilateral of the Q-net. The compatibility of these difference equations within any elementary hexahedron is guaranteed by the consistency of the construction of the vectors Wi. Of course, compatibility of equations (2.15) can also be checked analytically; see Exercise 2.5. Thus, the normalization coefficients 9i can be specified by prescribing their values on all edges of the corresponding coordinate axes 13 i . We end up with the following alternative description of Q-nets: (2.17)

oil

(2.18)

OiWj

(2.19)

Oi9j

9i W i,

i

,jiWi,

(Tj9d,ij,

i= j, i i= j.

Rotation coefficients satisfy a closed system of difference equations (discrete Darboux system), which replaces system (2.7), and which can be derived as compatibility conditions of the linear difference equations (2.18): (2.20) This system is considerably simpler than (2.7), since it consists of three separate systems with only two unknowns each. Equations (2.19) and (2.20) are implicit, but can be easily solved for the shifted variables, resulting in (2.21) (2.22)

9j

+ 9i/ij

1 -,ij/ji ,kj

i '

+ ,ki/ij

1 -,ij/ji

'

i= j, i

i= j i= k i= i.

The last formula defines an explicit rational 3D map {,kj} f----+ {Ti/kj}. Like the map {Ckj} f----+ {TiCkj} for the local plaquette coefficients, the map (2.22) is 4D consistent, but now this can be checked via an easy computation by hand (see Exercise 2.6).

2.1.4. Continuous limit. Observe that equations (2.1), (2.7) are quite similar to equations (1.1), (1.2) characterizing smooth conjugate nets. We will demonstrate in Chapter 5 that the status of this similarity can be raised to that of a mathematical theorem about approximation of smooth conjugate nets by discrete ones. More precisely, we will show how to choose initial data

2.2. Discrete line congruences

43

for a discrete system (with a small mesh size E) so that it approximates a given m-dimensional smooth conjugate net as E ---- O. Analogously, equations (2.9)-(2.12) are similar to (1.11)-(1.14). Accordingly, initial data of a discrete system with m + 1 independent variables can be chosen so that, keeping one direction discrete, one arrives in the limit at a given smooth conjugate net and its F-transform. From discrete systems with m + 2 and with m + 3 independent variables, one proves the permutability properties of F-transformations formulated in Theorem 1.3, keeping the last two, resp. three, directions discrete. Thus, permutability of F-transformations, which is a nontrivial theorem of differential geometry, becomes an obvious consequence of the multidimensional consistency of discrete conjugate nets, combined with the convergence result mentioned above.

2.2. Discrete line congruences Closely related to Q-nets are discrete line congruences. We will discuss them in the setting of projective geometry from the very beginning. Let [.;N be the space of lines in lRlP'N; it can be identified with the Grassmannian Gr(N + 1,2) of two-dimensional vector subspaces of lRN+1. Definition 2.9. (Discrete line congruence) A map £ : zm ____ [.;N is called an m-dimensional discrete line congruence in JRlP'N (N 2: 3) if every two neighboring lines £, £i (for each u E zm and for each 1 ~ i ~ m) intersect (are coplanar).

For instance, the lines £(i) = (J fi) connecting neighboring points f = f(u) and fi = f(u + ed of an arbitrary Q-net f : zm ---- lRlP'N constitute a discrete line congruence. Likewise, the lines £ = (J f+) connecting the corresponding points f = f (u) and f+ = f+ (u) of two Q-nets f, f+ : zm ---lRlP'N related by an F-transformation build a discrete line congruence. A two-dimensional discrete line congruence is called generic if any two diagonally neighboring lines (that is, £1 and £2, as well as £ and £12) do not intersect. Each of these two pairs of lines span a three-dimensional space V12 , where all four lines £, £1, £2 and £12 lie. See Figure 2.8. Analogously, an m-dimensional discrete line congruence is called generic if for every vertex of an m-dimensional elementary cube the m lines corresponding to all its m neighboring vertices span a space V12 ... m of maximal possible dimension m + 1. This space contains all lines assigned to the vertices of the mdimensional cube. All discrete line congruences we deal with are assumed to be generic. Our construction of line congruences is similar to that of Q-nets. We start with the case m = 2. Given three lines £, £1, £2 of a congruence, one

44


Figure 2.S. Four lines of a generic discrete line congruence.

has a two-parameter family oflines admissible as the fourth line £12: connect by a line any point of £1 with any point of £2. Thus, given any two sequences of lines £ : Z x {O} ---t £N and £ : {O} x Z ---t £N with a common line £(0,0), such that any two neighboring lines are coplanar, one can extend them to a two-dimensional line congruence £ : Z2 ---t £N in an infinite number of ways: on each step of the induction procedure one has a freedom of choosing a line from a two-parameter family. The next theorem shows that generic line congruences are described by a discrete 3D system:

Theorem 2.10. (Discrete line congruences are described by a 3D system) Given seven lines £, ii, £ij E £N (1 S i < j S 3) satisfying the conditions for a generic line congruence, there is a generically unique line £123 that intersects all three £ij. Proof. All seven lines, and therefore also the three-dimensional spaces 7i Vjk = span( £ij, £ik), lie in V 123 . A line that intersects all three of £ij should lie in the intersection of these three three-dimensional spaces. But a generic intersection of three three-dimensional spaces in V123 is a line:

(2.23) It is now not difficult to realize that this line does, indeed, intersect all three of £ij' For instance, 71 V 23 n 72 V 13 = span(£12' £13) n span(£12' £23) is a plane containing £12; therefore its intersection with 73 V12 (the line £123) intersects £12. 0

2.2. Discrete line congruences

45

A similar argument shows (see Exercise 2.10):

Theorem 2.11. (Discrete line congruences are 4D consistent) The 3D system governing generic discrete line congruences is 4D consistent. As in the case of Q-nets, this theorem yields the existence of transformations of discrete line congruences with remarkable permutability properties.

Definition 2.12. (F-transformation of line congruences) Two m-dimensional line congruences C, C+ : zm -. £.,N are said to be related by an Ftransformation if for every u E zm the corresponding lines C = C(u) and C+ = C+(u) intersect, i.e., if the map L : zm X {0,1} -. £.,N defined by L(u, 0) = C(u) and L(u, 1) = C+(u) is a two-layer (m + l)-dimensionalline congruence. It follows from Theorem 2.10 that, given a line congruence C, its Ftransform C+ is uniquely defined as soon as its lines along the coordinate axes are suitably prescribed. According to Definition 2.9, any two neighboring lines C = C(u) and Ci = C(u + ei) of a line congruence intersect at exactly one point F = Cn Ci E JRJP>N, which is thus combinatorially associated with the edge (u, u + ei) of the lattice zm: F = F(u, u+ ei). It is, however, sometimes more convenient to use the notation F(u, u + ei) = F(i)(u) for this point, thus associating it to the vertex u of the lattice (and, of course, to the coordinate direction i).

Definition 2.13. (Focal net) For a discrete line congruence C : zm -. r.,N, the map F(i) : zm -.JRJP>N defined by F(i)(u) = C(u) n C(u + ei) is called its i-th focal net. This is illustrated in Figure 2.8.

Theorem 2.14. (Focal nets are Q-nets) For a nondegenerate discrete line congruence C : zm -. £.,N, all its focal nets F(k) : zm -. JRJP>N, 1 :S k :S m, are Q-nets. Proof. The proof consists of two steps. 1. First, one shows that for the k-th focal net F(k), all elementary quadrilaterals (F(k), Fi(k) , Fi~)' F?») are planar. This is true for any line congruence. Indeed, both points F(k) and F~k) lie on the line Ck, while both points F?) and Fi~k) lie on the line Cik. Therefore, all four points lie in the plane spanned by these two lines Ck and Cik , which intersect by the definition of a line congruence. 2. Second, one shows that for the k-th focal net F(k), all elementary quadrilaterals (F(k), Fi(k) , Fi~k), F?\ with both i # j different from k, are

46


planar. Here, one uses essentially the assumption that the line congruence f is generic. All four points in question lie in each of the three-dimensional spaces Vij

= span(f,fi,fj,£ij)

and

TkVij

= span(fk,£ik,fjk,£ijk)

(see Figure 2.9). Both 3-spaces lie in the four-dimensional space Vijk span( £, £i, £j, £k), so that generically their intersection is a plane. D

Figure 2.9. Elementary (ij) quadrilateral of the k-th focal net.

Corollary 2.15. (Focal net of F-transformation of a line congruence) Given two generic line congruences £, £+ : zm -+ J..,N in the relation of F-transformation, the intersection points F = £ n £+ form a Q-net F : zm -+ ~JlDN . A different aspect of the close relations between Q-nets and line congruences is given in the following theorem. Theorem 2.16. (Extending a Q-net to a line congruence) Given a Q-net f : zm -+ ~N, there exist discrete line congruences £ : zm -+ J..,N such that f (u) E £( u) for every u E zm. Such a congruence is uniquely determined by prescribing the lines £ I'Bi through the points f I'Bi along the coordinate axes. Proof. We present the construction for one elementary 3D cube; the rest of the claim will follow from the 4D consistency. We start with a 2D face. Given two skew lines £1, £2 and a point h2 in the three-dimensional space V12 spanned by these lines, there exists a unique

47

2.3. Discrete Koenigs and Moutard nets

line £12 through !I2 intersecting £1 and £2 (this line is the intersection of the planes through !I2, £1 and through !I2, £2), Having constructed the three lines £12, £13, £23, Theorem 2.10 provides us with the unique line £123 given by (2.23). The point !I23 is the intersection of three planes (2.6). The incidence !I23 E £123 follows from TiITjk C Ti Vjk' 0 Q-nets with line congruences introduced in Theorem 2.16 play an important role in the theory of Q-surfaces; see Section 4.5. They are closely related to discrete Combescure transformations. The following definition is a straightforward discretization of Definition 1.4. Definition 2.17. (Discrete Combescure transformation) Two m-dimensional Q-nets f, f+ : zm --+ ]RN are said to be related by a Combescure transformation if at every u E zm and for every 1 :S i :S m the edges 8d and 8d+ are parallel. The net f+ is called parallel to f or a Combes cure transform of f. The following result is straightforward. Theorem 2.18. (Combescure transformations and line congruences) Given a Q-net f : zm --+ ]RN with a discrete line congruence £ : zm --+ £N, i.e., f(u) E £(u) for every u E zm, there exists a one-parameter family of parallel Q-nets ft : zm --+]RN such that ft(u) E £(u). Such a net is uniquely determined by its one vertex ft(uo) E £(uo). Conversely, a Combescure pair of Q-nets f, f+ determines a line congruence by £(u) = (f(u)f+(u)) for every u E zm.

2.3. Discrete Koenigs and Moutard nets 2.3.1. Notion of dual quadrilaterals. Definition 2.19. (Dual quadrilaterals) Two quadrilaterals (A, B, C, D) and (A*, B*, C*, D*) in a plane are called dual if their corresponding sides are parallel: (2.24) (A* B*) II (AB), (B*C*) II (BC), (C* D*) II (CD), (D* A*) I (DA), and the noncorresponding diagonals are parallel:

(2.25)

(A*C*)

I

(BD),

(B* D*)

I

(AC).

Lemma 2.20. (Existence and uniqueness of dual quadrilateral) For any planar quadrilateral (A, B, C, D), a dual one exists and is unique up to scaling and translation.


48

c B*

M

D

A*

Figure 2.10. Dual quadrilaterals.

Proof. Uniqueness of the form of the dual quadrilateral can be argued as follows. Denote the intersection point of the diagonals of (A, B, e, D) by M = (AC) n (BD). Take an arbitrary point M* in the plane as the designated intersection point of the diagonals of the dual quadrilateral. Draw two lines £1 and £2 through M* parallel to (AC) and (BD), respectively, and choose an arbitrary point on £2 to be A *. Then the rest of construction is unique: draw the line through A* parallel to (AB); its intersection point with £1 will be B*; draw the line through B* parallel to (Be); its intersection point with £2 will be e*; draw the line through C* parallel to (CD); its intersection point with £1 will be D*. It remains to see that this construction closes, namely that the line through D* parallel to (DA) intersects £2 at A *. Clearly, this property does not depend on the initial choice of A * on £2, since this choice only affects the scaling of the dual picture. Therefore, it is enough to demonstrate the closing property for some choice of A *, or, in other words, to show the existence of one dual quadrilateral. This can be done as follows. Denote by el and e2 some vectors along the diagonals, and introduce the coefficients a, ... ,(j by (2.26)

so that ----t

AB = (3e2 - ae1, ---+ CD = (je2 - ,el,

(2.27)

---+

Be ---+ DA

= ,el = ae1 -

Construct a quadrilateral (A * , B* , C* , D*) by setting (2.28)

M*JP =

_ e2 ,

a

M*i? =

_ e1 (3'

(3e2' (je2.


49

Its diagonals are parallel to the noncorresponding diagonals of the original quadrilateral, by construction. The corresponding sides are parallel as well: -------+ 1 1 1 -4 A*B* --el + -e2 = -AB, f3 a af3 -------+ 1 1 1 ----+ B*C* - - e2 + - el = - BC, "I f3 f3'Y -------+ 1 1 1 ----+ C*D* --el + -e2 = -CD,

8

"1 8

"I

1 1 1 ----+ --e2+-el=-DA.

-------+

D*A*

a

8

8a

Thus, the quadrilateral (A*, B*, C*, D*) is dual to (A, B, C, D).

D

Note that the quantities a, . .. ,8 in (2.26) are not well defined by the geometry of the quadrilateral (A, B, C, D), since they depend on the choice of the vectors el, e2. Well defined are their ratios, which can be viewed also as ratios of the directed lengths of the corresponding segments of diagonals, say "I: a = l(M, C) : l(M, A) and 8 : f3 = l(M, D) : l(M, B). It is natural to associate these ratios with directed diagonals:

Definition 2.21. (Ratio of diagonal segments) Given a quadrilateral (A, B, C, D), with the intersection point of diagonals M = (AC) n (BD), we set -4 l(M,C) (i3D) = l(M, D) (2.29) q(AC) = l(M, A) , q l(M,B)· Changing the direction of a diagonal corresponds to inverting the associated quantity q. Note that (2.30)

(A, B, C, D) convex

{:::}

-4

--+

q(AC) < 0 and q(BD)
0 --+ -4--+ and q(BD) < 0; and a nonembedded one with q(AC) > 0 and q(BD) > o.

2.3.2. Notion of discrete Koenigs nets. Definition 2.22. (Discrete Koenigs net) A Q-net f : zm -+ ]RN is called a discrete Koenigs net if there exists a Q-net 1* : zm -+ ]RN, called Christoffel dual to f, such that any elementary quadrilateral of the net 1* is dual to the corresponding quadrilateral of the net f:

118d, 821* 11821, fi2 - 1* II h - 12, fi - fi II h2 - f. 8d*

(2.31 )


50

M

D D

A

B

A

Figure 2.11. Different forms of planar quadrilaterals.

This definition can be seen as a discretization of conditions (1.27).

In order to understand restrictions imposed on a Q-net by this definition, we start with the following construction. Each lattice zm is bi-partite: one can color its vertices black and white so that each edge connects a black vertex with a white one (for instance, one can call vertices u = (Ul' ... ,um ) with an even value of lui = Ul + ... + U m black and those with an odd value of lui white). Each elementary quadrilateral has a black diagonal (the one connecting two black vertices) and a white one. One can introduce the black graph Z:en with the set of vertices consisting of black vertices of zm and the set of edges consisting of black diagonals of all elementary quadrilaterals of zm, and the analogous white graph Z~d' The geometry of the elementary quadrilaterals of a Q-net f : zm -+ ]RN induces, according to Definition 2.21, the quantities q (ratios of directed lengths of diagonal segments) on all directed diagonals, white and black. Definition 2.23. (Discrete one-form) Let G be a graph with the set of vertices V and with the set of directed edges E. Let W be a vector space.

(i) A function p : E -+ W is called a discrete (additive) one-form on G if p( -e) = -p( e) for every directed edge e E E. It is called exact if for every cycle of directed edges the sum of the values of p along this cycle vanishes.

(ii) A function q : E -+ ]R* is called a multiplicative one-form on G if q ( - e) = 1/ q (e) for every directed edge e E E. It is called exact if for every cycle of directed edges the product of the values of q along this cycle is equal to one.


52

Figure 2.12. Four quadrilaterals around a vertex of a two-dimensional net.

Similarly, we find: A2 A3

a202 a3{J3'

A3 A4

a303 a4{J4'

A4 Al

a404 al{Jl·

All four edges adjacent to I can be matched if and only if the cyclic product of expressions for the quotients of scaling factors is equal to one. This condition reads: alOl a2 02 a303 a404 --·--·--·--=1 a2{J2 a3{J3 a4{J4 al{Jl ' or (2.32)

01 02 03 04 _ 1 {Jl . {J2 . {J3 . {J4 - .

This is nothing but the exactness condition of the form q for an elementary quadrilateral of the white graph. All other white and black cycles are sums of elementary ones; therefore conditions (2.32) for all elementary white and black cycles are necessary and sufficient for the exactness of the form q. On the other hand, it is easy to see that if the exactness condition is fulfilled for all white and black cycles, then no closed chain of quadrilaterals can lead to an obstruction during the construction of the dual net.

m = 3. In this case the simplest closed chain of quadrilaterals is given by three faces of any elementary hexahedron of the net, sharing a (black, for definiteness) vertex I; see Figure 2.13. The further arguments are completely analogous to the two-dimensional case. Matching the edges shared by the dual quadrilaterals 1 and 2, by the dual quadrilaterals 2 and 3, and by


53

Figure 2.13. Three quadrilaterals around a vertex of a threedimensional net.

the dual quadrilaterals 3 and 1, we find the relations between their scaling factors: Al

cwh

A2

CY282

A3

CY3 83

A2

CY2fi2'

A3

CY3fi3'

Al

CYlfil·

All three edges adjacent to f can be matched simultaneously if and only if the cyclic product of expressions for the quotients of scaling factors is equal to one, which condition after cancellations reads: (2.33) This is nothing but the exactness condition for the elementary cycle of the white graph of the lattice Z3, which is a triangle. All cycles of the white and of the black graphs (including those encountered in the case m = 2, i.e., the squares of the two-dimensional slices of the white and the black graphs of Z3) are sums of elementary triangles; see Figure 2.14. Again, exactness condition for all white and black cycles guarantees that no closed chain of quadrilaterals leads to an obstruction.

m > 4. Also in this case any white or black cycle is a sum of elementary 0 triangles, as for m = 3; therefore no additional conditions appear. Remark. Note that it is impossible for three quadrilaterals of a discrete Koenigs net adjacent to a common vertex, like in Figure 2.13, to be convex simultaneously, since in the latter case the product of the corresponding three quantities q would be negative.

54


Figure 2.14. The elementary square in the black graph of 7L,2 is a sum of four elementary triangles in the black graph of ;Z3.

2.3.3. Geometric characterization of two-dimensional discrete Koenigs nets. The definition of discrete Koenigs nets obviously belongs to affine geometry, since it relies on the notion of parallelism. It turns out, however, that the class of discrete Koenigs nets is projectively invariant. The proof of the corresponding projectively invariant characterizations relies on the generalized Menelaus theorem (Theorem 9.12), which has a similar flavor: its hypotheses are of affine-geometric nature, while its conclusions are projectively invariant. In the following considerations, we use the negative indices -1, -2 to denote the downward shifts T 11, T;;I. We characterize two-dimensional Koenigs nets in terms of the four elementary quadrilaterals adjacent to the point f = f(u), i.e., the quadrilaterals (I, h fij, fj) with (i,j) E {(±1, ±2)}. Theorem 2.26. (Discrete 2D Koenigs nets; characterization in terms of intersection points of diagonals) Let f : 7i} ----'> }RN be a Q-net such that for every point f = f(u) its four neighbors f±1' f±2 are not coplanar. Then f is a discrete Koenigs net if and only if for every point f = f (u) the intersection points of diagonals of the four quadrilaterals adjacent to fare coplanar, that is, if the intersection points of diagonals build a Q-net. Proof. This is an immediate consequence of equation (2.32) and the case n = 4 of the generalized Menelaus theorem (Theorem 9.12). D Sometimes another characterization is more convenient, where only the points of the net participate, and not the intersection points of diagonals of its quadrilaterals. Theorem 2.27. (Discrete 2D Koenigs nets; characterization in terms of vertices) Let f : Z2 ----,>}RN be a Q-net such that for every point f = f (u) its four neighbors f ±1, f ±2 are not coplanar.

I) If at every vertex the four adjacent quadrilaterals span a 4-dimensional space, then f is a discrete Koenigs net if and only if for every u E Z2 the


55

five points f and f±I,±2 lie in a three-dimensional subspace V = V(u) C]RN not containing the four neighbors f±l, f±2. Note that if one of the neighbors lies in V, then all four do. II) If N = 3, then f is a discrete Koenigs net if and only if for every u E 'Z} the three planes n(up) =

Uh2f-l,2),

have a common line n(left) =

e(l),

n(down) =

Uh,-d-l,-2),

n(1) =

Ufd-d

or, equivalently, the three planes

Uf-l,d-l,-2),

n(right) =

U!I,2!I,-2),

n(2) =

Uhf-2)

have a common line e(2) . Proof. I) First let f be a Koenigs net and therefore satisfy the property of Theorem 2.26. Then the space V through f and f ±l,±2 is clearly threedimensional. Conversely, let this space be three-dimensional. The four quadrilaterals U, Ii, /ij, fj) lie in a four-dimensional space through f, f±l' f ±2· The intersection points of their diagonals lie in the intersection of V with the three-dimensional space through f ±l, f ±2. The intersection of two three-dimensional subspaces of a four-dimensional space is generically a plane.

II) Let Mij be the intersection point of diagonals of the quadrilateral U, Ii, /ij, fj), with (i,j) E {(±1, ±2)}. Coplanarity of the four points Mij is equivalent to the statement that the lines (MI,2M-l,2) and (MI,-2M-I,-2) intersect. Denote their intersection point by p(l). These two lines lie in the planes (hhf-l), Ud-2f-l), respectively; therefore their intersection point must belong to the intersection of these planes: pel) E Ud-1). Thus, coplanarity of the points Mij is equivalent to the fact that three lines (Ml,2M-I,2), (MI,-2M-I,-2), and (hf-d have a common point p(l); see Figure 2.15. Now the planes n(u p ), n(down) and n(1) can be viewed as the planes through the point f and the lines (MI,2M-l,2), (MI,-2M-I,-2), and Ud-l), respectively. Therefore their intersection is the line e(1) through f and p(l). 0

Remark. The equivalence of two conditions in part II of Theorem 2.27 follows, of course, from the fact that in the notion of discrete Koenigs nets there is no asymmetry between the coordinate directions 1 and 2. However, it is worthwhile to see a more direct argument in favor of this equivalence. Such an argument is given by the (spatial version of) Desargues theorem. Indeed, if the triangles 6UIMI,2MI,-2) and 6U-lM_l,2M-l,-2) are perspective from the point p(l), then they are also perspective from the line which is the intersection of the planes containing these two triangles. Clearly, this line is nothing but (hf-2), which therefore contains the intersection point p(2) of the lines (M1,2Ml,-2) and (M-I,2M-I,-2).


56

Figure 2.15. Four quadrilaterals of a Koenigs net around a vertex.

Remark. It is not difficult to see that in the dimension N 2: 4 the condition in part I of Theorem 2.27 automatically yields the condition in part II (in the general position case). Indeed, for N 2: 4 all nine points f, f±l, !±2 and !±1,±2 generically lie in a four-dimensional subspace of ~JP>N. In this subspace one can consider, along with the three-dimensional subspace V, the three-dimensional subspaces V(up) containing the two quadrilaterals (f,fI,fI2,h), (f,f-1,f-1,2,h), and v(down) containing the quadrilaterals (f,f1,!I,-2,f-2), (f,f-1,f-1,-2,f-2). Obviously, one has: rr(up)

= V(up) n V,

rr(down)

= V(down) n V,

rr(1)

Generically, three three-dimensional subspaces V, four-dimensional space intersect along a line £(1).

= V(up) n V (down) .

V(up)

and

v(down)

of a

2.3.4. Geometric characterization of three-dimensional discrete Koenigs nets. Theorem 2.28. (Discrete 3D Koenigs nets; characterization in terms of intersection points of diagonals) A three-dimensional Q-net f : Z3 --+ ~N is a discrete Koenigs net if and only if for every point f = f (u) and for every elementary hexahedron with a vertex f, the intersection points of diagonals of the three hexahedron faces adjacent to f are collinear. Proof. This is nothing but the reformulation of (2.33) in terms of Menelaus' 0 theorem (n = 3 case of Theorem 9.12). A characterization in terms of the net points only is as follows.

Theorem 2.29. (Discrete 3D Koenigs nets; characterization in terms of vertices) A Q-net f : Z3 --+ ~N is a discrete Koenigs net if and only if


57

for every elementary hexahedron of the net its four white vertices are coplanar, or its four black vertices are coplanar (each of these conditions implies the other one). Proof. Consider an elementary hexahedron with the vertices f, Ii, lij, iI23' Denote the intersection points of diagonals of the quadrilaterals (J, Ii, fij, fy) by M ij . Clearly, if the points Mij are collinear, then the four points f and lij (the black ones) are coplanar. Similarly, the four white vertices of the elementary hexahedron are coplanar.

We show next that for a Q-net, coplanarity of the four black points of an elementary hexahedron yields coplanarity of the four white points. Suppose that the four black points f, lij lie in a plane IIo. Let III be the plane through the three points iI, 12, h. Set C = IIo n III. Then the intersection points Mij of diagonals of the quadrilaterals (J, Ii, fij, fj) belong to C. Denote by Oij the intersection points of the lines (Jikfjk) C IIo with C. Then the classical Pappus theorem (Theorem 9.16) yields that the three lines (iI023), (120 13 ), (J30l2) c III intersect in one point. This point is clearly iI23 E III, so that the four points Ii, iI23 are coplanar. The points Oij = (Jikfjk) n (fkiI23) are the intersection points of diagonals of the quadrilaterals (ik, lib iI23, fyk). See Figure 2.16, which is nothing but Figure 9.12 with our present notation.

f ITo

Figure 2.16. Elementary hexahedron of a 3D Koenigs net.

Now we can finish the proof of Theorem 2.29 as follows. Suppose that the black vertices of an elementary hexahedron of a Q-net are coplanar. Then also the white vertices of this hexahedron are coplanar. Then the


58

intersection points of diagonals of all six faces of the hexahedron are collinear (they belong to the common line of the "black" and the "white" planes). According to the characterization of Theorem 2.28, the net is Koenigs. 0

2.3.5. Function v and Christoffel duality. We start with the following statement, which is a direct consequence of the algebraic characterization of discrete Koenigs nets given in Theorem 2.25, and of Theorem 2.24. Theorem 2.30. (Function v for a discrete Koenigs net) A Q-net f : zm -> jRN is a discrete Koenigs net if and only if there exists a real-valued function v : zm -> jR* with the following property: for every elementary quadrilateral (f, Ii, fij, fj), Vj Vi

(2.34) where M

= (f~f.) = q

t

J

l(M, fj) l(M, fi) ,

= (f fij) n (fdj) is the intersection point of the diagonals.

On both the black and the white graphs of zm such a function v is defined up to a multiplicative constant. This freedom is fixed by prescribing values of v arbitrarily at one black and at one white point. Equations (2.34) are equivalent to (2.35)

1 ---+ -Mfij Vij

1~

= -Mf, v

which can be re-written also as (2.36) It follows that (2.37) This formula can be used for an elegant representation of the dual Koenigs net for f.

Theorem 2.31. (Christoffel duality) Let f : zm -> jRN be a discrete Koenigs net, and let v : zm -> jR* be the function defined by the property (2.34). Then the jRN -valued discrete one-form 81* defined by (2.38)

8d*

=

8d

VVi

is exact. Its integration defines (up to a translation) the Christoffel dual Koenigs net 1* : zm -> jRN.


59

Proof. Equation (2.37) can be equivalently rewritten as

(2.39)

Iij - Ii ViVij

+ fi

- f VVi

= fij - fj + fj - f. VjVij

VVj

This is equivalent to the exactness of the discrete form IS1*. Note that equation (2.38) says that the corresponding sides of elementary quadrilaterals of the nets f and 1* are parallel. It remains to show that the noncorresponding diagonals of elementary quadrilaterals of f and 1* are also parallel, so that these quadrilaterals are dual in the sense of Definition 2.19. To this end, we prove the following two formulas: (2.40) where

(2.41)

aij

= (V:j - t)/(:j - :).

Indeed, using (2.37) and (2.41) we find that

~ Iij - f

aij

VijV

o

Theorem 2.31 is completely proven.

For future reference, we note here that after some manipulations formula (2.37) can be transformed into the point equation of discrete Koenigs nets (2.42) The following data determine a solution to this equation and therefore a discrete Koenigs net uniquely:

(Kf) the values of f on the coordinate axes

'Bi

s: i s: m; for 1 s: i < j s: m.

for 1

(K~) the values of v on the coordinate planes 'Bij

60


2.3.6. Moutard representative of a discrete Koenigs net. Constructions of the previous subsection (functions 1/ and aij for a given Koenigs net) can be used also in a different spirit.

Theorem 2.32. (Discrete Koenigs nets = discrete Moutard nets in homogeneous coordinates) A Q-net f : zm - t ]RN is a discrete Koenigs net if and only if there exists a function 1/ : Y : zm - t ]RN+l,

zm

-t

]R*

such that the points

(2.43) satisfy the discrete Moutard equation with minus signs

(2.44)

TiTjY - Y

= aij(TjY -

TiY)

with aij E]R given by (2.41). The net Y = I/-l(f, 1), considered as a special lift of f : zm - t ]RpN to the space of homogeneous coordinates, will be called the Moutard representative of the discrete Koenigs net f.

Proof. First let f : zm - t ]RN be a discrete Koenigs net. Define the function 1/ : zm - t ]R* according to Theorem 2.30. Then equation (2.36) holds, with M being the intersection point of the diagonals of the quadrilateral (f, Ii, /ij, fj)· Denoting Y = I/-l(f, 1), we immediately arrive at (2.44) with the coefficients aij defined by (2.41). Note that the quantities aij are naturally assigned to elementary squares of zm parallel to the coordinate plane 'B ij . Conversely, given a solution Y : zm - t ]RN+l of the Moutard equation (2.44) in ]RN+l, define 1/ : zm - t ]R* and f : zm - t ]RN by Y = I/-l(f, 1). In other words, let 1/-1 denote the last component of y, and let f be the vector in ]RN obtained by multiplying the first N components of Y by 1/. Then, inverting the previous arguments, it is easy to show that f is a discrete Koenigs net. Indeed, one immediately finds expression (2.41) for the coefficient aij of the Moutard equation, and then from Yij - Y

=

aij(Yj - Yi)

follows (2.37). This allows us to define the point M by (2.36). The latter equation is equivalent to (2.35); therefore M is nothing but the intersection point of the diagonals of (f, Ii, fij,!j). Equation (2.34) holds, so f is a Koenigs net by Theorem 2.30. 0 2.3.7. Continuous limit. In order for a Q-net to admit a continuous limit, all its quadrilaterals should be of a reasonable shape. Anyway, they must be convex. As mentioned in Subsection 2.3.2, diagonals of convex quadrilaterals carry negative quantities q (ratios of segments of diagonals). Theorem 2.25 shows that a discrete Koenigs net cannot consist of convex quadrilaterals (and thus cannot admit a continuous limit) for m ~ 3. However, there are


61

no obstructions in the case m = 2. This is in a good agreement with the existence of two-dimensional smooth Koenigs nets only. Equations (2.34) show that in the case m = 2 with all convex quadrilaterals, we can assume, without losing generality, that the sign of v( u) at u = (Ul,U2) E'l} is either (-1)u l or (_1)U 2 • Clearly, such a wildly oscillating function cannot have a well-behaved continuous limit. However, upon redefining (2.45) we get a positive function, which turns out to be a proper discrete analog of the function v for smooth Koenigs nets. Note that this redefinition is equivalent to changing (2.34) to (2.46)

V12

v

1(f12,M) l(M, J) ,

1(12, M) l(M,!I) .

We mention also that equation (2.42) with the redefined v becomes (2.47) with (1.25) as a continuous limit. Likewise, formulas (2.38) turn into

(2.48)

8d* = 8d ,

82 1* = _ 82 f ,

VVI

VV2

where the second possible redefinition of v in (2.45) has been used, for definiteness (the first would result in changing signs of both fractions). Under continuous limit these formulas turn into (1.26). For the Moutard representative y : 'I} ----> discrete Koenigs net the change (2.45) leads to

]RN+l

of a two-dimensional

(2.49) These points satisfy the discrete Moutard equation with the plus signs:

(2.50)

TIT2Y

+ Y = a12(TIY + T2Y)·

A smooth counterpart of this equation is the Moutard equation (1.29).

2.3.8. Notion and consistency of T-nets. The nets satisfying the discrete Moutard equation are of interest by themselves, independently of the interpretation as special lifts of discrete Koenigs nets. Thus, we introduce the following notion. Definition 2.33. (T-net) A map Y : zm ---->]RN is called an m-dimensional T-net if for every u E zm and for every pair of indices i :/= j, the discrete Moutard equation with minus signs, (2.51)


62

holds with some aij : zm --t ]R; in other words, if all elementary quadrilaterals (y, TiY, TiTjY, TjY) are planar and have parallel diagonals. The definition of T-nets, considered irrespectively of the Koenigs nets, refers to the notion of parallelism and thus belongs to affine geometry. Of course, the coefficients aij must be skew-symmetric, aij = -aji. As usual, we will consider these functions as attached to elementary squares of zm parallel to the coordinate plane 'B ij . Any T-net is a Q-net. For a given Q-net with coefficients Cij, the T -condition (parallel diagonals) is expressed as Cij + Cji + 2 = 0, and then the skew-symmetric coefficients aij in (2.51) are given by aij = Gij + 1. m = 2 : T-surfaces. To define a two-dimensional T-net Y : Z2 --t ]RN, one can arbitrarily prescribe two coordinate curves, Y f'Bl and Y f'B2 , and a real-valued function a12 on elementary squares of Z2.

=

m 3 : basic 3D system. We show that three-dimensional T-nets are described by a well-defined 3D system. An inductive construction step of the net f is as follows.

Theorem 2.34. (Elementary hexahedron of a T-net) Given seven points y, Yi, and Yij (1 ::; i i=- j ::; 3) in ]RN satisfying (2.51), there exists a unique point Y123 such that all three quadrilaterals (Yi, Yij, Y123, Yik) have parallel diagonals. Proof. Three equations (2.51) for the faces of an elementary cube in Figure 2.2 adjacent to Y123, give: TiYjk =

(1 + hajk) (aij + aki))Yi

- (Tiajk)aijYj - hajk)akiYk.

They lead to consistent results for Y123 for arbitrary initial data if and only if the following conditions are satisfied: 1 + (Tl a23) (a12

+ a31) 1 + (T2a3t)(a23 + a12) 1 + (T3aI2)(a31 + a23)

-(T2 a31)aI2 -(T3aI2)a23

-(T3 a I2)a31' -(Tl a23)aI2,

- (Tl a23)a31

- (T2 a3t) a23.

These conditions constitute a system of 6 (linear) equations for 3 unknown variables Tiajk in terms of the known ajk. A direct computation shows that this system is not overdetermined but admits a unique solution: (2.52)

1

With Tiajk so defined, equations (2.51) are fulfilled on all three quadrilaterals 0 adjacent to Y123.


63

Equations (2.52) define a birational map {ajd r--t {Tiajk}, which can be considered as the fundamental 3D system related to T-nets. It is sometimes called the "star-triangle map". The local statement of Theorem 2.34 immediately yields the m = 3 case of the following theorem.

Theorem 2.35. (T-reduction ofQ-nets) lfthe coordinate surfacesyf'Bij of a Q-net y : zm --* ~N are T-surfaces, then y is aT-net. Proof. Let the quadrilaterals (y, Yi, Yij, Yj) be planar and have parallel diagonals. The planarity of the the quadrilaterals (Yi, Yij, Y123, Yik) defines the point Y123 as the intersection point of three planes TiIIjk. Then these three quadrilaterals automatically have parallel diagonals. Indeed, by Theorem 2.34 there exists a point Y123 with this property. It satisfies the planarity condition, and therefore it must coincide with the unique point defined by the planarity condition. 0 m > 4 : consistency. The 4D consistency of T-nets is a consequence of the analogous property of Q-nets, since the T-constraint propagates in the construction of a Q-net from its coordinate surfaces. As a consequence, Theorem 2.35 holds for m 2: 4. On the level of formulas we have for T-nets with m 2: 4 the system (2.51), while the map {ajd r--t {Tiajd is given by (2.53)

1

Tiajk _ ajk

aijajk

+ ajkaki + akiaij

All indices i,j, k vary now between 1 and m, and for any triple of pairwise different indices (i,j, k), equations involving these indices solely, form a closed subset. Initial data which allow for a unique solution of this system consist of: (T~) the values of Y on the coordinate axes 13 i for 1 :::; i :::; m; (T~) the values of m( m - 1) /2 functions aij on all elementary squares of the coordinate planes 13 ij for 1 :S i < j :::; m.

2.3.9. Transformations of T -nets. Because of the multidimensional consistency, transformations of T-nets do not differ from the nets themselves. Definition 2.36. (Discrete Moutard transformation) A pair of mdimensional T-nets y, y+ : zm --* ~N is said to be related by a Moutard transformation if for every u E zm and for every 1 :::; i :::; m the diagonals of the quadrilateral (y, TiY, TiY+, y+) are parallel. The net y+ is called a Moutard transform of the net y. Thus, the net Y : zm X {O, I} --* ~N defined by Y(u,O) = y(u) and Y(u, 1) = y+(u), is an (m+ I)-dimensional T-net. Moutard transformations


64

of T -nets are governed by

1 ::; i ::; m,

(2.54)

where the quantities bi = aiM, M = m + 1, defined on the "vertical" plaquettes of zm x {O, 1}, parallel to 'BiM, can be also associated to the edges of zm parallel to the coordinate axes 'B i . Equations (2.53) with one of the indices equal to M express compatibility of (2.54) with (2.51):

a~

Tibj bj

(2.55)

aij

1 (bi - bj )aij

+ bibj .

A Moutard transform y+ of a given T-net y can be specified by prescribing the following data: (MT~) the value of y+(O); (MT~) the values of bi on "vertical" elementary squares attached to all edges of the coordinate axes 'Bi for 1 ::; i ::; m.

The multidimensional consistency of T-nets immediately leads to the following theorem.

Theorem 2.37. (Permutability of discrete Moutard transformations) 1) Let y be a T-net, and let y(1) and y(2) be two of its Moutard transforms. Then there exists a one-parameter family of T-nets y(12) that are Moutard transforms of both y(l) and y(2). The corresponding points of the four nets satisfy the following condition: y(12) - y is parallel to y(2) _ y(l).

2) Let y be a T-net. Let y(l), y(2) and y(3) be three of its Moutard transforms, and let three further T-nets y(12), y(23) and y(13) be given such that y(ij) is a simultaneous Moutard transform of y(i) and y(j). Then generically there exists a unique T-net y(123) that is a Moutard transform of y(12), y(23) and y(13). The net y(123) is uniquely determined by the condition that for every permutation (ij k) of (123) the corresponding points of y(i), y(ij), y(ik) and y(123) are coplanar.

Proof. In part 1) of the theorem, one constructs an (m T-net Y: zm x {0,1}2 such that Y(u,O,O)

= y(u),

Y(u, 1,0)

= y(1)(u),

Y(u,O, 1)

+ 2)-dimensional = y(2)(u),

and then one sets y(12)(U) = Y(u, 1, 1). For this, an initial condition y(12) (0) must be chosen so that y(12)(0) - y(O) is parallel to y(2)(O) - y(l)(O). This leads to a one-parameter family of y(12). In part 2), we are dealing with the initial data which uniquely define an (m + 3)-dimensional T-net Y : zm x {a, 1}3, which gives y(123)(u) = Y(u, 1, 1, 1). D


65

2.3.10. Discrete M-nets. Recall that under the change of signs (2.49) the discrete Moutard equation (2.51) with m = 2 turns into the discrete Moutard equation with the plus signs (2.50). If one would like to consider solutions of such an equation in ]RN, disregarding their interpretation as special representatives of discrete Koenigs nets in the space of homogeneous coordinates, one arrives at the following notion. Definition 2.38. (Discrete M-net) A map y : 'Z} ----+ ]RN is called a discrete M-net if it satisfies the discrete Moutard equation with plus signs: (2.56) or, equivalently,

(2.57) with some a12 = 1 +

! q12 : 7!}

----+

lR..

Clearly, the continuous limit of (2.57) is the classical Moutard differential equation (1.29). As in the smooth context, discrete M-nets are not a particular case of Q-nets, as long as they are not considered as homogeneous coordinates of points of a projective space. Initial data that can be used to determine a discrete M-net are: (M~) the values of y on the coordinate axes '13 1 , '13 2 ;

(M~) a real-valued function a12 defined on the elementary squares of 'Z}.

As in the smooth case, there are no m-dimensional discrete M-nets with m 2: 3. However, a close relation with multidimensionally consistent T-nets immediately leads to Moutard transformations of discrete M-nets. Start wi th a three-dimensional T -net Y : Z2 x {O, I} ----+ ]R N, interpreted as the Moutard pair of T-surfaces y = Y(',O) and y+ = Y(-,l), and perform the change of variables

This is equivalent to the change of variables

leading to two discrete M-nets y, y+. Under this change of variables, and upon renaming b2 (u) I---> -b2 (u), equations (2.54), (2.55) turn into the following ones: (2.58) (2.59)

1


66

which thus describe Moutard transformation of discrete M-nets. Note that (2.58) can be equivalently represented as

(2.60)

81 y+

+ 81 y = Pl(Y+ -

TIY),

82y+ - 82y = P2(Y+

+ T2Y),

where Pi = bi - 1. A Moutard transform y+ of a given discrete M-net Y can be specified by prescribing the data (MT~2) with m = 2. Theorem 2.37 holds literally for discrete M-nets.

2.4. Discrete asymptotic nets 2.4.1. Notion and consistency of discrete asymptotic nets. Definition 2.39. (Discrete A-net) A map f : zm ---+ IR3 is called an m-dimensional discrete A -net (discrete asymptotic net) in IR3 if for every u E zm all the points f(u ± ei), i E {1, ... ,m}, lie in some plane :?(u) through f(u). A discrete A-net f is called non-degenerate, if all its elementary quadrilaterals (j, Til, TiTjf, Tjf) are non-planar. Note that Definition 2.39 belongs to the projective geometry and could equally well be formulated for the ambient space IR1P'3. In principle, it would be possible to consider discrete A-nets in IRN with N > 3, however this would not lead to an essential generalization. Indeed, fix any u E zm and any pair of indices i i- j from {1, ... , m}, and consider the three-dimensional affine subspace V of IRN through the four points f, Til, Tjf and TiTjf. A simple induction shows that the whole net f lies in this subspace. Indeed, one argues that, with an arbitrary index k from {1, ... , m}, f,TiI,Tjf E V f, Til, TiTjf E V Til, Tjf, TiTjf E V

::::}:? ::::}

::::}

c V::::}

Ti:? C V TiTj:? C V

::::}

::::}

Tkf E V, TiTkf E V, TkTiTjf E V.

Thus, for any k, the four points Tkf, TkTiI, TkTjf and TkTiTjf belong to V. This justifies the inductive conclusion. m = 2 : discrete A-surfaces serve as discrete counterparts of surfaces parametrized along their asymptotic lines. For a construction of a discrete A-surface f : Z2 ---+ IR3 , one can start with two arbitrary discrete coordinate curves f l'Bi (i = 1,2), satisfying two conditions: first, all four neighbors f(±ed of the point f(O) lie in one plane :?(O), and second, for every u E 13 i , the three points f (u) and f (u ± ei) are not collinear. The latter condition assures that the planes :?(u) are well defined for all u E 13 i . Now these data can be recursively extended to a discrete A-surface: one step of this procedure for the quadrant Z~, say, consists of choosing f(u+el +e2) on the straight line through f (u) which is the intersection of the planes :?( u + q)

2.4. Discrete asymptotic nets

67

and P(u + e2). Clearly, there is one free real parameter on each step of this extension procedure. As usual, it is convenient to think of this parameter as attached to the corresponding elementary squares of 'I}. m = 3 : basic 3D system. Given any eight points f, Ii, fij, fI23 in lR.3 (1 :S i < j :S 3), define the eight planes P, Pi, Pij , P 12 3 as follows: P,

the plane through fI, 12, 13,

Pi, the plane through f, fij, fik' Pij , the plane through Ii, fj, fI23, P 123 , the plane through f12' fI3, 123. Thus, one obtains two tetrahedra, one with the vertices (f, fI2, 123, 131) and with the face planes (PI, P 2, P3, P 123 ), and the second with the vertices (fI, 12, 13, fI23) and with the face planes (P, P 12 , P 23 , P 31 ). Now the condition that the eight points are vertices of an elementary hexahedron of a discrete three-dimensional A-net is expressed as the following eight incidence relations: (2.61) Thus, each vertex of each tetrahedron lies in the corresponding face plane of the other one. In other words, the two tetrahedra are inscribed in each other. Such pairs of tetrahedra are called Mobius pairs. We see that the geometry of an elementary hexahedron of a discrete three-dimensional A-net is exactly that of a Mobius pair of tetrahedra. Mobius demonstrated that eight conditions (2.61) are not independent: anyone of them follows from the remaining seven.

Theorem 2.40. (Mobius pair of tetrahedra) If the four vertices of one tetrahedron lie respectively in the four face planes of another, while three vertices of the second lie in three face planes of the first, then the remaining vertex of the second lies in the remaining face plane of the first. Proof. A construction of a Mobius tetrahedra pair may be performed as follows. Let f be some point in lR.3 , and let P and Pi (i = 1,2,3) be four planes through f such that any three of them are in general position. For i = 1,2,3, let fi be an arbitrary point on the line P n Pi, and, for (ij) = (12), (23), (31), let fij be an arbitrary point on the line Pi n Pj. Now, construct the planes Pij through Ii, fj, fij, and the plane P 123 through fI2, 123, 131. Then Theorem 2.40 claims that the last point fI23, uniquely defined as the intersection of the planes P 12 , P 23 , P 31 , automatically lies in the plane P123 . The proof of this claim is based on the Pappus theorem (Theorem 9.16 and Figure 9.12) that has already been used in Theorem 2.29, but with a

68


different combinatorial interpretation indicated by notation in Figure 2.17. The upper half of this figure represents the plane :P, and the lower half represents the plane :P 123 . Each plane :Pi is represented by a pair of lines :Pi n:P = (f Ii) and :Pi n :P 123 = (fijIik) meeting on e = :P n :P 123 . Similarly, each plane :Pij is represented by a pair of lines :Pij n:P = (filj) and :Pij n:P 123 , which is a line passing through Iij and meeting (filj) on e. The claim of the theorem can be now interpreted as follows: three lines in :P123 which are traces of :P 12 , :P23 , :P31 in this plane, are concurrent (with the common point being h23). But this is nothing but the Pappus theorem. D

f

Figure 2.17. Mobius theorem.

According to Theorem 2.40, we can think of discrete A-nets as being governed by a 3D system in the sense of Figure 2.2. The fields attached to the vertices of Z3 are pairs (f, :P) consisting of a point I E 1R3 and a plane :P through this point. The system is characterized by the defining property of discrete A-nets, according to which all neighbors of a point I belong to the corresponding plane:P. Theorem 2.40 assures that the fields at any seven vertices of an elementary cube of Z3 determine the eighth one uniquely. The construction of the Mobius tetrahedra performed in Theorem 2.40 can also be given a different, somewhat unusual combinatorial interpretation. In this interpretation, we assign points I E 1R3 and planes :P c 1R3 to the even, resp. odd, sub lattice of Z4, i.e., to u E Z4 with the even (resp. odd) values of lui = U1 + U2 + U3 + U4. An edge of Z4 connects a point I and a plane :P if and only if I E :P. It is not hard to realize that this condition determines a three-dimensional system (the fields at any seven vertices of a combinatorial 3D cube determine the eighth one uniquely); see Figure 2.18.


69

For instance, the left diagram in Figure 2.18 corresponds to the construction

P23

P123

123

P3

13

iI3

P13

I I

P2 • /

I I

- - - -- - -

iI2

12.----- - /

/

I

1123

P12

/

/

P

/

II

Figure 2.18. 3D system with points assigned to the even sublattice and planes assigned to the odd sublattice.

with the following seven initial data: a point I, three planes Pi in general position through I, and three points Iij E Pi n Pj' Then the eighth field is the plane P123 through iI2, 123 and iI3' The right diagram in Figure 2.18 corresponds to a projectively dual construction. Clearly, the fields at the eight vertices of any combinatorial 3D cube correspond to the four vertices and four face planes of a tetrahedron in JR3. In particular, the cubes in Figure 2.18 encode both tetrahedra of a Mobius pair. Now, the defining properties of a Mobius tetrahedra pair and its construction are encoded in the hypercube in Figure 2.19, where the points I and the planes P with equal indices are connected by edges parallel to the fourth coordinate direction. Thus, in this interpretation Theorem 2.40 is nothing but the statement about the 4D consistency of the latter 3D system.

,,'..

iI'" P .r----------------~.

Figure 2.19. 4D consistency of the 3D system with points on the even sub lattice and planes on the odd sublattice.


70

The previous, conventional combinatorial interpretation of the Mobius tetrahedra pairs (or, what is the same, of the discrete A-nets), is reobtained upon a contraction of all edges of the fourth coordinate direction in Figure 2.19. As a result of this operation, the 4D hypercube turns into a 3D cube, whose vertices carry pairs (j, P) with equal indices such that f E P. m 2: 4 : consistency. A nonstandard interpretation of the Mobius theorem (Theorem 2.40) as a statement about 4D consistency of a certain 3D system, together with Theorem 2.6, yields the (m + I)-dimensional consistency of the latter for any m 2': 4. Returning back to the standard interpretation (by contraction of one of the coordinate directions), we see that the following statement holds. Theorem 2.41. (Consistency of discrete A-nets) Discrete A-nets are multidimensionally consistent. This justifies Definition 2.39. 2.4.2. Discrete Lelieuvre representation. At each point u E zm, the tangent plane P(u) of a discrete A-net defines also its normal direction. A remarkable way to fix a certain normal field is given by the discrete Lelieuvre representation which states: Theorem 2.42. (Discrete Lelieuvre representation) For a nondegenerate discrete A -net f : zm ----.. lR 3 , there exists a normal field n : zm ----.. lR3 such that i = 1, . ..

(2.62)

,m,

called a Lelieuvre normal field. It is uniquely defined by a value at one point Uo E zm. All other Lelieuvre normal fields are obtained by n( u) I----t an( u) for lui = Ul + ... + U m even, and n(u) I----t a-1n(u) for lui odd, with some a E lR (black-white rescaling).

Proof. Let v : zm ----.. lR3 be some normal field to elementary quadrilateral (j, fi, fij, fj) we have: (2.63) (2.64)

Ii - f fij - fj

= aVi

x v,

= ajVij x Vj,

fj - f

f.

For every (nonplanar)

= (3Vj

X

v,

Iij - Ii = (3i Vij

X

Vi,

with some coefficients a, (3, aj, (3i E lR* (associated with the edges of the quadrilateral). We will show that (2.65) This will prove the theorem, because relation (2.65) is equivalent to the existence of the function p : zm ----.. lR*, associated with the vertices of zm,


71

such that (2.66)

0

= PPi,

f3

= PPj,

OJ

= PjPij,

f3i

= PiPij·

A solution P of the latter system is completely determined by its value at one point, and any two solutions differ by a black-white rescaling. Comparison of (2.66) with (2.63), (2.64) shows that the vectors n = pv satisfy (2.62). To prove (2.65), build the scalar product of the first equation in (2.63) with Vj, and of the second with Vi. Taking into account that (Vi x V, Vj) = - (Vj X V, Vi)' and that Ij - I is orthogonal to Vj, we find: o f3

(Ii - I, Vj) (Ij - I, Vi)

(Ii - Ij, Vj) (Ii - Ij, Vi) .

Similarly, from (2.64) we derive: OJ

f3i

(Iij - Ij, Vi) (Iij - Ii, Vj)

(Ii - Ij, Vi) (Ii - Ij, Vj)'

From the last two formulas, the relation (2.65) follows.

o

Theorem 2.43. (Lelieuvre normals of discrete A-nets are T-nets) Discrete A-nets in JR.3 (modulo parallel translations) are in a one-to-one correspondence, via the discrete Lelieuvre representation (2.62), with T-nets in JR.3 (modulo black-white rescalings). Proof. It follows immediately from (2.62) that hTjn-n) x hn-Tjn) = 0, that is, the Lelieuvre normal field of a discrete A-net satisfies the discrete Moutard equation (2.67)

TiTjn - n = aij(Tjn - Tin)

with some aij : tl m -; R Conversely, given aT-net n : tl m -; JR.3, formula (2.62) defines an exact form whose integration produces a discrete A-net I : tl m -; JR.3. 0 Theorem 2.43 gives another justification of the multidimensional consistency of A-nets. In particular, the initial data which determine (up to translations) a discrete A-net I : tl m -; JR.3 are just the data (Mt2) for the Lelieuvre normal field n : tl m -; JR.3: (A~) the values of n on the coordinate axes

'Bi

for 1 ~ i ~ m;

(A~) the values of m(m - 1)/2 functions aij on all elementary squares of the coordinate planes 'Bij for 1 ~ i < j ~ m.

We will be mainly interested in the case m = 2, which is an immediate discretization of the smooth A-surfaces. The change of variables (2.49) for the Lelieuvre normal field, say (2.68)

72


leads to the replacement of the general Lelieuvre formulas (2.62) by (2.69)

2.4.3. Transformations of discrete A-nets. A natural generalization of Definition 1.19 would be the following one. Definition 2.44. (Discrete Weingarten transformation) A pair of discrete A-nets f, f+ : 7/.,m -> ~3 is related by a Weingarten transformation if, for every u E 7/.,m, the line (J (u) f+ (u)) lies in both tangent planes to f and f+ at the points f (u) and f+ (u), respectively. The net f+ is called a Weingarten transform of the net f. But, as usual, this definition means simply that the net F : 7/.,m X {O, 1} -> ~3 with F(u,O) = f(u) and F(u,1) = f+(u) is an M-dimensional discrete A-net, where M = m+ 1. So, once again, transformations of discrete A-nets do not differ from the nets themselves. In particular, Lelieuvre normals n+ of the net f+ can be uniquely fixed by the requirement that, together with the Lelieuvre normals n of the net f, they build a Lelieuvre representation of the net F:

(2.70)

f+ - f

= n+ x n.

From Theorem 2.43 we immediately derive:

Theorem 2.45. (Discrete Weingarten transformation = discrete Moutard transformation for Lelieuvre normals) The Lelieuvre normal fields n, n+ of a pair of discrete A-nets f, f+ related by a Weingarten transformation, are Moutard transforms of one another:

(2.71)

1 :S i :S m.

Here the coefficients bi are nothing but aiM, M = m + 1, for the net F, and are therefore attached to the "vertical" elementary squares of 7/.,m x {a, 1} parallel to the coordinate plane 'BiM, or simply to the edges of 7/.,m parallel to the coordinate axis 'B i . In order to specify a Weingarten transform f+ of a given discrete A-net or, what is the same, a Moutard transform n+ of the Lelieuvre normal field n, one can prescribe the following data:

f,

(Wf) the value of n+(O); (W~) the values of bi on the "vertical" elementary squares attached to all edges of the coordinate axes 'Bi for 1 :S i :S m.

Permutability of discrete Weingarten transformations is a direct consequence of the multidimensional consistency of A-nets.

2.5. Exercises

73

Theorem 2.46. (Permutability of discrete Weingarten transformations)

1) Let f be a discrete A -net, and let f( 1) and f(2) be two of its Weingarten transforms. Then there exists a one-parameter family of discrete A-nets f(12) that are Weingarten transforms of both f(1) and f(2).

2) Let f be a discrete A-net. Let f(1), f(2) and f(3) be three of its Weingarten transforms, and let three further discrete A-nets f(12), f(23) and f(13) be given such that f(ij) is a simultaneous Weingarten transform of f(i) and f(j). Then generically there exists a unique discrete A-net f(123) that is a Weingarten transform of f(12), f(23) and f(13). The net f(123) is uniquely determined by the condition that each of its points lies in the tangent planes to f(12), f(23) and f(13) at the corresponding points.

In the case m = 2, upon the change of variables (2.49) for the Lelieuvre normal field, formulas (2.71) for a Moutard transformation of the Lelieuvre normal field turn into: (2.72) or (2.73) where Pi = bi - 1. Permutability of smooth Weingarten transformations (Theorem 1.15) will follow if one combines the discrete permutability with the convergence results of Section 5.5.

2.5. Exercises 2.1. Let the function

j : zm

satisfy equation (2.3). Assume that it is generic in the sense ~h~t f~r each u E zm and for each pair of indices i =I j the three vectors f, Ii, fj are linearly independent. Prove that the net f = p-1 j with a scalar function p : zm ---+ ]R* is a Q-net in ]RN if and only if the function p satisfies the same equation as j, i.e., if ---+ ]RN

(2.74) 2.2.* Consider eight points in a plane with the combinatorics of a 3D cube. Prove that if for some pair of opposite faces the four intersection points of the four corresponding pairs of edge lines are collinear, then the same is true for each of the other two pairs of faces. Give a construction of the eighth point of such a configuration, provided seven points are given (so that such a configuration is described by a 3D system in the sense of Figure 2.2). Demonstrate the 4D consistency of this system. This is a reasonable definition of an elementary hexahedron of a Q-net in a plane

74


(note that Definition 2.1 does not make sense in the case of the ambient space of dimension N = 2). 2.3.* Let f, f(1), f(2), f(12) be four Q-nets related as in part 1) of Theorem 2.8, i.e., f(1) and f(2) are F-transforms of f, and f(12) is a simultaneous F-transform of f(l) and of f(2). For an arbitrary elementary quadrilateral of the net f consider the corresponding quadrilaterals of the other three nets f(1), f(2), f(12). Prove that the four planes of these elementary quadrilaterals meet at one point. 2.4.* Prove a discrete analog of the classical Eisenhart formulation of the F -transformation:

(2.75) where p : zm

-+

JRN is a Q-net parallel to f in the following sense:

(2.76) with certain functions Cti : zm -+ JR on the edges parallel to compatible system of difference equations

~i,

solving the

(2.77) further, cjJ : zm -+ JR is an additional scalar solution of (2.1), and'ljJ : zm corresponds to cjJ in the same way as p corresponds to f:

-+

JR

(2.78) Hint: The construction is similar to that of Subsection 1.1.4 and uses addi-

tional scalar solutions cjJ, cjJ+ : zm -+ JR of the point equations (2.1) for the nets f, f+ which can be defined by the compatible systems (2.79)

TicjJ

cjJ

= 1+

bi ai

TicjJ+

' A . + =1+bi' 'I-'

1:Si:Sm.

2.5.* Check that the system of difference equations (2.15) is compatible. 2.6. Check the 4D consistency of the map discrete Darboux system (2.22).

{'Ykj}

f-t

{Ti'Ykj}

defined by the

2.7.* Let 'Yji be the discrete rotation coefficients of a given Q-net, defined by equation (2.16) and satisfying the discrete Darboux system (2.22). Prove the consistency of the difference equations

(2.80)

Ti()j

-

()j

Tj()i

= -

()i

= 1 - 'Yij"fji

for functions ()i on the edges (u, u + ei) of zm parallel to the coordinate axes ~i. Show that the first equation in (2.80) yields the existence of a realvalued function p on vertices of zm such that ()i(U, u + ei) = p(u + ei)/ p(u).

2.5. Exercises

75

Thus, (2.80) takes the form (2.81)

(

(T(Tjp)p )( ) = 1 -

TiP

TjP

'Yij/ji.

2.8. Consider the following geometric system on zm. The fields Xi : zm --t ]RN are attached to the edges (u, u + ei) parallel to the coordinate axes ~i; see Figure 2.20. For any elementary quadrilateral of zm, it is required that

T3 X I

TIT2 X 3

T2 X 39

I I I

X3

}------oT2 X I

/ /

•

X2

Xl

TIX2

Figure 2.20. 3D system on an elementary cube: fields on edges.

the four points attached to its four edges be collinear, that is, for any u E zm and for any 1 :S i 01- j :S m, the four points Xi, Xj, TiXj and TjXi lie on a common straight line. (This forces the 12 points corresponding to the edges of any elementary 3D cube to be coplanar; see Figure 2.21.) Show that

Figure 2.21. Geometry of the elementary cube in Figure 2.20.


76

this is a 3D system with fields on edges, in the sense that the fields on six suitable chosen edges of a cube determine the other six fields uniquely. 2.9. Given a Q-net f : zm -+ JRN+1 and a generic hyperplane II in JRN+l. Intersect the edge lines of the Q-net with II. Show that the resulting points Xi = (f fi) n II build in II a geometric system of Exercise 2.8. Describe the reverse construction. Show that all the points x E II jointly with arbitrarily chosen f r13i on the coordinate axes determine the corresponding Q-net f uniquely. This implies the 4D consistency of the system of Exercise 2.8. 2.10. Prove Theorem 2.11 about the 4D consistency of discrete line congruences. 2.11. Let f : Z2 -+ JRJlDN be a two-dimensional Q-net. One looks for a line congruence £ : Z2 -+ £., N such that f (u) E £(u) for each u E Z2. Show that generically such a congruence is uniquely defined by its lines along the coordinate axes, i.e., by £ : Z x {O} -+ £.,N and £ : {O} x Z -+ £.,N. 2.12. Let f : zm -+ JRJlDN be an m-dimensional Q-net. One looks for a line congruence £ : zm -+ £.,N such that f(u) E £(u) for each u E zm. Show that generically such a congruence is uniquely defined by its lines along the coordinate axes, i.e., by £ : 13 i -+ £.,N for 1 ::; i ::; m. Hint: Apply the construction of Exercise 2.11 and prove that h23 E £123. 2.13. Let f : Z2 -+ JRJlDN be a two-dimensional Q-net. For i = 1,2 the lines £i (u) = (f (u) f (u + ei)) constitute a discrete line congruence - the so-called i-th tangent congruence of f. Define the i-th Laplace transform £.,(i) f of the net f as the j-th focal net of the congruence ii, where i ::/: j. Thus one has £.,(1) f(u) = £l(U) n £l(U + e2) and £.,(2) f(u) = £2(U) n £2(U + eI); see Figure 2.22. Show that the Laplace transforms of fare Q-nets .

.e(1)

f

Figure 2.22. Laplace transformations of a Q-net: a face of the net is mapped to a vertex of its Laplace transform ,c(i) f.

2.14. Let

f : Z2

-+

f

lRJIDN be a two-dimensional Q-net with the equation

f12 =

Ah + B 12 + C f

2.5. Exercises

77

(in homogeneous coordinates). Show that its Laplace transforms are given by

£.,(1) f = Aft

+ Cf =

ft2 -

B12,

£.,(2) f =

B12 + Cf =

ft2 -

Aft.

f : Z2 ---+ JRJlDN be a two-dimensional Q-net. Each line £i(U) = (f( u)f(u+ei)) of the i-th tangent congruence carries four points: two points of the original net, as well as two points £.,(i) f(u) = £i(U) n £i(U + ej) and £.,(i) f( u - ej) = £i( u) n £i( u - ej); cf. Exercise 2.13. The cross-ratio of these 2.15. Let

four points is called the Laplace invariant of the net:

+ ed, £.,(1) f(u = q(J(u), £.,(2) f(u), f(u + e2), £.,(2) f(u -

h(u)

q(J(u), £.,(1) f(u), f(u

k(u)

e2)), e1))'

It is natural to assign the Laplace invariants h(u) to the edges [u, u + ed, and the Laplace invariants k(u) to the edges [u, u + e2] of the lattice. Show that in the notation of the previous exercise,

k = _ BA_1 h = _ AB_2 C ' C . These expressions do not depend on the choice of homogeneous coordinates. The combinatorial assignment of the quantities in these expressions is illustrated in Figure 2.23.

112

12

h2

h

1

C h A

B_2

B h

k A-I C

1

h

Figure 2.23. Laplace invariants of a Q-net.

2.16.* Prove that two quadrilaterals in C whose corresponding edges are parallel are dual if and only if their complex cross-ratios are equal. 2.17. Consider two mutually corresponding quadrilaterals (f, ft, ft2' h) and (f+, ft, f0., fi) of two Q-nets f, f+ related by a fundamental transformation. The lines £ = (ff+), £1, £2, £12 build a piece of a line congruence.


78

Denote by F(l) =.e n .e 1, F(2) = .e n .e2, FF) and Fi 2) the focal points of the line congruence (see Figure 2.24). Generalized Menelaus' theorem (Theorem 9.12) says that the following products of ratios of directed lengths are equal to one:

l(F(l), f) . l(F(2), h) . l(Fj1), f12) . l(Fi 2), fI) _ 1 l(f, F(2)) l(12, Fj1)) l(f12, Fi 2)) l(h, F(1)) - , l(F(1), f+) . l(F(2), fi) . l(FF) ,112) . l(Fi 2), fn _ 1 l(f+, F(2)) l(fi, Fj1)) l(f12, Fi 2)) l(ft, F(l)) - . Build similar products involving only black, resp. only white points on the four lines:

l(F(1), f) l(F(2), fi) l(Fj1), h2) l(Fi 2), ft) l(f,F(2)) ·l(fi,Fjl)) ·l(f12,Fi 2)) ·l(ft,F(l))' AW =

l(F(l),f+) l(F(2),12) l(FF),f12) l(Fi 2),fI) l(f+,F(2)) . l(12,FF)) . l(f12,Fi 2)) . l(h,F(l))"

Show that AB = 1 if and only if AW = 1. According to the generalized Menelaus theorem, this means that the black points of an elementary hexahedron of a Q-net are coplanar if and only if the white ones are coplanar. This is an alternative proof of a statement in Theorem 2.29.

Figure 2.24. Two corresponding elementary quadrilaterals of Q-nets related by an F-transformation.

2.5. Exercises

79

2.18. In the situation of the previous exercise, introduce cross-ratios of the quadruples of points on each line of the congruence £:

q = q(F(l), f, F(2), f+), q2

f F(2) ' f+) = q(F,2(1) ,2, 2'

Show that the following relations hold:

(2.82)

q1q2 = A+ = 1 qq12 A ' Conclude that the configuration in Figure 2.24 is an elementary hexahedron of a discrete Koenigs net if and only if (qq12) 2 = (q1 q2) 2 = l. 2.19. Let a, b, c, d : 'Z} ~ C be four complex-valued functions on the square lattice. Prove that a four-point difference hyperbolic equation

(2.83)

aX12

+ bX1 + CX2 + dx = 0

for x : 'll} ~ C yields a five-point equation on the even (and on the odd) sublattice of Z2,

AX12

+ BX-1,2 + CX-1,-2 + DX1,-2 + Ex =

0,

if and only if equation (2.83) is gauge equivalent to the discrete Moutard equation. The latter means that there exists a gauge transformation y = ax with a : Z2 ~ C* bringing equation (2.83) for x into the Moutard equation (2.50) for y. 2.20.* Apply the projective duality to Theorems 2.26, 2.27 to define a notion of "dual Koenigs nets" as special Q* -nets. 2.21.* a) Let f : Z2 ~ IRlP'N be a discrete Koenigs net. Show that its Laplace invariants (defined as in Exercise 2.15) satisfy the relation hh-1 = kk_2.

b) Let M : Z2 ~ IRlP'N be a Q-net built by the intersection points of the diagonals of a discrete Koenigs net. Show that its Laplace invariants satisfy the relation hh2 = kk 1 . The combinatorial assignment of Laplace invariants in these statements is illustrated in Figure 2.25. Hint: Use the result of Exercise 2.15 and Theorem 2.32. 2.22.* Consider a planar quadrilateral (J fI!1212) , and denote L1 = (J fd n (12h2) and L2 = (J h) n (JI!12). Consider four further points L3 E (J h), L4 E (J2112), L5 E (J h), L6 E (JI!12). Show that the six points L 1, ... , L6 belong to a conic (see Figure 2.26) if and only if the following relation is satisfied:


80

h

h-l

f k-2

Ok, h2

k k

f

Figure 2.25. Laplace invariants of a discrete Koenigs net and of a net built by intersection points of diagonals of a discrete Koenigs net.

This claim admits the following interpretation in terms of discrete differential geometry: the Q-net M : Z2 ----) lRlP'N built by the intersection points of the diagonals of a discrete Koenigs net possesses the following geometric property: for any u E Z2, the six points ,e(1) M(u), ,e(l) M(u±e2), ,e(2) M(u), ,e(2)M(u ± ed lie on a conic.

Figure 2.26. A discrete analogue of the Koenigs conic for the net built by the intersection points of diagonals of a discrete Koenigs net.

2.23.* Consider a two-layer discrete Koenigs net F : Z2 x {O, I} ----) lR 3 , and set f(u) = F(u,O), f+(u) = F(u, 1). Denote the intersection points of the diagonals of the elementary quadrilaterals of f, f+ by M, M+, respectively. Prove that the lines £ = (M M+) form a discrete line congruence, that is, any two neighboring lines intersect. Moreover, the sextuple of points on £, consisting of M, M+, £ n £±1, and £ n £±2, constitutes a quadrangular set.

2.5. Exercises

81

2.24. Check by a direct computation the 4D consistency of equations (2.53). 2.25. Show that equations (2.53) allow one to introduce, for a given m,dimensional T-net, a real-valued function cr defined on the vertices of zm and such that

(2.84)

i

> j,

which satisfies the equation

(2.85) cr(TiTjTkcr) = hcr)(TjTkcr) - (Tjcr)hTkcr)

+ (Tkcr)hTjcr),

i

<j {oriented planes in ]R3} is conical if and only if the corresponding points Q-net in JPl(]R3,1,1).

f3 : zm

--> JPl(lL 3 ,1,1)

form a

Proof. Representatives f3 of the planes P form a Q-net if and only if they satisfy (2.1), that is, if v : zm --> §2 and d : zm --> ]R satisfy this equation. Equation (2.1) for v yields that v : zm --> §2 is a Q-net in §2, so that any quadrilateral (V,Vi,Vij,Vj) in §2 is planar and therefore circular. Equation (2.1) for (v, d) yields that the (unique) intersection point of the three planes P, Pi, P j lies on P ij as well, so that all four planes intersect in one point. Thus, we arrived at a characterization of P as a net of planes for which every quadruple of planes (P, Pi, P ij , P j ) is concurrent and every quadrilateral (v, Vi, Vij, Vj) of unit normal vectors is circular. According to Definition 3.20 and Theorem 3.21, P is a conical net. 0

Thus, conical nets constitute a further example of multidimensional Qnets restricted to a quadric (the absolute quadric in the projective model of Laguerre geometry).

3. Discretization Principles. Nets in Quadrics

106

3.5. Principal contact element nets We have encountered two classes of nets (circular and conical nets) which can be interpreted as discrete analogs of surfaces parametrized along curvature lines. Circular nets are objects of Mobius geometry, while conical nets belong to Laguerre geometry. Generally, there are several ways to describe surfaces and to discretize this notion, and circular and conical nets exemplify just two of them. A surface can be viewed simply as built of points. This makes sense in each geometry where points are distinguished space elements, such as projective geometry, Euclidean geometry, etc. In particular, this is the case for Mobius geometry. A discrete surface in one of these geometries is a map

f : 'l}

---t

1R3.

On the other hand, a surface can be viewed as the envelope of the system of its tangent planes. This makes sense as soon as planes play the role of distinguished space elements, e.g., in projective geometry, in Euclidean geometry, and also in Laguerre geometry. A discrete surface in such a geometry should be understood as a map

P : 71}

---t

{oriented planes in 1R3}.

A substantial part of such a description of a surface is its Gauss map v : 71}

---t

§2,

consisting of unit normals v to the planes P = {x E 1R3 : (v, x) = d}. There are also geometries where contact elements are distinguished. A contact element can be interpreted as a pair (1, P) consisting of a point f of the surface along with an (oriented) tangent plane P through f (or, equivalently, a normal vector v to P at 1). A surface is then viewed as built of its contact elements. For instance, in Lie geometry, a contact element can be understood in terms of oriented spheres and their oriented contact, as a one-parameter family (pencil) of all spheres S through f which are in oriented contact with P (and with one another), thus sharing the normal vector v at f; see Figure 9.1 in Section 9.2. Another instance is Plucker line geometry, where a contact element is understood in terms of lines in 1R]fD3 and their intersections, as a one-parameter family (pencil) of lines in P through f. Clearly, the description of a surface in terms of its contact elements contains more information than the description in terms of points only or the description in terms of tangent planes only; actually, it combines them. This description can be discretized in a natural way: a discrete surface is a map (1, P) : Z2 ---t {contact elements in 1R3}.

3.5. Principal contact element nets

107

The stage where the projective model of Lie geometry takes place is the Lie quadric IP'(JL4,2) C 1P'(]R4,2). In particular, contact elements are modelled as isotropic lines (lines in 1P'(]R4,2) which lie in IP'(JL4,2)). We will denote the set of isotropic lines of 1P'(]R4,2) by ,(,6,2. A discretization of the Lie-geometric surface theory can and should be based on both our fundamental principles, which dictates studying line congruences and Q-nets in the Lie quadric.

Definition 3.23. (Principal contact element nets, projective model of sphere geometry) A map £ : 'l} -+ ,(,6,2 is called a principal contact element net if it is a discrete congruence of isotropic lines in 1P'(]R4,2), that is, if every two neighboring lines intersect: (3.28) Naturally associated to a congruence of isotropic lines £ are two focal nets (3.29) cf. Definition 2.13. The points sCi) E IP'(JL 4,2) represent oriented spheres SCi) in ]R3.

A direct translation of Definition 3.23 into the language of the geometry of spheres in ]R3 looks as follows.

Definition 3.24. (Principal contact element net, Euclidean model of sphere geometry) A map

(j, P) : 71,2

-+

{contact elements in ]R3}

is called a principal contact element net if every two neighboring contact elements (j, P), (Td, TiP) have a sphere SCi) in common, that is, a sphere in oriented contact with both planes P, TiP at the corresponding points f,

Td· Thus, the normals to the neighboring planes P, Pi at the corresponding points f, fi intersect at a point c( i) (the center of the sphere S( i)), and the distances from c(i) to f and to fi are equal; see Figure 3.9. The spheres Sci) are naturally assigned to the edges of 71,2 parallel to the i-th coordinate axis. They will be called principal curvature spheres of the discrete contact element net. This is a discretization of the Lie-geometric description of curvature line parametrized surfaces, according to which two infinitesimally close contact elements (sphere pencils) belong to the same curvature line if and only if they have a sphere in common. Adding one extra dimension to the lattice of independent variables (or, equivalently, considering multidimensional congruences of isotropic lines in IP'(JL 4,2)) results in the following definition.

108


Figure 3.9. Principal curvature sphere.

Definition 3.25. (Ribaucour transformation, projective model of sphere geometry) Two principal contact element nets £, £+ : 'I} - t ,C~,2 are called Ribaucour transforms of one another if these discrete congruences of isotropic lines are related by an F-transformation, that is, if every two corresponding lines intersect: (3.30)

Again, a direct translation of this into the conventional geometric terms reads: Definition 3.26. (Ribaucour transformation, Euclidean model of sphere geometry) Two principal contact element nets

(f, P), (f+, p+): Z2

-t

{contact elements in ]R3}

are called Ribaucour transforms of one another if every two corresponding contact elements (f, P) and (f+, p+) have a sphere S in common, that is, a sphere in oriented contact with both planes P, p+ at the corresponding points f, f+· Spheres S of a Ribaucour transformation are naturally assigned to the vertices u of the lattice Z2, or, better, to the "vertical" edges connecting the vertices (u,O) and (u,l) of the lattice Z2 x {O,l}. In the projective model, their representatives (3.31)

109

3.5. Principal contact element nets

s (f,P) Figure 3.10. Ribaucour transformation.

build the focal net of the three-dimensional line congruence for the third coordinate direction. According to Theorem 2.14, both focal nets s(i) of a principal contact element net are Q-nets in JID(JL4,2) C JID(]R4,2), and the same holds for the spheres s of a generic Ribaucour transformation. This motivates the following definition. Definition 3.27. (Discrete R-congruence of spheres) A map

S : Zm -----; {oriented spheres in ]R3} is called a discrete R-congruence (Ribaucour congruence) of spheres if the corresponding map

(3.32)

s : Zm -----; JID(JL4,2),

S = c + eo

+ (lcl 2 -

r2)e oo

+ re6,

is a Q-net in JID(]R4,2). Thus, we can formulate: Corollary 3.28. (R-congruences of principal curvature spheres and of a Ribaucour transformation)

a) For a discrete contact element net, the principal curvature spheres of the i-th coordinate direction (i = 1,2) build a two-dimensional discrete R-congruence. b) The spheres of a generic Ribaucour transformation build a discrete R-congruence. The R-congruence of principal curvature spheres S(i) of the i-th coordinate direction is degenerate in the sense that the plane of every its elementary quadrilateral (s(i),s~i),s~y,sy)) contains two isotropic lines €i,€ij,


110

so that the three-dimensional vector space I;

= span(s(i) , sci) sCi) s(i)) t , tJ' J

C ~4,2

contains two two-dimensional isotropic subspaces. On the contrary, the R-congruence of spheres of a generic Ribaucour transformation is non degenerate: the three-dimensional vector spaces I; of its elementary quadrilaterals do not contain two-dimensional isotropic subspaces. A geometric characterization of nondegenerate discrete R-congruences of spheres will be given in Section 3.7.

3.6. Q-congruences of spheres With a view towards a geometric characterization of discrete R-congruences of spheres, we observe that from (3.32) it follows immediately that a map

S : Zm

--t

{oriented spheres in ~3}

is a discrete R-congruence if and only if the centers c : Q-net in ~3, and the two real-valued functions,

Icl 2 -

r2 : Zm

--t

~

and

r: Zm

--t

zm

--t

~3 form a

~,

satisfy the same equation of the type (2.1) as the centers c. By omitting the latter requirement for the signed radii r, one comes to a less restrictive definition than that of R-congruence, namely to Qcongruences of spheres. Those are naturally a subject of Mobius geometry rather than of Lie geometry. Recall (or see Section 9.3) that spheres in ~3 can be represented in the Mobius-geometric formalism as elements of JP>(~~~t), where ~4,1 out

(3.33)

= {s

E ~4,1 ..

(s , s) > o}

is the space-like part of ~4,1. In the rest of this section, we use the symbol s exclusively in the Mobius-geometric sense!

Definition 3.29. (Q-congruence of spheres) A map (3.34)

S : Zm

--t

{nonoriented spheres in ~3}

is called a Q-congruence of spheres if the corresponding map (3.35)

s: zm --t JP>(~!~t),

S = C+ eo

+ (lcl2 - r 2)e oo ,

is a Q-net in JP>(~4,1).

Thus, a map (3.34) is a Q-congruence if and only if the centers c : --t ~3 of the spheres S form a Q-net in ~3, and the function 1c1 2 - r2 satisfies the same equation (2.1) as the centers c. Clearly, as r --t 0 the latter characterization turns into the criterion of Theorem 3.7. A geometric description of Q-congruences is given in the following theorem.

zm

3.6. Q-congruences of spheres

111

Theorem 3.30. (Three types of Q-congruences) Four (nonoriented) spheres (81 ,82 ,83 ,84 ) in lR 3 constitute an elementary quadrilateral of a Qcongruence if and only if they satisfy one of the following three conditions:

(i) they have a common orthogonal circle, or (ii) their intersection consists of a pair of points (a O-sphere), or else (iii) their intersection consists of exactly one point. Case (iii) can be regarded as a degenerate case of both (i) and (ii).

Conceptual proof. For a Q-congruence, the vector subspace 'E

= span(8l' 82, 83, 84)

C

lR 4,1

is three-dimensional, so its orthogonal complement 'E..l is two-dimensional. If 'E..l lies in lR~~t' i.e., if the restriction of the Minkowski scalar product to 'E..l is positive definite (of signature (2,0)), then 'E..l represents a I-sphere (a circle) orthogonal to our four spheres, and we have case (i). If, on the contrary, the restriction of the scalar product to 'E..l has signature (1, 1), so that 'E lies in lR~~lt, then 'E represents a O-sphere which is the intersection of our four spheres, and we have case (ii). Finally, if the restriction of the scalar product to 'E..l is degenerate, then 'E n 'E..l is an isotropic one-dimensional vector subspace, which represents the common point of our four spheres, and we have case (iii). 0 Computational proof. For a Q-congruence, the quadrilateral in lR 3 with vertices at the sphere centers Cl, C2, C3, C4 is planar; denote its plane by II. In the same way as in the proof of Theorem 3.7 we show that there is a point CEIl such that (3.36)

Ict - CI 2 - r~ =

h - 01 2 -

r~ = IC3 - CI 2 -

d=

IC4 - CI 2 -

r;

d·

Indeed, let the centers Ci and the quantities ICil2 satisfy one and the same linear relation of the type (3.1). Then the third equality in (3.36) is automatically satisfied as soon as the first two equalities hold. But the first two are equivalent to

(2C - Cl - C2, Cl - C2) = r~ - r~,

(2C - C2 - C3, C2 - C3) = r~ - r~ ,

from which the point C is uniquely determined as the intersection of two lines £'12 and £'23 in II, where £'ij

= {x

E

II : (2x - Ci -

Cj,

Ci -

Cj)

= r] - rl}.

Now, if the common value of all four expressions in (3.36) is positive, say equal to R2, then the four spheres under consideration are orthogonal to the circle in the plane II with center C and radius R, so that we have case (i); see Figure 3.11. If the common value of (3.36) is negative, say equal to - R2, then the pair of points on the line through C orthogonal to II, at the

112


distance R from C, belongs to all four spheres, so that we have case (ii). Finally, if the common value of (3.36) is equal to 0, then C is the intersection point of all four spheres, and we have case (iii). D

Figure 3.11. Elementary quadrilateral of a Q-congruence of spheres, the orthogonal circle case.

Clearly, case (i) of Q-congruences reduces to circular nets if the radii of all spheres become infinitely small; cf. Figure 3.11. Q-congruences with intersections of type (ii) are natural discrete analogs of principally parametrized sphere congruences, because four infinitesimally neighboring spheres of such a congruence intersect this way, the pairs of intersection points comprising two enveloping surfaces of the congruence (Section 1.5). Q-congruences of spheres are multidimensionally consistent, with the following reservation: given seven points 8, 8i, 8ij in lP'(JR~~t), the Q-property (planarity condition) uniquely defines the eighth point 8123 in lP'(JR 4 ,1), which, however, might get outside of lP'(JR~~t), and therefore might not represent a real sphere. Thus, the corresponding discrete 3D system is well defined only on an open subset of the space of initial data. As long as it is defined, it can be used to produce transformations of Q-congruences, with usual permutability properties. Theorem 3.31. (Elementary hexahedron of a Q-congruence of circular type) Consider a three-dimensional Q-congruence of spheres with all faces of type (i), i.e., with orthogonal circles. For an elementary hexahedron of such a congruence, the six orthogonal circles corresponding to the six faces lie on a 2-sphere which is orthogonal to all eight spheres assigned to the vertices.

3.7. Ribaucour congruences of spheres

113

Proof. The existence of the orthogonal circle for the face e12 (say) means that ~f2 = span(8, 81, 82)1- is a two-dimensional vector subspace lying in JR~~t. Therefore, ~t23

= span(8, ,h, 82, 83)1-

is a one-dimensional vector subspace lying in JR~~t. This subspace is contained in all ~f2' ~f3' ~r3' as well as in (T3~12)1-, (T2~13)1-, (Tl~23)1-. It represents a sphere which contains all six circles assigned to the faces and which is orthogonal to all eight spheres assigned to the vertices. 0 A more elementary geometric proof of this statement is sketched in Exercise 3.17.

3.7. Ribaucour congruences of spheres We now return to the study of R-congruences of (oriented) spheres, i.e., of Q-nets in the Lie quadric JP>(JL4,2) C JP>(JR4,2). Thus, in the present section the notation 8 again refers to the Lie-geometric formalism! We restrict ourselves to nondegenerate R-congruences, for which the subspaces spanned by its elementary quadrilaterals ~ = span(81, 82, 83, 84) do not contain two-dimensional isotropic subspaces. Thus, we leave aside the principal sphere congruences. The case when the signature of (', ')I~L is (3,0), so that the spheres 81,82,83,84 have no common touching spheres, has no counterpart in the smooth differential geometry, and is therefore less significant from the point of view of discrete differential geometry. Therefore we only consider here the cyclidic case, when the signatures of both (-, .) I~ and (., ')I~L are (2,1), so that to any elementary quadrilateral (81,82,83,84) of a discrete R-congruence there corresponds a Dupin cyclide (see Definition 9.4 in Subsection 9.2.4).

Theorem 3.32. (Geometric characterization of R-congruences of cyclidic type) Consider four (oriented) spheres (81,82,83,84) in JR3, such that there exists a nonpoint sphere 8 0 in oriented contact with three of them, say with 8 1,82,83. The four spheres (81 ,82,83,84) constitute an elementary quadrilateral of an R-congruence if and only if they constitute (as nonoriented spheres) an elementary quadrilateral of a Q-congruence, and 8 0 is in oriented contact with the fourth sphere, 84, as well. Proof. Let a sphere 8 0 in oriented contact with the three spheres 8 1,82,83 have center Co and (finite) oriented radius ro i= O. This means that the following conditions are satisfied: (3.37)

114


(tangency of 8 i with 8 0 ; cf. (9.10)). Now, using the fact that Ci and ICil 2-rf satisfy one and the same linear dependence of the type (3.1), we conclude that (3.37) is fulfilled for (C4' r4) if and only if ri satisfy the same linear dependence (3.1) as Ci and ICil2 - rf do. This proves the theorem in the case when the common tangent sphere 8 0 for the three spheres 8 1 ,82 ,83 has a finite radius. The case when 80 has an infinite radius, i.e., is actually a plane, is treated similarly, with the help of equation

(3.38) which comes to replace (3.37).

D

Remark. We have seen that, generically, if three oriented spheres 8 1 ,82 ,83 have a common sphere in oriented contact, then they have a one-parameter (cyclidic) family of common touching spheres, represented by a three-dimensionallinear subspace ~~ of ]R4,2. It is easy to see that if the projection of ~~ onto e~ is nonvanishing, then the family of spheres represented by ~~ contains exactly two planes. (For a conical cyclidic family ~ all elements have vanishing eo-component and represent planes, while the family ~~ contains no planes.) Therefore, in all cases but the conical one, the four spheres of an elementary quadrilateral of a cyclidic R-congruence can be characterized by the properties of being, as nonoriented spheres, a quadrilateral of a Qcongruence, and possessing a common tangent plane (actually, two common tangent planes). Note the following difference between Q-congruences and R-congruences: given three spheres 8 1 ,82 ,83 of an elementary quadrilateral, one has a twoparameter family for the fourth sphere 84 in the case of a Q-congruence, and only a one-parameter family in the case of an R-congruence. This reflects the fact that ]R~~t is an open set in ]R4,1, while lL 4 ,2 is a hypersurface in ]R4,2 (not containing isotropic planes).

Theorem 3.33. (Common tangent spheres of two neighboring quadrilaterals of an R-congruence) For two neighboring quadrilaterals of a discrete R-congruence of spheres, carrying cyclidic families, there exist generically exactly two spheres in oriented contact with all six spheres of the congruence. Proof. Let the quadrilaterals in question belong to the planar families generated by the subspaces ~1 and ~2 of signature (2,1). These quadrilaterals share two spheres, which span a linear space of signature (1,1). Generically, each of the planar families ~1 and ~2 adds one space-like vector, so that the linear space ~1 U ~2 spanned by all six spheres of the congruence is four-dimensional and has signature (3,1), so that its orthogonal complement (~1 U~2)~ is two-dimensional and has signature (1,1). Intersection of

3.8. Discrete curvature line parametrization in various geometries

115

lL 4 ,2 with a two-dimensional linear subspace of signature (1,1) gives, upon projectivization, exactly two spheres: indeed, if el, e2 form an orthogonal basis of (~l U ~2)1. with (el' el) = -(e2' e2) = 1, then the spheres in this space correspond to aIel + a2e2 with (aIel +a2e2, aIel +a2e2)

=0

{:}

at

=

a~

{:}

al: a2

= ±1. D

In particular: a) For any two neighboring quadrilaterals of a circular net, considered as an R-congruence of spheres, there is one nonoriented sphere (hence two oriented spheres) containing both circles. Its center is the intersection point of the lines passing through the centers of the circles orthogonallly to their respective planes; see Figure 3.2. b) For any two neighboring quadrilaterals of a conical net, considered as an R-congruence of spheres, there is a unique oriented sphere in oriented contact with all six planes of the net (the second such sphere is the point at infinity). The center of this sphere is the intersection point of the axes of the cones; see Figure 3.8. Theorem 3.34. (Common tangent spheres of an elementary hexahedron of an R-congruence) For an elementary hexahedron of a discrete R-congruence of spheres, with all faces carrying cyclidic families, there are generically exactly two spheres in oriented contact with all eight spheres assigned to the vertices. Proof. This proof goes along the same lines as the proof of Theorem 3.33, with additional use of Theorem 3.32. D

3.8. Discrete curvature line parametrization in Lie, Mobius and Laguerre geometries We have seen that principal contact element nets are discrete analogues of curvature line parametrized surfaces. Now, we turn to the study of the geometry of an elementary quadrilateral of contact elements of a principal contact element net that consists of four isotropic lines £, £1, £2, £12 (in the projective model), or of four contact elements (/, P), (II, PI)' (12, P2 ), (/12, P 12 ) (in the Euclidean model). If all four lines have a common point and span a four-dimensional space, one is dealing with a degenerate (umbilic) situation. Geometrically, this means that all four contact elements contain a certain sphere S c ]R3. In this situation, one cannot draw any further conclusions about the four points f, II, 12, f12 on the sphere S: they can be arbitrary. We will restrict our attention to the nonumbilic situation, when the space spanned by the four lines £, £1, £2, £12 is three-dimensional. Then the following statement, illustrated in Figure 3.12, holds.


116

Figure 3.12. Geometry of a principal contact element net. The four neighboring contact elements are represented by points and (tangent) planes. The points are concircular, and the planes are tangent to a cone of revolution. Neighboring normal lines intersect at the centers of principal curvature spheres.

Theorem 3.35. (Points and planes of principal contact element 2, or nets) For a principal contact element net £ : '1'.2 --t

£'6.

(j, P) : '1'.2

--t

{contact elements in ]R3},

its points f : '1'.2 --t ]R3 form a circular net, while its planes P {oriented planes in ]R3} form a conical net.

Z2

--t

Proof. In the nonumbilic situation, the four points i, iI, i2, i12 E P(JL 4,2) obtained as the intersection of the four isotropic lines £, £1, £2, £12 with the projective hyperplane P( et) in p(]R4,2) lie in a two-plane. They correspond to the points f, h, 12, h2 E ]R3. Moreover, omitting the inessential (vanishing) e6-component, we arrive at a planar quadrilateral in the Mobius sphere p(JL4,1) C p(]R4,1), so that the corresponding quadrilateral in ]R3 is circular, according to Theorem 3.9. Analogously, the four points p,P1,ih,P12 E p(JL4,2) obtained as the intersection of the four isotropic lines £, £1, £2, £12 with the projective hyperplane P( e~) in p(]R4,2) lie in a two-plane. They correspond to the planes P, PI, P2, P12 C ]R3. Omitting the inessential (vanishing) eo-component, we arrive at the Laguerre-geometric description of planes as points in P(JL 3,1,1) c p(]R3,1,1). According to Theorem 3.22, a planar quadrilateral there correD sponds to a conical quadruple of planes in ]R3. The proof of Theorem 3.35 is illustrated in Figure 3.13.

3.8. Discrete curvature line parametrization in various geometries

~-...... P12

117

LAGUERRE

LIE

MOBIUS

Figure 3.13. An elementary quadrilateral of a principal contact element net with vertices I and tangent planes P in the projective model. The vertices I build a circular net (Mobius geometry), and lie in the planes P building a conical net (Laguerre geometry). Contact elements (/, P) are represented by isotropic lines e (Lie geometry). Principal curvature spheres S(i) pass through pairs of neighboring points I, Ii and are tangent to the corresponding pairs of planes P, Pi.

In view of Theorem 3.35, it is natural to ask whether, given a circular net f : Z2 ~ 1R3, or a conical net P : Z2 ~ {oriented planes in 1R3}, there exists a principal contact element net

(J, P) : Z2 ~ {contact elements in 1R3 }, with prescribed half of the data (J or P). Theorem 3.36. (Extending circular and conical nets to principal contact element nets) i) Given a circular net f : Z2 ~ :IR3, there exists a two-parameter family of conical nets P : Z2 ~ {planes in 1R3} such that f E P for all u E Z2, and

(J, P) : Z2 ~ {contact elements in 1R3} is a principal contact element net. Such a conical net is uniquely determined by prescribing a plane P(O,O) through the point f(O, 0). ~ {oriented planes in 1R3 }, there exists a two-parameter family of circular nets f : Z2 ~ 1R3 such that f E P for all u E Z2, and

ii) Given a conical net P : Z2

(J, P) : Z2 ~ {contact elements in 1R3}


118

is a principal contact element net. Such a circular net is uniquely determined by prescribing a point f(O, 0) in the plane P(O, 0).

Proof. i) Here we have to solve at each construction step the following problem: Given a contact element (j, P) and a point Ii, find a plane Pi through fi such that there exists a sphere S(i) tangent to both planes P, Pi at the points f, Ii, respectively. Solution: Pi is obtained from P by refiection in the bisecting orthogonal plane of the edge [f, fiJ. The center c( i) of the sphere S( i) is found as the intersection of the normal to P at f with the bisecting orthogonal plane of the edge [f, fiJ. Closing of this construction around a quadrilateral (j, iI, iI2, h) follows from the fact that the four bisecting orthogonal planes to its edges intersect along a common line, which is a consequence of the circularity. ii) Similarly, here the elementary construction problem is the following: Given a contact element (j, P) and a plane Pi, find a point fi in Pi such that there exists a sphere S(i) tangent to both planes P, Pi at the points f, Ii, respectively. Solution: The point fi is obtained from f by refiection in the bisecting plane of the dihedral angle formed by P, Pi. The center c(i) of the sphere S(i) is found as the intersection of the normal to P at f with this bisecting plane. Again, the construction closes around a quadrilateral, 0 due to the conical condition. Theorem 3.36 admits a far going generalization (recall that in the framework of Lie geometry both circular and conical nets are particular cases of R-congruences of spheres, as introduced in Definition 3.27). Theorem 3.37. (Enveloping principal contact element nets for a discrete R-congruence of spheres) Given a generic discrete R-congruence of spheres S : zm -+ {oriented spheres in ]R3}, there exists a two-parameter family of principal contact element nets

(j, P) : Zm such that, for every u E

-+

{contact elements in

]R3}

zm,

the sphere S belongs to the contact element E S. Such a principal contact element net is uniquely determined by prescribing a contact element (j, P)(O) containing the sphere S(O).

(j, P), i.e., P is the tangent plane to S at the point f

Proof. This is a reformulation of Theorem 3.19 in the present context.

0

3.9. Discrete asymptotic nets in Pliicker line geometry A contact element in Plucker line geometry is understood as a family of lines through a point f E ]R3 lying in the plane P c ]R3. A contact element can

3.9. Discrete asymptotic nets in PlUcker line geometry

119

be identified with a pair (f, P) such that f E P. In the present section, we will only consider contact elements in the sense of Plucker line geometry. It is not difficult to realize that Definition 2.39 of a discrete asymptotic net allows for the following reformulation: Definition 3.38. (Discrete A-net, Euclidean model of line geometry) A map (f, P) : Zm -+ {contact elements in ffi.3} is called an A-net if each pair of neighboring contact elements (f, P), (k Pi) has a line in common, that is, if the line (f fi) belongs to both planes P, Pi.

This can be immediately translated into the language of the projective model of PlUcker line geometry, where contact elements are represented by the set ,c~,3 of isotropic lines in the PlUcker quadric lP(lL3,3) C lP(ffi.3,3). Definition 3.39. (Discrete A-net, projective model of line geometry) A map £ : zm -+ .c~,3 is called an A -net if it is a discrete congruence of isotropic lines in lP(ffi.3,3), that is, if every two neighboring lines intersect:

(3.39) £(u)

n £(u + ei)

= [(i)(u) E lP'(lL3,3),

Vu E Zm, Vi E {I, 2, ... , m}.

The elements of the focal nets [(i) : zm -+ lP'(lL3,3) represent the lines (f fi) of the A-net in ffi.3. A comparison of Definitions 3.23 and 3.39 shows that the only difference between the principal contact element nets and discrete asymptotic nets is the signature of the basic quadric of the projective model of the corresponding geometry. This is an instance of the famous Lie correspondence between spheres and lines in ffi.3. We proceed with a reformulation of Definition 2.44: Definition 3.40. (Discrete W -congruence, Euclidean model of line geometry) Two discrete A -nets (f, P), (f+, p+) : Zm

-+

{contact elements in ffi.3}

are called Weingarten transforms of each other if for every pair of corresponding contact elements (f, P), (f+, P+) the line I = (f f+) belongs to both tangent planes P, P+. The connecting lines l : zm -+ {lines in ffi.3} of a Weingarten pair are said to constitute a discrete W-congruence.

In the language of the projective model this reads: Definition 3.41. (Discrete W-congruence, projective model of line geometry) Two discrete A-nets £, £+ : zm -+ .c~,3 are called Weingarten transforms of each other if these discrete congruences of isotropic lines are related by an F-transformation, that is, if every two corresponding lines intersect: (3.40)


120

The intersection points W- congruence.

i : zm -----)

lP'(lL 3,3) represent the lines of a discrete

In the situation of Definition 3.40, both A-nets (1, P) and (1+, P+) are said to be focal nets of the W-congruence l = (1 f+). More generally, a discrete A-net (1, P) is called focal for a discrete W-congruence l if each line l belongs to the corresponding contact element (1, P), that is, f E l and l c P. It is important to note a terminological confusion which is unfortunately unavoidable for historical reasons: a discrete W-congruence is not a discrete line congruence in the sense of Definition 2.9, and a focal A-net of a discrete W-congruence is not a focal net in the sense of Definition 2.13. A characterization of discrete W-congruences which does not refer to their focal A-nets follows immediately from Theorem 2.14: Corollary 3.42. (W-congruences are Q-nets in the PlUcker quadric) A generic W-congruence of lines is represented by a Q-net in the Plucker quadric lP'(lL3,3).

In particular, four vectors ([,ii ,iij,ij) in ]R3,3 representing the four lines of an elementary quadrilateral of a generic W-congruence are linearly dependent. This means that the four lines (I, Ii, lij, lj) in ]R3 belong to a regulus (a hyperboloidic family of lines). In a complete analogy to Theorem 3.37, the following statement holds. Theorem 3.43. (Focal A-nets of a discrete W-congruence) Given a generic discrete W-congruence

I : Zm -----) {lines in ]R3}, there exists a two-parameter family of discrete A-nets

(1, P) : Zm -----) {contact elements in

]R3}

zm,

such that, for every u E the line I belongs to the contact element (1, P), that is, passes through the point f and lies in the plane P. Such a discrete A-net is uniquely determined by prescribing a contact element (1, P)(O) containing the line l (0). Proof. This is a translation of the content of Theorem 3.19 into the lan0 guage of Plucker line geometry.

3.10. Exercises 3.1. The following theorem can be considered as a spatial generalization of the Miquel theorem (Theorem 9.21): Consider a tetrahedron with vertices

3.10. Exercises

121

h, 12, 13, f4, and choose a point fij on each side (filj)· Then the four spheres TiSjkl through (fi, fij, fik' fiz) intersect at one point h234. Prove this theorem using the 4D-consistency of circular nets. 3.2. Prove that a quadrilateral its cross-ratio is real:

(h, 12, 13, f4)

in C is circular if and only if

h - 12 13 - f4 f· f 4 - fER 2- 3 1

q(h,12,h,f4) = f

3.3. Let the quadrilateral (h, 12, 13, f4) in C be circular. Prove that it is embedded (i.e., its opposite edges do not intersect) if and only if its (real) cross-ratio is negative: q(h, 12, 13, f4) < o. 3.4. Consider an elementary hexahedron of a circular net. Prove that one can choose the cross-ratios q1, ... ,q6 of the six circular faces (by choosing a certain permutation of the vertices) so that their product be equal to 1. Use this result for a new proof of Theorem 3.2 about a circular hexahedron. 3.5. Generalize the statement of Exercise 3.4 for a circular closed oriented quad-surface which is a topological sphere, i.e., a cell decomposition of a sphere with all faces being quadrilaterals inscribed in circles. Find a corresponding generalization of the Miquel theorem. 3.6.* Give a geometric proof of the Clifford-algebraic circularity criterion (3.22), based on the consideration of reflections in the orthogonal bisecting planes of the sides of a circular quadrilateral (f, fi' lij, fj)·

3.7. Show that if a quadric contains three collinear points, then it contains the whole line through these three points. 3.8. Prove that for any three lines in lR1P'3, there exists a quadric containing these three lines. Hint: Choose a triple of points on each of the lines; the quadric through these nine points will contain all three lines, according to the previous exercise. 3.9.* Prove Theorem 2.40 about Mobius pairs of tetrahedra with the help of Theorem 3.12 on the eighth associated point. 3.10.* Prove that discrete A-nets admit a restriction to an arbitrary quadric Q C lRIP'N.

3.11. The following three theorems constitute the beginning of an infinite sequence (Clifford's chain of theorems). CLIFFORD'S FIRST THEOREM. Let G1, G2, C3, C4 be four circles in general position in a plane, with a common point f. Let fij be the second intersection point of the circles Gi and Cj . Let Gijk denote the circle through fij, fik' fjk. Then the four circles C 123 , G124 , G134 , C 234 all pass through one point h234; d. Figure 3.14.

122


Figure 3.14. Clifford':; first theorem.

Let C 1 , ... ,C5 be five circles in general position in a plane, with a common point f. Then the five points !l234, !l235, !l245, !l345 , 12345 all lie on one circle C 12345· CLIFFORD'S THIRD THEOREM. Let C I , ... , C6 be six circles in general position in a plane, with a common point f. Then the six circles C12345, C 12346, C12356, C12456, C l 3456 , C 23456 all pass through one point !l23,156. Prove these theorems, by restricting Cox's chain of theorems to a sphere. CLIFFORD'S SECOND THEOREM.

3.12. Consider the 3D system with fields on edges from Exercise 2.8. Any choice of two diametrically opposite vertices of an elementary cube defines six "white" edges (those not incident to the chosen vertices). Thus, one has four "white" point sextuples. Show, with the help of Pascal's theorem, that if the six points of one "white" sextuple lie on a conic, then the same holds for the other three "white" sextuples, as well. 3.13. Let H, ... ,P4 be the four intersection points of a conic with a circle. Prove that the principal axes of the conic are parallel to the bisectors of the angles built by the lines (P1 P 2 ) and (P3 P4 ). Hint: Consider a pencil (oneparameter family) of conics through PI"'" P4. Comment: If PI,"" P4 lie on a surface, then the conic approximates the intersection of the surface with


123

the slightly shifted tangent plane (Dupin's indicatrix), and the principal axes of the conic approximate the principal curvature directions on the surface. 3.14. Prove that through every point of the hyperboloid H = {(x,y,z) E 1R3 : x 2 + y2 - z2 = I} there pass two lines lying on H (isotropic lines). They are organized in two families so that each line of the first family intersects each line of the second family, and no two lines of one and the same family intersect. 3.15. Consider four quadrilaterals of a two-dimensional Q-net sharing a vertex. Denote the angles of the quadrilaterals adjacent to this vertex by WI, W2, W3, W4 (in cyclic order). Show that this vertex is conical (Le., the oriented planes of the four quadrilaterals are tangent to a common oriented cone of revolution) if and only if WI + W3 = W2 + W4. 3.16. Let a sphere S and a circle C intersecting S orthogonally be given. Show that any sphere containing C is orthogonal to S. Hint: Perform a Mobius transformation sending one of the intersection points of C with S to infinity. 3.17.* a) Show that all the circles orthogonal to two given nonintersecting spheres pass through two common points.

b) Use this fact to give a geometric proof of Theorem 3.31 in the case when for every edge of the elementary cube the two spheres assigned to its vertices are disjoint. c) According to the previous part of this Exercise, to every elementary cube of Z3 there corresponds a sphere on which all six face circles lie. Show that these spheres comprise a Q-congruence of type (ii): four spheres corresponding to four elementary cubes sharing an edge intersect in two points.


Section 3.1: Circular nets. Circular nets as discrete analogs of curvature line parametrized surfaces were mentioned in Nutbourne-Martin (1988). Their main point of interest was a surface approximation by cyclidic patches, and they observed that the corner points of such a patch are concircular. The circular discretization of triply orthogonal coordinate systems was presented in 1996 and published in Bobenko (1999). The next crucial step in the development of the theory was made in Ciesliriski-Doliwa-Santini (1997). They considered circular nets as a reduction of Q-nets, generalized them to arbitrary dimension and proved their multidimensional consistency based on the classical Miquel theorem (Miquel (1838a,b)). Special circular nets

124


(discrete isothermic surfaces) were investigated earlier in Bobenko-Pinkall (1996b). The analytic description of circular nets of Subsection 3.1.3 is due to Konopelchenko-Schief (1998). Analytic methods of soliton theory were applied to circular nets in Doliwa-Manakov-Santini (1998) (a-method) and in Akhmetshin-Vol'vovskijKrichever (1999) (algebro-geometric solutions). A Clifford algebra description of circular nets was given in Bobenko-Hertrich-Jeromin (2001). Some fragments of the general picture of circular nets were known within the theory of transformations of smooth surfaces; see Eisenhart (1923). In his study of Ribaucour transformations of orthogonal coordinate systems, Bianchi (1923) observed that the corresponding points of the permutability diagram of two Ribaucour transformations (Bianchi quadrilateral) are concircular. A three-dimensional permutability diagram of Ribaucour transformations was discovered in Ganzha-Tsarev (1996). A modern presentation of this topic is in Rogers-Schief (2002). Section 3.2: Q-nets in quadrics. The theorem on the associated point is classical; see, for example, Pedoe (1970). The observation that Q-nets can be consistently restricted to an arbitrary quadric is due to Doliwa (1999). Section 3.3: Discrete line congruences in quadrics. We are not aware of previous presentations of the general theory of line congruences in quadrics. For special cases see the bibliographical notes to Sections 3.5 and 3.9. Section 3.4: Conical nets were introduced in Liu-Pottmann-WallnerYang-Wang (2006) as Q-nets admitting the normal shift (parallel face offset). This property was interpreted as a discrete analog of curvature line parametrization. Conical nets can also be characterized in terms of angles of the quadrilateral faces, see Wang-Wallner-Liu (2007). Section 3.5: Principal contact element nets were introduced in Bobenko-Suris (2007a). Section 3.6: Q-congruences of spheres were introduced in BobenkoSuris (2005). Section 3.7: Ribaucour congruences of spheres. The analytic Definition 3.27 is due to Doliwa (2001b). The geometric characterization of Theorem 3.32 was given in Bobenko-Suris (2007a). It should be mentioned that the spheres assigned to elementary hexahedra of a Ribaucour pair of circular nets do not form a discrete R-congruence, contrary to the main claim in Doliwa (2001b).


125

Section 3.8: Discrete curvature line parametrization in Lie, Mobius and Laguerre geometries. Our presentation follows Bobenko-Suris (2007a). Theorem 3.36 was independently obtained in Pottmann-Wallner (2008). Section 3.9: Discrete asymptotic nets in Plucker line geometry. The treatment of discrete asymptotic nets within the Plucker quadric is due to Doliwa (200la). Theorem 3.43 is new. Section 3.10: Exercises. Ex. 3.1: This theorem can be found already in Miquel (1838b). Ex. 3.5: See Below-Krummeck-Richter-Gebert (2003). Ex. 3.6: See Bobenko-Hertrich-Jeromin (2001). Ex. 3.11: The first theorem of the chain is actually due to Miquel (1838b). For the whole chain of theorems see Clifford (1871) and, for example, Coxeter (1969). This is another instance of the relation of classical chains of incidence theorems, see Coolidge (1916) and Berger (1987), to the multidimensional consistency (cf. Exercise 2.28). Note that the old papers Godt (1896), Neville (1926), and Ziegenbein (1940), devoted to the Clifford configurations, even label the points and circles by the vertices of the n-dimensional cube without mentioning this. Ex. 3.12: The 3D system with fields on edges and conical condition was studied in Schief (2003b) under the name of discrete CKP system. A well-posed Cauchy problem and the 4D consistency for this system were established in King-Schief (2006). Ex. 3.13: see Bobenko-Tsarev (2007) where this statement is used for the construction of principal curvature directions on a surface. Ex. 3.15: See Liu-Pottmann-Wallner-Yang-Wang (2006) and WangWallner-Liu (2007).

Chapter

4

Special Classes of Discrete Surfaces

We now turn to the study of special discrete surfaces, which are described by discrete two-dimensional equations. A discrete equation is treated as two-dimensional if it admits a well-posed initial value problem on an elementary square (2-cell) of the regular square lattice. For example, if the fields (dependent variables) of such an equation are assigned to vertices, then it is required that any three vertices of an elementary quadrilateral determine the fourth vertex in a unique way. Such equations naturally arise in the context of Q-nets, if one imposes two admissible reductions simultaneously. The multidimensional consistency is then inherited by the resulting two-dimensional equations, and is treated as their integrability. The most important class of such nets appears by imposing the Koenigs property and simultaneously restricting to quadrics. Thus, we come to the study of T-nets in quadrics.

4.1. Discrete Moutard nets in quadrics Let ]RN be equipped with a nondegenerate symmetric bilinear form (.,.) (which does not need to be positive definite), and let

be a quadric in

]RN.

We study T-nets

f : 7/.,m

-----+ Q.

m = 2 : basic 2D system. Knowing two coordinate curves f r131' f r132 of a two-dimensional T -net f : 7/.,2 -----+ Q allows one to extend the net f to the

-

127

128

4. Special Classes of Discrete Surfaces

whole of 7!}. The induction step consists in computing TIT2f = f

+ a12(T2f -

Td),

where the coefficient a12 (attached to every elementary square of Z2) is determined by the condition Tl T2f E Q, provided f, Td, T2f E Q. A simple computation using the formula

(.f + a12(T2f - Td), f

+ aI2(T2f -

TI.f))

= ~o

shows that this condition is equivalent to aI2 =

(.f, Tlf - T2.f) ~o - (Td, T2.f)

.

This elementary construction step, i.e., finding the fourth vertex of an elementary quadrilateral from the known three vertices, is symbolically represented in Figure 4.1. It is this picture that we have in mind when speaking about discrete 2D systems (or equations) with fields on vertices.

h--------{

f------h Figure 4.1. 2D system on an elementary quadrilateral.

m ;:::: 3 : consistency. Turning to the case m 2': 3, we see that one can prescribe all coordinate lines of aT-net n in Q, i.e., 13; for all 1 :s; i :s; m. Indeed, these data are independent, and one can, by induction, construct the whole net from them. The induction step is essentially two-dimensional and consists in determining TiTjf, provided f, Td and Tjf are known. In order for this induction process to work without contradictions, equations

n

(4.1)

TiTjf - f

= aij(Tjf -

Td),

aij

= (.f, Td ~o -

- Tj.f) (Td, Tj.f)

must have a very special property. To see this, consider in detail the case of m = 3; higher dimensions do not add anything new. From f and fi one determines all fij uniquely. After that, one has, in principle, three different ways to determine h23, from three squares adjacent to this point; see Figure 4.2. These three values for h23 have to coincide, independently of initial conditions. Definition 4.1. (3D consistency) A 2D system is called 3D consistent if it can be imposed on all two-dimensional faces of an elementary cube of Z3.

129

4.1. Discrete Moutard nets in quadrics

h----'----------{ I

12 • - - - - - -

iI2

/

/ /

f------iI Figure 4.2. 3D consistency of 2D systems.

As in the case of the 4D consistency of 3D systems, this definition is not restricted to systems with fields on vertices, and makes sense, for instance, for systems with fields on edges. This case will be considered in more detail in Chapter 6. A quite general theorem, analogous to Theorem 2.6, holds. Theorem 4.2. (3D consistency yields consistency in all higher dimensions) Any 3D consistent discrete 2D system is also m-dimensionally consistent for all m > 3. Proof. The proof goes by induction on m and is analogous to the proof of Theorem 2.6. 0 Theorem 4.3. (3D consistency of T-nets in quadrics) The 2D system (4.8) governing T-nets in Q is 3D consistent. Proof. This can be checked by a tiresome computation, which, however, can be avoided by the following conceptual argument. The T -nets in Q are the result of imposing two admissible reductions on Q-nets, namely the T-reduction and the restriction to a quadric Q. This reduces the effective dimension of the system by 1 (allows one to determine the fourth vertex of an elementary quadrilateral from the three known vertices), and transfers the original 3D equation into the 3D consistency of the reduced 2D equation. Indeed, after finding iI2, 123 and iI3, one can construct iI23 according to the planarity condition (as intersection of three planes). Then both the T -condition and the Q-condition are fulfilled for all three quadrilaterals adjacent to iI23. Therefore, these quadrilaterals satisfy our 2D system. 0

To formulate the next important property of T-nets in quadrics often used in the sequel, the following definition will be convenient. Definition 4.4. (Labelling of edges) A system of functions Cti defined on the edges of 7J.,m parallel to the coordinate axes 'Bi is called a labelling of edges if these functions satisfy

(4.2)

i

1= j,


130

i. e., iJ in every elementary square the opposite edges caTTY equal values oj the corresponding ai.

If one assigm; the value of O'i on the edge ('IL, II + ei) to the lattice point II E zm, then the labelling property is expressed as ai = ni (Hi) for i = 1, ... ,m. To determine a labelling, one can prescribe it on the coordinate axes 13 i .

Theorem 4.5. (Labelling property for T-nets in quadrics) For a T-net J : zm

-+ Q,

the Junctions

(4.3)

O:i

defined on the edges oj

zm

= (1,Td),

pamllel to 13 i , have the labelling p7'Operty (4.2).

Proof. It follows directly from (4.1) that (TiTjJ, TjJ) = (Td, J),

hTjJ, Td) = (TjJ, J),

o

which is equivalent to (4.2).

With the notation (4.3), the expression in (4.1) for the coefficients aij of the discrete Moutard equations takes the form ni - aj (4.4 ) aij = -------"---""0 - hf. TjJ)

4.2. Discrete K-nets 4.2.1. Notion of a discrete K-net. In discretizing K-surfaces and their transformations, we take as a starting point the characterization given in Theorem 1.24.

zm

IR3 is called an m-dimensional discrete K-net iJ Jor any elementary quadTilateml (1, Td, TiTjJ, TjJ),

Definition 4.6. (Discrete K-net) A discTete A-net J :

(4.5)

hTjJ -

-r:dl =

lTd - JI

and

hTjJ - Tdl

-+

= hJ - JI;

in otheT 'Words, iJ the Junctions {3i = Ib"dl, defined on the edges pamllel to the coordinate axes 13 i JOT i = 1, ... ,m, have the labelling p7'Operty (depend on Hi only).

The property (4.5) of a net J is known as the Chebyshev p7'Operty, so a quadrilateral (1, Td, TiTjJ,TjJ) satisfying (4.5) can be called a Chebyshev quadrilateml. Thus, a Chebyshev quadrilateral can be considered as a parallelogram bent in space along one of its diagonals.

Lemma 4.7. (Chebyshev quadrilateral) A Chebyshev quadTilateml is symmetric under the 180 0 7'Otation about the line thTough the midpoints oj its diagonals.

4.2. Discrete K-nets

131

Figure 4.3. A discrete K-surface

Proof. Let 0 1 and O 2 denote the midpoints of the diagonals [j, TiTjfl and hf, Tjf], respectively (see Figure 4.4). It is enough to show that this line is orthogonal to both diagonals. But, as it follows from considering the congruent triangles 6(j, Til, TiTjf) and 6(j, Tjf, TiTjf) , the point 0 1 is equidistant from Til and Tj f, and therefore belongs to the plane through O 2 orthogonal to hf, Tjfl· Hence, the line (0 1 0 2 ) is also orthogonal to hf, Tjfl. For similar reasons, this line is orthogonal to the second diagonal as well. 0

,

f

-

-

-

-

-

-

-

-

01 : - -,, I

-

-

-

-

-

-

-

-

-

, I

Figure 4.4. Geometry of a Chebyshev quadrilateral.

A characterization of the Lelieuvre normal field of a discrete K-net is analogous to the smooth case. Theorem 4.8. (Gauss map of a discrete K-net = T-net in a sphere) The Lelieuvre normal field n : 7L,m ----+ ]R3 of a discrete K-net f : 7L,m ----+ ]R3 takes values, possibly upon a black-white rescaling, in some sphere S2 C ]R3 .• thus being proportional to the Gauss map.


132

Conversely, any T-net in the unit sphere n : zm --7 §2 is the Gauss map and the Lelieuvre normal field of a discrete K-net f : zm --7 ]R3. The functions

(4.6) have the labelling property (depend on Ui only), which therefore holds also for

(4.7)

Ii

= 15inl = 2\ sin ~i \

and

{3i

= 15ill = Isinail.

Proof. The definition of K-nets is equivalent to the following conditions for the Lelieuvre normals:

Because of the symmetry formulated in Lemma 4.7 (which clearly yields the rotational symmetry also for the directions of normal vectors), we derive:

As a consequence,

Thus, the Lelieuvre normal field of a discrete K-surface forms, possibly after a black-white rescaling, a T-net in a sphere, being an instance of the class considered in Section 4.1. This proves the first claim of the theorem. Turning to the second claim, we start with aT-net n in the unit sphere §2, described by the equations (4.8)

aij

=

(n, Tin - Tjn)

1 - (Tin, Tjn)

.

Due to Theorem 4.5, the edge functions cos ai = (Tin, n) depend on Ui only, and therefore "If = 15inl2 = 2(1-cosai) = 4sin2(ad2) also depend only on Ui. Define the discrete A-net f : zm --7 ]R3 by (2.62). Then {3i2 = 15ill 2 = 1 - (Tin, n) 2 = 1 - cos 2 ai · =2 sm ai, which proves that (4.5) is fulfilled.

o

According to Theorem 4.8, the discrete K-nets f (modulo scalings and translations) are in a one-to-one correspondence with the T-nets n in §2. A set of initial data which determines a net of this class can be chosen as (K.6.) the values of n on the coordinate axes 13 i for 1 :S i :S m, i.e., m discrete curves niB; in §2 through a common point n(O).


133

4.2.2. Backlund transformation. Definition 4.9. (Discrete Backlund transformation) Two discrete Knets f, f+ : zm ~ ]R3 with corresponding edges of equal length, i = 1, ... ,m, are related by a Backlund transformation if they are related by a Weingarten transformation and the distance If+ - fl is constant, i.e ... does not depend on u E zm. The net f+ is called a Backlund transform of f.

Comparing this with Definition 4.6, we see that the net F : zm X {O, I} ~ with F(u, 0) = f(u) and F(u, 1) = f+(u) is an M-dimensional discrete K-net, where M = m + 1. For discrete K-nets, once again, transformations do not differ from the nets themselves. In particular, to specify a Backlund transform f+ of a given m-dimensional discrete K-net f, or, equivalently, a Moutard transform n+ of the Gauss map n, one can prescribe the following data:

]R3

(B~) the value of n+(O) E §2.

Permutability of Backlund transformations for discrete K-nets is a direct consequence of the 3D consistency of T-nets in §2. Theorem 4.10. (Permutability of discrete Backlund transformations) Let f be a discrete K-net, and let f(l) and f(2) be two of its Backlund transforms. Then there exists a unique discrete K-net f(12) which is simultaneously a Backlund transform of f(1) and of f(2). The points of the fourth surface f(12) lie in the intersection of the tangent planes to f(1) and to f(2) at the corresponding points, and are uniquely defined by the properties If(12) - f(1)1 = If(2) - fl and If(12) - f(2)1 = If(1) - fl, or, in terms of the Gauss maps, (n(1),n(12)) = (n,n(2)) and (n(2),n(12)) = (n,n(l)). The four Gauss maps are related by the discrete M outard equation with the minus sign, that is, n(12) - n is parallel to n(l) - n(2).

4.2.3. Hirota equation. For a convenient analytic description of discrete K-nets and their Gauss maps, we will use the following matrix formalism. The space ]R3 can be identified with the Lie algebra su(2), (4.9)

(

-ZX3 X2 - zXl

The vector product in as follows:

-X2. - iX1) ZX3

]R3

E su(2)

+-+

( Xl, X2, X3 )T Em,.. 1TJ)3

and the matrix commutator in su(2) correspond

[X,y] = 2x x y. This isomorphism makes it unnecessary to distinguish between vectors in ]R3 and matrices in su(2). In other words, we use the following basis of the


134

linear space su(2): e1

=

( 0 -i). 0 = -20"1, -i

e2

=

(01 -1). 0 = -20"2,

e3

=

(-i0 0) = -20"3, . i

where O"j are the Pauli matrices. We supply su(2) with the scalar product (.,.) induced from 1P1 3 . It is easy to see that in the matrix form it may be represented as (4.10)

(x, y) =

-~ tr(xy) = ~ tr(xy*),

where x* stands for the Hermitian conjugate of x. Rotations in 1P13 are conveniently described by the adjoint action of the Lie group SU(2), which consists of complex 2 x 2 matrices satisfying the condition * = * = 1, where 1 is the 2 x 2 unit matrix (the group unit), so that -1 = *. In terms of components:

=

(~ ib

ib)

a '

a, bEe,

The property which makes SU(2) suitable for a description of rotations in 1P1 3 is the following.

Lemma 4.11. (Rotations in terms of 2x2 matrices) An arbitrary E SU(2) can be written as:

(4.11)

= cos 0 . 1

+ sin 0 . xo,

with

Xo E su(2), (xo, xo) = 4.

In this notation, the action -1 x on an arbitrary vector x E su(2) zs nothing but the rotation of x around the vector Xo by the angle 20.

For a given discrete K-net f : zm --+ 1P13 with the Gauss map n : zm --+ §2, we introduce the frame (u) E SU(2) at each point u E zm by the formula (4.12)

Clearly, the frame is defined not uniquely but rather up to transformations of the type (4.13)

with real-valued functions 1\,. Admissible transformations (4.13) of the frame are called gauge transformations. We define also the transition matrices (4.14)

naturally assigned to the edges (u, u + ej) of zm. The action of a gauge transformation (4.13) on the transition matrices is given by

(4.15)

Uj(u)

I---'

exp(il\,j0"3)Uj(U) exp( -iI\,0"3) ,

I\,

= I\,(u),

I\,j = I\,(u+ej).


135

By definition, transition matrices satisfy the identity (4.16)

Theorem 4.12. (Hirota equation for a T-net in a sphere) Let n : zm ----;. §2 be a T-net in §2 with the labelling (n'J' n) = cos cx j. Then the frame ([> : zrll ----;. SU(2) can be gauged so that all transition matrices Uj = Uj(u) have the form

(4.17)

Uj =

(

COS( CXj /2)ei~j

-i sin( CXj /2)

-i sin( ui/2) )

cos( CXj /2)e-i~J

with some numbers ~j = ~j(u) E 1R/(27rZ) assigned to the edges (u o u There e:r'ists a function ¢ : zm ----;. IR/ (47rZ) such that 1

~j = 2(¢j -

where. as us'ual, ¢ = ¢(u) and ¢j = ¢(u so-called Hirota equation

(4.18)

. 1 sm - (¢jk 4

+ ¢k -

/2

:

tan(Qj/2) 'Wj tan(Qj/2) 'WA:'

We derived this equation from a 3D consistent geometric system (T-nets in a sphere), which assures the 3D consistency of (4.28). However, this property can be verified also directly; see Exercise 4.1.


139

4.2.4. Discrete zero curvature representation. Observe that the dependence of equation (4.18) on parameters is through the ratios tan(aj/2)/tan(ak/2) only. Therefore, this equation remains invariant under the change of parameters aj ~ aj(>') such that tan (aj(>')/2) = >. tan(aj/2). Consider matrices Uj(>') E SU(2) depending on the parameter>. (called spectral parameter) which are obtained from Uj by replacing aj ~ aj(>'):

(4.29)

U, >. = J( )

(coSC~j~>'»)ei(¢j-¢)/2 -isin(CtjY»)

-isin(Ctj~>'»)

) .

cos(CtjY»)e-i(¢j-¢)/2

By construction, equations (4.16) hold in view of the Hirota equation (4.18) also with these matrices Uj (>') for any fixed >.. Considering>. as a formal parameter allows us to make the converse claim. Theorem 4.14. (Discrete zero curvature representation of Hirota equation) Equation (4.16) with the spectral parameter dependent matrices Uj(>') given in (4.29) is equivalent to the Hirota equation (4.18). Proof. Direct computation; see Exercise 4.2. Note that in this computation it is convenient to factor out the respective scalar normalizing factors cos(aj(>.)/2) and to deal with transition matrices

(4.30)

Uo(>.) = J

ei(¢j-¢)/2 (

-i>. tan(~)) 2

-i>. tan(¥)

e- i (¢j-¢)/2

with a simpler (rational) dependence on the spectral parameter.

0

Equation (4.16) with spectral parameter dependent matrices Uj(>') (or UJ (>.)) is known as the discrete zero curvature representation of the Hirota equation, and is a classical integrability attribute. In Chapter 6 we will show how one can derive the zero curvature representation of a 3D consistent equation from the equation itself, with an illustrative example of the Hirota equation. 4.2.5. Discrete K-surfaces. For discrete K-surfaces, that is, for discrete K-nets with m = 2, one can, as usual, modify the notion of Lelieuvre normals so that the passage to the continuous limit can be performed in a straightforward way. Indeed, after the usual change of variables (2.68), the Gauss map n : 71} -> §2 will turn into a discrete M-net in §2 satisfying the discrete Moutard equation with plus signs: (4.31)

140


Figure 4.5. Discrete K-surfaces of revolution.

Alternatively, such nets are called discrete Lorentz-harmonic nets in §2. Of course, the change of variables (2.68) must be supplied with the Lelieuvre formulas (2.69): (4.32)

This will also yield the change of sign cos a:2 f---t - cos a:2, that is, for the labelling of edges of the second coordinate direction.

a:2

f---t

7r-a:2

Modifications in Theorem 4.12 for the case of discrete Lorentz-harmonic nets in a sphere, necessary for a well-behaved continuous limit, are given in the following statement. Theorem 4.15. (Hirota equation for M-nets in a sphere) Let n : -+ §2 be an M-net in §2 with the labelling (nj, n) = cos a:j. Then the frame AS at black points, S f---> J-lS at white points), which can be fixed by prescribing S arbitrarily at one black and at one white point. There exists a labelling Ct of edges of zm such that (4.67)

Ifi - fl2 = CtiSSi

(i = 1, ... , m).

A black-white rescaling of the function S results in the rescaling Ct of the labelling Ct.

f--->

(AJ-l) -lCt

Proof. For a circular quadrilateral (j, Ii, fij, fj) with M the intersection point of diagonals, one has two pairs of similar triangles,

153

4.3. Discrete isothermic nets

cf. Figure 4.12. Hence,

(4.68)

IMfijl

IMfjl

IMfil

IMfl

lfij - fjl

IMfijl

Ifi - fl '

IMfjl

IMfil IMfl

Ifij - fil Ifj - fl .

It follows that

(4.69)

1M fiji. 1M fjl IMfl

IMfil

This can be written as (4.70) I(M, fij) . I(M, fj) I(M, f) I(M, fi)

lfij - fjl2

IMfij I . IMfil IMfl IMfjl

Ifi - fl2 '

lfij - fjl2 Ifi - fl2 '

I(M, fij) . I(M, fi) I(M, f) I(M, fj)

Ifij - fil2 Ifj - fl2 .

lfij - fil2 Ifj - fl2 .

Indeed, contemplating Figure 4.12, it is not difficult to realize that the fractions on the left-hand side of each of the two equations in (4.70) are either both negative (for an embedded quadrilateral), or both positive (for a non-embedded quadrilateral), so that the replacement of the quotients of lengths in (4.69) by the quotients of directed lengths in (4.70) is legitimate. Substitute the defining relations (4.66) of the function 8 into (4.70):

Ii

Ii

Figure 4.12. Embedded (left) and nonembedded (right) circular quadrilaterals.

(4.71)

8j8ij 88i

lfij - fjl2 Ifi - fl2 '

8i8ij _ Ifij - fil2 88j Ifj - fl2

But this is equivalent to the claim that the functions (4.72) possess the labelling property,

Tj (}:i

= (}:i·

o


154

The notation ai for edge labellings in Theorems 4.25 and 4.29 is the same for a reason. Theorem 4.30. (Origin of the edge labelling for factorized crossratios) If the edge labelling ai for a discrete isothermic net f : zm ----t IRN is introduced according to equation (4.67), then the cross-ratios of its elementary quadrilaterals are factorized as in equation (4.63). Proof. For a circular quadrilateral (f, Ii, fij, fj) one has:

q(f, Ii, fij, fj) =

Iii E

fl· llij - fjl Ifj - fl·lfij -iii'

where E = -1 for an embedded quadrilateral and one. Thus,

(f f f f) q , i,. ij, j =

E

Ifi - fl2 - f 12

Ifj

Ifij - fjl

Iii - fl

E

= 1 for a nonembcdded Ifj - fl Ifij -iii'

Upon using equations (4.67) and (4.68), the previous formula can be rewritten as (f f · f·· f·) - aiSi .IAIfjl_ O'.iSi .l(M,fj) q , t, tJ' J - E ajSj IMfi I - ajSj l(M , f) , i and finally, due to (4.66), we arrive at

q(f, Ii, lij, fj) = aisi . 8j = ai , aj8j 8i aj

o

which proves the theorem.

Remark. Theorem 4.29 as it stands cannot be reversed: the existence of a function 8 satisfying (4.67) does not yield the Koenigs property. Indeed, from (4.67) and (4.70) one finds: (4.73)

l(M, fij) . l(M, fj) l(M, f) l(M, fi)

8j S ij

l(AI, fij) . [(AI, Ii) l(AI,.f) l(M, fj)

88j

which are equivalent to (4.74)

l(M,fij) = ± 8ij, l(M, f) 8 .

(with the same sign ± in both equations). The latter equations are somewhat weaker than (4.66), which is necessary and sufficient for the net f to be Koenigs. However, assuming some additional information about f, it is possible to force the plus signs in the latter formulas. For instance, if it is known that all elementary quadrilaterals of a two-dimensional circular net f are embedded, then property (4.67) is sufficient to assure that f is Koenigs. Indeed, in this case a2/a1 < 0, so that (4.67) yields 82/81 < and 812/8 < 0, and then the plus sign has to be chosen in (4.74).

°


155

Remark. For a Darboux pair of discrete isothermic nets, formula (4.67) for the (m + 1)-st direction reads:

(4.75)

If+ - fl2 = css+,

and literally coincides with the corresponding formula (1.76) for the smooth case. 4.3.5. Moutard representatives of discrete isothermic nets. The metric coefficient of a discrete isothermic net f can be used to produce its Moutard representative by a suitable rescaling of its lift j into the light cone of jRN+1,l. This leads to a new characterization of discrete isothermic nets, which is manifestly Mobius invariant, since it is given entirely within the formalism of the projective model of Mobius geometry. The following statement is a discrete analog of Theorem 1.32. Theorem 4.31. (Discrete isothermic net

= T-net

in light cone) If

f : zm --; jRN is a discrete isothermic net, then its lift s = s-l j : zm --; IL N +1,l to the light cone of jRN+1,l satisfies the discrete Moutard equation

(4.76) Conversely, given a discrete T-net 8 : zm --; IL N +1,l in the light cone, let the functions s : zm --; jR and f : zm --; jRN be defined by

(4.77)

8 = s-l(f + eo

+ Ifl2e oo )

(so that s-l is the eo-component, and s-l f is the jRN -part of 8 in the basis

e1,··· ,eN,eo,e oo )' Then f is a discrete isothermic net. Proof. This follows from Theorem 2.32 and the fact that for a Koenigs net f in jRN the net j = f + eo + Ifl2e oo is also a Koenigs net in the light cone IL N +1,l C jRN+l,l. 0 Thus, we found an interpretation of discrete isothermic nets as an instance of T-nets in a quadric, governed by (4.76) with

(8, T'8 J - T8) t

(4.78)

aij

=

( ' TiS,

')

TjS

.

The edge labelling of a discrete isothermic net f (which provides the factorization (4.63) of its cross-ratios) is already encoded in its lift 8 to the light cone. Indeed, ai

= Iii - fl2 = -2(8, Ti 8 ), SSi

and Theorem 4.5 assures that these quantities depend on

Ui

only.

Theorem 4.3 says that T-nets in a quadric are 3D consistent. This proves the 3D consistency of discrete isothermic nets (Theorem 4.26). In particular,


156

Darboux transformations are nothing but Moutard transformations in the light cone. They are governed by the discrete Moutard equations

(4.79) and are specified by prescribing the value 8+(0) at one point. The parameter c of a Darboux transformation is encoded in the quantity c -__ 2('s, s'+) -_

If+ -+ fl2, ss

which is independent of u E zm, so that c = -2(8(0),8+(0)).

4.3.6. Christoffel duality for discrete isothermic nets. Specializing the notion of Christoffel duality from general Koenigs nets to circular ones, the first essential observation is: the dual net for a discrete isothermic net is discrete isothermic, as well. Indeed, any quadrilateral with sides parallel to the corresponding sides of a circular quadrilateral is, obviously, also circular. A more detailed description of duality for discrete isothermic nets is contained in the following theorem. Theorem 4.32. (Christoffel dual of a discrete isothermic net) Let f : zm ----7]RN be a discrete isothermic net, with factorized cross-ratios (4.80)

and with metric coefficient s : zm one-form c5 1* defined by (

4.81)

~f

*

c5d = ai lc5dl2 =

----7

~f SSi '

Then the

]R*.

i

]RN -valued

discrete

= 1, ... ,m,

is exact. Its integration defines (up to a translation) a net 1* : Z2 ----7 ]RN, called Christoffel dual of the net f. The net 1* is discrete isothermic, with cross-ratios

(4.82)

q

ai (f *' f* i ' f*ij' f*) j =-;:;:J

and with metric coefficient s* = s-l : zm ----7 ]R*. Conversely, if for a given Q-net f : zm ----7 ]RN there exists an edge labelling ai such that the discrete one-form (4.83)

c5 f* i

c5d = ai lc5dl 2

is exact, then f is a discrete isothermic net, with cross-ratios as in (4.80).


157

Proof. The first part of the theorem is a consequence of the general construction of dual Koenigs nets. Equation (4.82) follows directly from (4.80), (4.81). To prove the converse part, identify the plane of the quadrilateral U, fi, iij, fj) with C. Then the exactness condition for an elementary quadrilateral is equivalent to (the complex conjugate of) _a_j_ _

fj - f

aj

fij - ii .

Upon clearing denominators the latter equation turns into the cross-ratio equation (4.80) (in the generic situation, when fij - ii - fj + f i- 0). Thus, the exactness of the form (4.83) actually characterizes discrete isothermic nets. 0

Corollary 4.33. The noncorresponding diagonals of any elementary quadrilateral of a discrete isothermic net f and of its Christoffel dual are related by (4.84)

f i* - f*j

=

( ai -

aj

) fij - f Ifij _ fl2 '

Proof. We start with putting (4.63) into several equivalent forms; these computations hold not only in the Clifford algebra e£(JRN ) but also in an arbitrary associative algebra with unit A. Written as (4.85) this equation displays the symmetry with respect to the diagonal flips of an elementary quadrilateral, expressed as fi +--> fj and f +--> fij, respectively (both have to be accompanied by the change ai +--> aj). Writing (4.85) as

and dividing from the left by fij - f, we arrive at the so-called three-leg form of the cross-ratio equation: (4.86) Observe that, according to (4.81), the right-hand side of (4.86) is equal to -Ut - 1*) + Uj - 1*) = fj - ft- This proves the first equation in (4.84), if one takes into account the inversion formula e- 1 = -e/lel 2 for vectors E JRN C e£(I~N). The second equation in (4.84) is analogous. 0

e

Note that by multiplying (4.86) by fij - f from the right, we eventually arrive at (4.87)


158

which is thus demonstrated to be equivalent to (4.85), a fact which is not obvious because of the noncommutativity. Finally, the diagonal flip f ~ lij turns (4.86) into (ai - aj)(fij - f)-I = ai(fij - fj)-1 - aj(fij - 1i)-I,

and, comparing the latter equation with (4.86), one finds an important consequence of (4.63): (4.88) Thus, we have found an alternative, noncommutative proof of the exactness of the one-form (j f*. 4.3.7. 3D consistency and zero curvature representation. We now turn to a direct algebraic proof of Theorem 4.26. This proof, like many other results on discrete isothermic nets, admits an immediate generalization to the case when the fields f in the cross-ratio equation (4.63) take values in an arbitrary associative algebra A with unit (over a field X), with the labellings ai taking values in X. In our geometric situation A = ee(Il~N) and X = R Proof of Theorem 4.26. We have to show that, imposing the cross-ratio equation on all six faces of an elementary cube e123 of £:3, the three values for iI23, coming from the faces Tiejk, coincide. In the case of commuting fields f this would be a result of a straightforward computation; however in the noncommutative context such a computation could hardly be performed without the aid of an ingenious matrix formalism. It is customary to represent Mobius transformations on ( v). Its corresponding bilinear symmetric form A(·,·) is of central importance for the following theory. Definition 4.41. (Mixed area) Let P and Q be two k-gons with parallel corresponding (possibly degenerated) edges. Their mixed area is given by the bilinear symmetric form

A(P, Q) = t(A(P + Q) - A(P) - A(Q». The area of a linear combination of two polygons is given by the quadratic polynomial (4.125)

A(P + tQ) = A(P)

+ 2tA(P, Q) + t 2A( Q).

We have a sort of scalar product A(-, .) on the space of polygons with parallel edges. It is natural to investigate which polygons are "orthogonal" with respect to the mixed area bilinear symmetric form. It turns out that this way one obtains the dual quadrilaterals from Section 2.3.l. Theorem 4.42. (Dual quadrilaterals via mixed area) Two quadrilaterals P = (PI, P2, P3, P4) and Q = (ql, q2, q3, q4) with parallel corresponding edges, Pi+! - Pi II qi+l - qi, i E Z (mod 4), have vanishing mixed area

A(P,Q) = 0 if and only if they are dual, i. e., if their noncorresponding diagonals are parallel:

Proof. Denote the edges of the quadrilaterals P and Q as in Figure 4.17. Formula (4.124) implies that the area of the quadrilateral P + tQ is given by 1

A(P + tQ) = "2([a + ta*, b + tb*]

+ [c + tc*, d + td*]).

172


P:,:,4_ _ _C~_-""7! P3

b

d PI

a

P2 Figure 4.17. Dual quadrilaterals.

Identifying the linear terms in t and using the identity a + b + c + d = 0, we get

4A(P,Q)

+ [a*, b] + [c, d*] + [c*, d] [a + b, b*] + [a*, a + b] + [c + d, d*] + [c*, c + d] [a + b, b* - a* - d* + c*]. [a, b*]

-

Vanishing of the last expression is equivalent to the parallelism of the non0 corresponding diagonals, (a + b) II (b* + c*). A quadrilateral with vanishing area, A(P) = 0, is self-dual, A(P, P) = 0, and has parallel diagonals. In the case A(P) of- 0, the existence and uniqueness of the dual quadrilateral pr~ven in Lemma 2.20 now follow from the fact that the corresponding space :1>( v) is two-dimensional. Moreover, a quadrilateral P wi~h nonvanishing area and its dual P* build an orthogonal basis of the space :1>( v): (4.126)

A(AP + f-LP*) = A2 A(P)

+ f-L2 A(P*).

We call the space :1>(v), v = (VI,V2,V3,V4) (and all the quadrilaterals in this space) nondegenerate if every two of its consecutive tangent_directions are different, Vi+I of- Vi. The signature of the area form A : :1>(v) ~ lR. depends on the quadruple v = (VI, V2, V3, V4) _ E (lR.IPI)4 and can be also characterized in terms of the quadrilaterals in :1>( v).

Theorem 4.43. (Signature of the area form) Suppose :1>(V) , v = (VI, V2, V3, V4), is a nondegenerate space of quadrilaterals with consecutive edges par~llel to the tangent directions VI, V2, V3, V4 E lR.IPl = §l / { ±I}, and let ~ E :1>( v) be a quadrilateral with nonvanishing edges. The area form A : :1>( v) ~ lR. is indefinite (resp. definite) if and only if two of the following equivalent conditions hold:

(i) the cross-ratio q(Vl,V2,V3,V4)
0),

4.5. Discrete surfaces with constant curvature

173

(ii) all vertices of P are extremal points of their convex hull (resp. one of the vertices of P lies in the interior of the convex hull of the other three vertices). See Figure 4.18.

2

A(P)

>0

3 -

1

A(P*) < 0

A(P) > 0

/1

A(P*) > 0

Figure 4.18. Left: The signature of the area form is indefinite; the vertices of the polygons lie on the boundary of their convex hull. Right: The signature of the area form is definite; for any quadrilateral of the family one vertex is in the interior of the convex hull of the other three.

The formulation of this theorem may require comments. The cross-ratio can be computed as

( ) _ [VI, V2] [V3, V4]

q v - [AV2, V3 A][AV4, VI A]'

where Vi E IR2 are some representatives of Vi E IRlP'I. Let us also mention that the property (ii) is independent of the choice of the quadrilateral P from the family. If A(P) -=1= 0, the diagonal form (4.126) allows us also to characterize two alternative cases described in Theorem 4.43 in terms of the areas of P and of its dual P*: the area form is indefinite (resp. definite) if A(P)A(P*) < 0 (resp. A(P)A(P*) > 0). Proof. The sign of the cross-ratio q( VI, V2, V3, V4) can be characterized combinatorially. The cross-ratio is positive if and only if the directions VI, V3 E IRlP'I are separated by the directions V2, V4 E IRlP'I (recall that VI -=1= V2 -=1= V3 -=1= V4 -=1= VI), and q < 0 in the opposite case. It is not difficult to see that in the last case, when the pairs of directions VI, v3 and V2, V4 do not intertwine, the edge lines can be realized by a convex quadrilateral. The whole corresponding family P( v) consists of convex and of nonembedded quadrilaterals, and for both types the vertices lie on the boundary of their convex hull. Similarly one can check that when t~e pairs of directions VI, v3 and V2, V4 intertwine, the corresponding family P(v) consists of embedded 0 nonconvex quadrilaterals. 4.5.3. Curvatures of a polyhedral surface with a parallel Gauss map. Consider a polyhedral surface f equipped with a congruence of lines £ such that every vertex has a line passing through it and the lines assigned to adjacent vertices are coplanar. Our main example is a Q-net f : Z2 ----+ IR3


174

with a line congruence £ : 'I} -+ £} such that f (u) E £( u) for all u E Z2. Let n : Z2 -+ ]R3 be the corresponding generalized Gauss map. If £ is simply connected, then the net n is determined up to a constant factor and is fixed as soon as the length of the normal at one vertex is prescribed. Theorem 4.44. (Parallel surface area) The area of the parallel surface ft = f + tn obeys the law (4.127)

A(ft)

=

L (1 - 2tHp + t 2Kp) A(f(P)), p

where

(4.128)

H

__ A(f(p),n(P)) p A(f(P)) '

Kp

=

A(n(P)) A(f(P))'

Here the sum is taken over all (combinatorial) faces P, and f(P) and n(P) are the corresponding faces of the surface f and its generalized Gauss map

n. Proof. Since the corresponding faces and edges of discrete surfaces n are parallel, the claim follows from formula (4.125).

f and 0

Having in mind Steiner's formula, we come to the following natural definition of the curvatures in the discrete case. Definition 4.45. (Mean and Gaussian curvatures of polyhedral surfaces) Let (f, n) be two parallel polyhedral surfaces. We consider n as the generalized Gauss map of f. The functions Hp and Kp on the faces given by (4.128) are the mean and the Gaussian curvatures of the pair (f, n), i.e., of the polyhedral surface f with respect to the Gauss map n. Note that, as in the smooth case, the Gaussian curvature is defined as the quotient of the areas of the Gauss image and of the original surface. Since for a given polyhedral surface with a line congruence the map n is defined up to a common factor, the curvatures at the faces are also defined up to multiplication by a constant. The principal curvatures /',;1, /',;2 at the faces are naturally defined using the formulas H = (/',;1 + /',;2)/2 and K = /',;1/',;2 as the zeros of the quadratic polynomial (4.129)

A(fd

= (1 -

2tH + t 2 K)A(f)

= (1 -

t/',;d(1- t/',;2)A(f).

Definition 4.46. (Principal curvatures of Q-nets) Let (f, n) : 'I} -+ ]R3 x ]R3 b~ a Q-net with a generalized Gauss map. Assume that the area forms A : P -+ ]R are indefinite for all the faces f (P). Then the functions /',;1,/',;2 of (4.129) are real-valued and are called the principal curvatures of the pair (f, n).


175

The results of Section 4.5.2 imply that the principal curvatures exist for quadrilateral faces with vertices on the boundary of the convex hull. In particular, for a circular net, principal curvatures exist for any Gauss map.

4.5.4. Q-nets with constant curvature. Let (j, n) : 'l} ----) IR3+3 be a Q-net with a generalized Gauss map, and let ft = f +tn : 7f} ----) IR 3, t E IR be the corresponding one parameter family of parallel Q-nets. We define special classes of surfaces as in the classical surface theory, the only difference is that the Gauss map is not determined by the surface. The treatment is similar to the approach in relative differential geometry. Definition 4.47. (Q-nets with constant curvature) We say that a pair (j, n) has constant mean (resp. Gaussian) curvature if the mean (resp. Gaussian) curvatures defined by (4.128) for all faces of the Q-net are equal. If the mean curvature vanishes identically, H == 0, then the pair (j, n) is called minimal. Although this definition refers to the Gauss map, the normalization of the length of n is irrelevant, and the notion of constant curvature nets is well defined for Q-nets equipped with line congruences.

Theorem 4.48. (Minimal Q-net with a line congruence) A pair (j, n) is minimal if and only if f : ,£2 ----) IR3 is a discrete Koenigs net and n : ,£2 ----) IR3 is 'its Christoffel dual '/I, = .f* . Proof. We have the equivalence

H = 0 {::} A(j, n) = 0 {::} n =

.f*.

D

Figure 4.19. DisCTete Koenigs nets interpreted as a Gauss image n, and its Christoffel dual minimal Q-net f = n*.

This result is analogous to the classical Theorem 1. 36. Figure 4.19 presents an example of a minimal Q-net constructed as the Christoffel dual of its Gauss image n, which is a discrete Koenigs net. The statement about surfaces with nonvanishing constant mean curvature resembles the corresponding facts of the classical theory; see Theorem 1.37.


176

Theorem 4.49. (Constant mean curvature Q-net with a line congruence) A pair (f, n) has constant mean curvature Ho if and only if f : Z2 ---+ ]R3 is a discrete Koenigs net and the parallel surface h/ Ho is the Christoffel dual of f:

1* = f + ~on. The mean curvature of this parallel surface (f + Ao n, -n) (with the reversed Gauss map) is also constant and equal to Ho. The mid-surface f + n has constant positive Gaussian curvature Ko = 4H5 with respect to the same Gauss map n.

210

Proof. We have the equivalence

A(f,n)=-HoA(f)

~

A(f,f+

~on)

=0

~

1*=f+

~on.

For the Gaussian curvature of the mid-surface we get K

1

2HO

= __A-,(,-n.:,.-)_ A(f +

210 n)

A(f +

A(n) 1) +

1 2Ho n,

1 4H6

= 4H2 A(n)

o·

o

It turns out that all surfaces parallel to a surface with constant curvature have remarkable curvature properties, in complete analogy to the classical surface theory; cf. Theorem 1.37.

Theorem 4.50. (Parallel of constant mean curvature Q-nets are linear Weingarten) Let (f, n) be a pair of Q-nets with constant mean curvature. Consider the family of parallel Q-nets ft = f + tn. Then for any t the pair (It, n) is linear Weingarten, i. e., its mean and Gaussian curvatures H t and K t satisfy a linear relation

(4.130)

CtHt

with constant coefficients

Ct,

+ (3Kt = 1

(3.

Proof. This can be checked by a direct computation; see Exercise 4.20.

0

We see that any discrete Koenigs net f can be extended to a minimal or to a constant mean curvature Q-net by an appropriate choice of the Gauss map n. Indeed, (f, n) is minimal for n =

1*;

(f, n) has constant mean curvature for n =

1* - f.

However, n defined in such generality can lead us too far away from the smooth theory. It is natural to look for additional requirements which bring it closer to the Gauss map of a surface. In the smooth case, n is a map to

177


the unit sphere. The following three discrete versions of this fact are natural to consider:

(1) n is a polyhedral surface with all vertices on the unit sphere §2. This implies that all faces of the Q-net n are circular. This condition holds also for any parallel surface. In particular, circular net.

f

is also a

(2) n is a polyhedral surface with all faces touching the unit sphere §2. This implies that for any vertex p there is a cone of revolution with the tip p touching all faces of n incident to p. This property holds true for any parallel surface. In particular, f is also a conical net.

(3) n is a polyhedral surface with all edges touching the unit sphere §2. Polyhedra with this property are called Koebe polyhedra. For any vertex p, all edges incident to p lie on a cone of revolution with the tip p. Also this property holds true for any parallel surface, in particular for f. Such nets are called nets of K oebe type. The implementation of these additional requirements into the theory makes it more intriguing. 4.5.5. Curvature of principal contact element nets. In this section we are dealing with the case when the Gauss image n lies in the two-sphere §2, i.e., is unitary, Inl = 1. It turns out that this is the case of principal contact element nets. Let f : 2,2 ----t ]R3 be a Q-net with a parallel unit Gauss map n : 7i} ----t §2. Introducing the oriented planes P which contain the corresponding vertices of f and are orthogonal to the corresponding normals n, we obtain a contact element net

(f, P) :

Z2

----t

{contact elements in

]R3};

see Section 3.5. It can be canonically identified with the map

(f, n) :

Z2

----t

]R3 X §2

considered previously. We will call the latter also a contact element net.

Theorem 4.51. (Q-nets with unit Gauss maps are principal contact elements) Let f : Z2 ----t ]R3 be a Q-net with a parallel unit Gauss map n : Z2 ----t §2. Then f is circular, and (f,n):

Z2

----t

]R3 X §2

is a principal contact element net. Conversely, for a principal contact element net (f, P), the net f is circular and the unit net n canonically corresponding to P via n J.. P, is a parallel Gauss map of f.

178


Proof. The circularity of f follows from the simple fact that any quadrilateral with edges parallel to the edges of a circular quadrilateral is also circular. Consider an elementary cube built by two parallel quadrilaterals of the nets f and f +n. All the side faces of this cube are trapezia, which 0 implies that the contact element net (f. n) is principal.

A(n)

Figure 4.20. Parallel q-nets f and f + n with tlIP unit Gaus!-i map n. All the nets are circular. The pair (f, 11) build a principal coutact element IWt.

The lllean and th(' Gauss curvatmes of the prillcipal contact dPlllcnt nets (f,n) are defined by formulas (4.128). In Section 4.5.:3 it is shown that in the circular case tlw principal curvatures always exist. 4.5.6. Circular minimal nets and nets with constant mean curvature. Minimal and constant mean curvature principal coutact elem('nt nets are defined as ill Section 4.5.4. Since circular Koenigs nets are isothcrmic nets, we obtain the following results from Theorems 4.48, 4.49. Corollary 4.52. (Minimal circular surfaces as principal contact element nets) A principal contact element net (f,n) : Z2 --'0 jK3 X §2 is minimal if and only if the net n : Z2 --'0 §2 is isothermic and f = n* is ds Christoffel dual. Discrete isothermic nets are descrilwd through the cross-ratios of their quadrilaterals. The stereographic projection (J : jK2 --'0 §2 preserVE'S thE' cross-ratios. Combined with til(' formula f = n*, this yields a Weierstrass representation for discrete minimal surfaces through isothermic nets in a plane. See Exercise 4.21. Corollary 4.53. (Circular surfaces with constant mean curvature as principal contact element nets) A p1"incipal contact dF7ne7lt net (f. n) : Z2 ---+ jK:3 x §2 has constant mean C7LT'7)aturc Ho i- 0 ~f and only if the circular net f : Z2 ---+ jK3 is isothennic and there e:rists its dual discrete isothennic S1t'ljace f* : Z2 ---+ jK3 at constant distance If - .f* I = ~(). The unit Gauss

4.6. Exercises

179

map n which determines the principal contact element net (j, n) is given by

n = Ho(j* - f).

(4.131)

J

The principal contact element net of the parallel surface (j + o ' -n) also has constant mean curvature Ho. The mid-surface (j + 0 , n) has constant Gaussian curvature 4H;5 .

21

Proof. Only the "if" part of the claim may require some additional consideration. If the discrete isothermic surfaces f and f* are at constant distance 1/ H o, then the map n defined by (4.131) is unitary and thus circular. Again, as in the proof of Theorem 4.51, this implies that the contact element net (j, n) is principal. Its mean curvature is given by

_ A(j, n) __ A(j, Ho(j* - f)) _ H A(j, f) A(j, f) - o·

D

Remark. We mention that there exists a theory of discrete minimal and constant mean curvature surfaces of Koebe type with discrete analogues of many famous classical surfaces. Its presentation lies beyond the scope of this book.

4.6. Exercises 4.1. Check the 3D consistency of the 2D equation (4.28).

4.2. Check by a direct computation the claim of Theorem 4.14. 4.3.* Work out the details of the proof of Theorem 4.15. 4.4.* Consider a Chebyshev quadrilateral (j, ft, f12' h). Let
.) =

1(0)); if el,···, en is a sequence of edges connecting 0 = (0,0) with

U = (Ul,U2) E Z2, then

II

w(U,>.) =

L(ei'>')w(O,>.).

l:S:i:S:n

One now defines the so-called Calapso transform 9 = T)J : Z2

w-l(u,>.)

G) ~

---+

~N by

(TA;(U)).

Here equivalence of 2 x 1 matrices with entries from ee(~N) is understood modulo simultaneous right multiplication of its entries with one and the same invertible element of the algebra, so that

>.

= 0 one gets

w(-u,O) =

(~) ~

G

(ab1-

1

-f1(U)), so that w-1(u,0)

).

(~)

Clearly, for =

(f~U)).

Therefore, for small >., TAf = f + 0(>'). Prove that the Calapso transform is a discrete isothermic surface with the cross-ratios

001/(1 - >'(01) q(g,gl,g12,g2) = 002/(1- >'(02) . 4.14. Prove that the Calapso transformation 9 = TAf acts on all stars of the net f by Mobius transformations. That is, for every u E Z2 there exists a ~1obius transformation 1I1A(u) : ~N ---+ ~N such that

g(u) = MA(U) . f(u)

and

gi(U) = MA(U) . fi(U)

(i = ±1, ±2).

4.15. Given three pairwise touching spheres, prove that the circle through the touching points is orthogonal to all three spheres. 4.16. Given four pairwise (cyclically) touching spheres, prove that the four touching points are concircular, and that the circle intersects all spheres under equal angles. 4.17. In the previous exercise: if the circle intersects the spheres orthogonally, then the spheres are linearly dependent (hence S-isothermic).

4.7. BibliogTaphicall1otcs

183

4.18. Prove the so-called "Touching coins lemma": whenever four circles in 3-space tOllch cyclically but do not lie on a common sphere, they intersect the sphere which passes through the points of contact orthogonally. 4.19. Complete the details in the proof of Theorem 4.43. 4.20.* Prove Theorem 4.50 and compute the coefficients n, ;1. 4.21. Derive the following discrete Weierstrass representation for circular minimal surfaces:

(1 02~ (1 n1~

611

gIg, -[(1 + 9U}) , g1 + g) , 091 - g g1 - Y g1 - 9

g2 g),

Y29, i(l + !J2Y) , + g2 - 9 g2 - g g2 - 9

621

where Y : ;;32 --> C is a discrete hololl1orphic map, i.e., a solution of the cross-ratio equation °1

q ( g, Y I • Y 12. g2 ) = - ; 0'2

see Chapter 8. Hint: The isothennic Gauss map n : ;;32 stereographic projection of g, (nl

.

+ In2,n:3)

=

(29 I')

1)

lyI 2 'I g-+l 9 12 +1

1

--> §2

is given by the

.

4.22. Derive an explicit formula for the discrete Enneper minimal surface via the discrete Weierstrass representation of Section 4.5.5 by applying it to the standard square grid. The latter is the simplest isothermic net in a plane.

4.7. Bibliographical notes Section 4.1: Discrete Moutard nets in quadrics. General Moutard nets in quadrics were introduced in Bobenko-Suris (2005) (the first online version of this book), along with the most prominent example of the l\loutard representatives of discrete isothermic nets. The latter example was generalized in Bobenko-Suris (2007b), where discrete isothenllic nets in various sphere geometries were investigated. Later the Harne class of nets (Koenigs nets in quadrics) was treated in Doliwa (20071». Section 4.2: Discrete K-nets. The notion of discrete K-nets is due to Sauer (1950) in the case rn = 2 and to Wunderlich (1951) in the case rn = 3. A study of discrete K-surfaces within the framework of the theory of integrable systems was performed ill Bobenko-Pinkall (1996a) geometrically, and in Bobenko-Pinkall (1999) analytically. A presentation in BobenkoMatthes-Suris (2005) is based on the notion of consistency. The study of the GausH map of K-Hurfaces leadH to the notioll of discrete Lorentz-harmonic

184


nets in §2, also introduced in Bobenko-Pinkall (1996a), where the m = 2 case of Theorem 4.8 was first observed. Special classes of discrete K-surfaces were constructed by Hoffmann (1999) (discrete Amsler surfaces; see also Exercise 4.8) and Pinkall (2008) (discrete K-cylinders that touch a plane along a closed curve and those exhibiting a cone point). Color images of discrete K-surfaces are included in the book Bobenko-Seiler (1999). Discrete surfaces in Figures 4.3, 4.5 were produced using a software implementation by Ulrich Pinkall. K-surfaces are reciprocal parallel to geodesic conjugate nets, called Voss surfaces. Discrete Voss surfaces were introduced in Sauer-Graf (1931) as Q-surfaces with the property that the opposite angles at each vertex are equal. Sauer (1950) has shown that the relation between Voss surfaces and K-surfaces is preserved in the discrete setup; see also a modern presentation in Schief-Bobenko-Hoffmann (2008). The study of the angle between the asymptotic lines on discrete Ksurfaces leads to the discretization of the sine-Gordon equation, performed in Bobenko-Pinkall (1996a). The closely related integrable discretization of the sine-Gordon equation was derived by Hirota (1977b) without geometric interpretation. Its symplectic structure was studied in Faddeev-Volkov (1994). Stationary solutions of the discrete sine-Gordon equation describe a discrete pendulum, which was studied in Suris (1989), Bobenko-Kutz-Pinkall (1993). Besides discrete K-surfaces, there exist further remarkable special classes of discrete asymptotic nets: discrete affine spheres studied in Bobenko-Schief (1999a, b), and discrete Bianchi surfaces studied in Doliwa-NieszporskiSantini (2001). Section 4.3: Discrete isothermic nets. Discrete isothermic surfaces were introduced in Bobenko-Pinkall (1996b).

Darboux transformations for discrete isothermic surfaces were introduced in Hertrich-Jeromin-Hoffmann-Pinkall (1999). In particular, Theorem 4.26 on the 3D consistency of the cross-ratio equation was given in this paper for the quaternionic cases N = 3,4 under the name "hexahedron lemma" with a computer algebra proof. An analytic description of the Darboux transformation as a dressing transformation was given in Cieslinski (1999). Three-dimensional discrete isothermic nets were introduced in Bobenko (1999) and Bobenko-Pinkall (1999). A conceptual proof of the 3D consistency in a more general context of an arbitrary associative algebra was given in Bobenko-Suris (2002b).


185

The Calapso transformation for discrete isothermic surfaces (see Exercises 4.13, 4.14) as well as permutability properties of various transformations are due to Hertrich-Jeromin (2000, 2003). Discrete isothermic surfaces in higher co dimensions were studied by Schief (2001). Besides the discrete isothermic nets, there exists another interesting special class of multidimensional circular nets. These are discrete analogs of Egorov metrics. They are characterized by the property that any elementary quadrilateral (j, Ii, fij, fj) has two right angles at the vertices fi and fj· (Note that for this definition it is essential to fix the directions of all coordinate axes.) The theory of discrete Egorov nets is due to Schief, AkhmetshinVol'vovskij-Krichever (1999) and Doliwa-Santini (2000). Section 4.4: S-isothermic surfaces. The presentation of this section essentially follows Bobenko-Suris (2007b). S-isothermic surfaces, along with their dual surfaces were originally introduced in Bobenko-Pinkall (1999) for the special case of touching spheres. The general class of Definition 4.34, together with Darboux transformations and dual surfaces, is due to Hoffmann. Section 4.5: Discrete surfaces with constant curvature. Circular minimal surfaces were introduced in Bobenko-Pinkall (1996b) as Christoffel duals of their isothermic Gauss maps. The discrete Weierstrass representation was also derived in this paper. Circular surfaces with constant mean curvature appeared for the first time in Bobenko-Pinkall (1999) and Hertrich-Jeromin-Hoffmann-Pinkall (1999) as isothermic nets with a Christoffel dual at constant distance. In the second paper it was shown that equivalently circular surfaces with constant mean curvature can be defined as isothermic surfaces with a Darboux transform at constant distance.

Curvatures of circular surfaces with respect to arbitrary Gauss maps n E §2 based on Steiner's formula were introduced in Schief (2003a, 2006), where it was also shown that the surfaces parallel to a surface with constant Gaussian curvature are linear Weingarten. A curvature theory for general polyhedral surfaces based on the notions of parallel surfaces and mixed area is developed in Pottmann-Liu-Wallner-Bobenko-Wang (2007) and BobenkoPottmann-Wallner (2008). In the circular case this theory yields the same class of surfaces with constant curvatures as originally defined in BobenkoPinkall (1999), Hertrich-Jeromin-Hoffmann-Pinkall (1999); see Corollaries 4.52, 4.53. Discrete surfaces in Figure 4.19 were produced using a software implementation by Peter Schroder. The theory of minimal surfaces of Koebe type was developed in BobenkoHoffmann-Springborn (2006). These surfaces are S-isothermic and their Gauss maps are Koebe polyhedra. Global results in this theory are based

186


on the remarkable fact that a Koebe polyhedron is essentially uniquely determined by its combinatorics. This theory is closely related to the theory of orthogonal circle patterns (see Chapter 8). Section 4.6: Exercises. Ex. 4.6, 4.7: See Bobenko-Pinkall (1996a).

Ex. 4.8: See Bobenko-Pinkall (1996a), Hoffmann (1999). Ex. 4.11: See Wallner-Pottmann (2008); the corresponding theorem for smooth surfaces can be found in Darboux (1914-27, §874). Ex. 4.13, 4.14: See Hertrich-Jeromin (2000, 2003). Ex. 4.18: See Bobenko-Hoffmann-Springborn (2006). Ex. 4.21, 4.22: See Bobenko-Pinkall (1996b).

Chapter 5

Approximation

We have already had several occasions to mention that the notions, constructions and results of discrete differential geometry have not just qualitative similarity with their much more sophisticated counterparts in the smooth theory. Rather, the latter can be obtained from the former through a wellestablished continuous limit. Strictly speaking, such a continuous limit has been established up to now only for those geometries which are described by hyperbolic systems of difference, resp. differential, equations. It is this class of equations for which a rather detailed approximation theory can be developed, which is similar to the corresponding theory for ordinary difference and differential equations. Actually, this hyperbolic theory covers a substantial part of the nets considered in this book.

5.1. Discrete hyperbolic systems To formulate the general scheme that covers the majority of situations encountered so far, we will put our hyperbolic systems into the first order form. It should be stressed that this is necessary only for general theoretical considerations, and will never be done for concrete examples.

Definition 5.1. (Hyperbolic system) A hyperbolic system of first order partial difference equations is a system of the form (5.1 ) for functions Xk : ZM - t X k with values in Banach spaces X k . For each Xk, equations (5.1) are posed for i E ek C {I, ... , M}, the nonempty set of evolution directions of Xk. The complement Sk = {I, ... ,M} \ ek consists of static directions of Xk·

-

187

188

5. Approximation

We think of the variable Xk(U) as attached to the elementary cell ek(U) of dimension #Sk adjacent to the point U E ZM and parallel to 'B Sk : e k = ek(U) = {u

+L

J-liei : J-li E

[0,1]}.

iESk

Here, we recall,

'B S = {u E Z M for an index set S c {I, ... , M}.

: Ui

= 0 if i 1:- S},

Definition 5.2. (Goursat problem) 1) A local Goursat problem for the hyperbolic system (5.1) consists in finding a solution Xk for all k and for all cells e k within the elementary cube of ZM at the origin from the prescribed values Xk(O). The system (5.1) is called consistent if the local Goursat problem for this system is uniquely solvable for arbitrary initial data Xk(O). 2) A global Goursat problem consists in finding a solution of (5.1) on ZM subject to the following initial data:

(5.2) where X k : 'BSk

---+

Xk are given functions.

The following rather obvious but extremely important statement holds: Theorem 5.3. (Well-posed Goursat problem) A Goursat problem for a consistent hyperbolic system (5.1) has a unique solution x on all ofZM. Consistency conditions read: bjbiXk = bibjXk for all i (5.1), one gets the following equations:

(5.3)

bj9k,i(X)

=

bigk,j(X),

i

1= j.

Substituting

1= j,

or 9k,i(X+ gj(x)) - gk,i(X) = gk,j (x+ gi(X)) - gk,j(X), where gi(X) is a vector function whose £-th component is equal to gC,i(X) if i E cp, and is undefined otherwise. Lemma 5.4. For a consistent system of hyperbolic equations (5.1), the function gk,i only depends on those components Xp for which Sp C Sk U {i}. Proof. Equations (5.3) must hold identically in x. This implies that the function gk,i can only depend on those components Xp for which bjxp is defined, i.e., for which j E cp. As (5.3) has to be satisfied for all j E Ck, j 1= i, one obtains that Ck \ {i} c cp for these £. 0 It follows from Lemma 5.4 that for any subset S C {I, ... , M}, equations of (5.1) for k with Sk c S and for i E S form a closed subsystem, in the sense that gk,i depend on Xp with Sp c S only.

5.1. Discrete hyperbolic systems

189

Definition 5.5. (Essential dimension) The number (5.4)

is called the essential dimension of system (5.1). If d = M, system (5.1) has no lower-dimensional hyperbolic subsystems. If d < M, then d-dimensional subsystems corresponding to S with #S = d are hyperbolic. In this case, consistency of system (5.1) is a manifestation of a very special property of its d-dimensional subsystems, which we treat as the discrete integrability (at least if one excludes certain noninteresting situations, such as trivial evolution in some of the directions). Section 6 will be devoted to an extensive treatment of integrability understood as consistency.

Example 1. Consider a difference equation with M = 3 independent variables:

(5.5)

616263x = F(x, 61 x, 62x, 63x, 6162x, 6163x, 6263x).

One can pose a Goursat problem by prescribing the values of x on the coordinate planes 1312,1313,1323. Equation (5.5) can be rewritten as a hyperbolic system of first order equations by introducing auxiliary dependent variables a, b, c, f, g, h: 61 x = a, 62 a = f, (5.6)

62 x = b, 63 a = g, 63 b = h,

61 b = f, 61 c=g, 62 c=h, 63f = 62g = 61 h = F(x, a, b, c, f, g, h).

It is natural to assume that the variable x lives on the points of the cubic lattice U E Z3; the variables a, b, c live on the edges C1(u), C2 (u), C3 (u) of the lattice adjacent to the points u and parallel to the coordinate axes 131,132,133, respectively; and the variables f, g, h are associated to two-cells (elementary squares) CI2 (U), CI3(U), C23(U) adjacent to the points u and parallel to the coordinate planes 1312,1313,1323, respectively. Thus, x has no stationary directions; the stationary directions of a, b, care {1}, {2}, {3}, while the stationary directions of f, g, hare {1, 2}, {1, 3}, {2, 3}, respectively. A Goursat problem for this system would be posed by prescribing the values of x at the point (0,0,0), the values of a, b, c on the axes 13 1,13 2,13 3, respectively, and the values of f, g, h on the planes 1312,1313,1323, respectively. The essential dimension of this discrete system is d = 3 = M. It is instructive to compare this construction with its continuous counterpart: equation (5.5) is a natural discretization of the partial differential equation

(5.7)

[hfhfhx = F(x, 81 x, 82x, 83x, 81 82x, 81 83x, 8283X).

190

5. Approximation

For the latter equation one can introduce auxiliary variables a, b, c, j, g, h via partial derivatives analogously to (5.6). All these variables would be on an equal footing, being defined just at the points U E ]R3.

Example 2. Consider the difference equations which govern M -dimensional Q-nets:

(5.8) Upon introducing auxiliary variables system of first-order equations: 6iX

(5.9)

{

Vi

they can be written as a hyperbolic

= Vi,

6jVi = CijVj

+ CjiVi,

6i Cjk = (TkCij )Cjk

i =I- j, + (TkCji)Cik

-

(TiCjk)Cik,

i =I- j =I- k =I- i.

The last equation is (2.7) from Section 2.1.1, where one can also find details about its origin, as well as about how one can put it in the form with the right-hand side depending only on the unshifted variables Cij. It is natural to assume that the variable x lives at the points of the cubic lattice U E 7l.M; the variables Vi live at the edges ei (u) of the lattice adjacent to the points u and parallel to the coordinate axes 'B i , and the variables Cij, Cji for i < j live oat two-cells (elementary squares) eij(u) adjacent to the points u and parallel to the coordinate planes 'B ij . Thus, x has no stationary directions, the stationary directions of Vi are {i}, and the stationary directions of Cij are {i,j}. A Goursat problem can be posed by prescribing the values of x at the point 0 E 7l. M , the values of Vi on the axes 'B i , and the values of Cij, Cji on the coordinate planes 'B ij . The essential dimension of this discrete system is d = 3, independently of M. In particular, it may well be d < A1. Consistency of this system for any M 2 3 is interpreted as its integrability.

5.2. Approximation in discrete hyperbolic systems To handle approximation results for discrete geometric models, we need to introduce small parameters into hyperbolic systems of partial difference equations. The domain of our functions becomes

'B€ =

El7l.

x ... x

EM7l..

If Ei = 0 for some index i, the respective component in 'B€ is replaced by lR.. For instance, if E = (0, ... ,0), then 'B€ = ]RM. Thus, the domains 'B€ possess continuous and discrete directions, with mesh sizes depending on the parameters Ei. The definitions of translations and difference quotients are modified for functions on 'B€ in an obvious way:

5.2. Approximation in discrete hyperbolic systems

If Ei = 0, then 5i is naturally replaced by the partial derivative multi-index cx = (CXI, ... ,CXM), we set 50: = 5r 1 •••

5't,r.

The definition of elementary cells ified as follows:

ek =

{n

+L

191

fk For a

ek , carrying the variables Xk, is mod-

JLiei: JLi E [0, EiJ}

iESk

(so that the cell size shrinks to zero in the directions with Ei = 0). We see how the discreteness helps to organize the ideas: in the continuous case, when all Ei = 0, all the functions Xk live at points, independently of the dimensions #Sk of their static spaces. In the discrete case, when all Ei > 0, one can clearly distinguish between functions living on vertices (those without static directions), on edges (those with exactly one static direction), on elementary squares (those with exactly two static directions), etc. Having in mind the limit E - t 0, we will only treat the case when the first m ::; M parameters go to zero in a uniform way, EI = ... = Em = E, while the other Joy! - m ones remain constant, Em+l = ... = EM = 1. In this case 13 E = (Ez)m X zM-m, and we set 13 = 13° =]Rm x zM-m. Assuming that the functions gk,i = gk,i on the right-hand sides of (5.1) depend on E smoothly and have limits as E - t 0, we will study the convergence of solutions XE of the difference hyperbolic system (5.1) to the solutions xO of the limiting differential (or differential-difference) hyperbolic system

n {1, ... , m}, Gk n {m + 1, ... , M}.

(5.10)

OiXk

gZ,i(X),

i E Gk

(5.11)

5iXk

gZ,i(X),

i E

Naturally, (5.10), (5.11) describe the respective m-dimensional smooth geometry with ]v! - m permutable transformations. Throughout this chapter, a smooth function 9 : 'D - t X is a function that is infinitely differentiable on its domain, 9 E CXl('D). For a compact set X c 'D, we say that a sequence of smooth functions gE converges to a smooth function gO with order 0;

ii) discrete system (5.1) approximates the differential(-difference) system (5.10), (5.11) with order 0 small enough, solution x k of the Goursat problem exists and is unique on 'BE(1'); moreover, solutions x k converge to smooth functions xZ with order Z is open and dense in Xk . In such a case, one requires in ii) the convergence gk,i ---+ 9£,i in Coo(1)O). Then conclusions of Theorem 5.7 hold for generic initial data.

5.2. Approximation in discrete hyperbolic systems

193

As for condition iii), smooth data X2 : 13sk ~ X k are usually given a priori, and discrete data X k are obtained by restriction to the lattice: X k = X2113 In such a situation, condition iii) is fulfilled automatically. f

"k

•

We will not present a complete proof of Theorem 5.7 here but rather a substantial part of it illustrating all necessary technical ideas. Namely, we will provide the arguments for the simplest situation M = m = 2, and in this particular case we will only demonstrate the convergence k ~ 0 for solutions themselves and omit the proof of 8Q (xk - xV ~ 0 for multi-indices a with lal :::: 1. In other words, we will prove the CO(13)-convergence instead of COO (13)-convergence. The proof of the COO(13)-convergence for general M requires technical care but no essentially new ideas. Thus, we consider the discrete hyperbolic system

x x2

(5.12) with smooth functions

r, gE : X x X

~

X, and with the Goursat data

(5.13) for

Ui E

(5.14)

13i( r). It is supposed that the functions r(a, b) = fO(a, b)

+ (9(E),

r, gE satisfy

gE(a, b) = gO(a, b)

+ (9(E),

uniformly on compact subsets of X x X, and that relation analogous to (5.14) holds for all partial derivatives of the functions gE. Further, it is assumed that the discrete Goursat data (5.13) also have smooth limits:

r,

(5.15) uniformly for Ui E 13i(r). Then the solutions (a E, bE) of the Goursat problems for system (5.12) converge uniformly in 13(1'), with a suitable l' E (0, r], to a pair of Lipschitz functions (a O, bO), (5.16)

a E (Ul' U2) = aO( Ul, U2)

+ (9( E),

bE ( Ul, U2) = bO( Ul, U2)

+ (9( E),

which constitute the unique solution in 13(1') of the Goursat problem for the system (5.17) with the Goursat data

(5.18) for Ui E 13i(r).

Lemma 5.S. (A priori estimate) Let the norms of Goursat data AE, BE be bounded by E-'independent constants. Then there exists some r E (0, r] such that the norms of the solutions (a E , bE) of the Goursat problem (5.12), (5.13) are bounded on 13 E (r) independently of E.

5. Approximation

194

Proof. Let

IA'I, IB'I 'S: Mo, r

and choose Ah > Mo arbitrarily. Define

(M] - AIo)/ sup E

sup {Ir(a, lal,lbl Ml and M > M 2. We now prove the estimate for 81aE. Proceeding from U2 to U2 + E, we find:

181aE(U1, U2

+ E)I
0 such that the canonical Q-nets r : 'Bf(r) ---t ]RN converge, as E ---t 0, to the unique conjugate net f : 'B(r) ---t ]RN with the initial data (Q1,2). Convergence is with order (')(E) in COO('B(r)). Proof. This follows directly from Theorem 5.7, since systems (2.1), (2.7) and (1.1), (1.2) are manifestly hyperbolic (and can be easily rewritten in the first order form). 0 Discretization of an F-transformation. Recall that an F-transform of a given conjugate net is determined by the initial data (F 1,2) (see Section 1.1). We now produce from these the initial data (Ft2) (see Section 2.1) for an E-dependent family of F-transforms of canonical Q-nets corresponding to the initial data (Q1,2). Take the point f+ (0) from (F d· Define the edge functions ail 'Bi' bi! 'Bi by restricting the functions ai I'B i , bi I'Bi to the lattice points, or, better, to the midpoints of the corresponding edges of 'Bi. This gives the data set (F~2); along with the data (Q~2) produced above this yields in a canonical wa; an E-dependent family of Q-nets FE : (Ez)m X {O, I} ---t ]RN, which will be called the canonical Q-nets for the initial data (Q1,2), (F 1,2)'

Theorem 5.12. (Convergence of discrete F-transformations) The canonical Q-nets (r)+ = F E(-, 1) : 'BE(r) ---t ]RN converge to the net f+ : 'B (r) ---t ]RN which is the unique F-transform of f with the initial data (F 1,2), Convergence is with order (')(E) in COO('B(r)). Proof. Again, this follows directly from Theorem 5.7 applied to the hyperbolic systems consisting of (2.10)~(2.12) in the discrete case and of (1.12)~ (1.14) in the smooth case. Note that the discrete equations are implicit, and their solvability for E small enough is guaranteed on the subset of the phase space, {aj -1= 0 : 1 :s; j :s; m}, which is open and dense. 0

5.4. Convergence of discrete Moutard nets Discretization of a Moutard net. Given a Moutard net y : ]R2 ---t ]RN defined by the initial data (M1,2) (see Section 1.2.1), we produce initial data

5. Approximation

198

(Mt2) for an E-dependent family of discrete M-nets yf : (EZ)2 ~ ]RN (see Section 2.3.10). Discrete curves yf r:13 ]R3 (see (fZ)2 ----> Section 2.4.2). Equations (2.69) define the discrete A-surfaces ]R3, called the canonical discrete A -surfaces corresponding to the initial data

r:

(A 1,2). Theorem 5.15. (Convergence of discrete asymptotic nets) Canonical discrete A -surfaces r : 13 E Cr) ----> ]R:3 converge, as f ----> 0, to the unique Asurface f : 13(r) ----> ]R3 w'ith the initial data (Au). Convergence is with order O(E) in C OO (13(r)). Proof. Equations (2.69) are hyperbolic and they approximate equations (1.38). Theorem 5.7 can be applied to prove the convergence of after the E convergence of n has been already proven. 0

r,

Discretization of a Weingarten pair. Initial data (W 1,2) for a Weingarten transformation (see Section 1.3) are nothing but initial data (MT 1,2) for a l\1outard transformation of the Lelieuvre normal field. The construction of Theorem 5.14 delivers the initial data for a family of Lelieuvre normal fields (n E )+ : (fZ)2 ----> ]R3, which are therefore seen as the data (W~2) for transformed A-surfaces (r)+ : (fZ)2 ----> ]R:3, obtained via (2.70) (see Section 2.4.3). Theorem 5.16. (Convergence of discrete Weingarten transformations) Canonical discrete A-nets (r)+: 13 E (r) ----> ]R3 converge to the unique Weingarten transform, f+ : 13(1') ----> ]R3 of f with the initial data (W 1 ,2). Convergence is with order C) ( f) in Coo (13 (1') ). Proof. This is proven by comparing the (identical) formulas (1.40) and (2.70), after the convergence of 11,( and (nE)+ has been established. 0

200

5. Approximation

5.6. Convergence of circular nets In this section, we address the problem of approximating smooth orthogonal nets by discrete circular nets. Recall that the former are governed by the system (1.44)-( 1.47) with constraint (1.48), while the latter are governed by similarly looking equations (3.12)-(3.14), (3.17) with constraint (3.18). We demonstrate that this analogy can be given a qualitative content, so that for a given orthogonal net one can construct an approximating family of circular nets. However, there is a substantial obstruction to accomplishing this, which can be seen by a careful comparison of the constraints (1.48) and (3.18). We think of smooth rotation coefficients as being approximated by discrete ones. But since the discrete rotation coefficients !3kj only have i =1= k, j as evolution directions (that is, they are plaquette variables attached to elementary squares parallel to 'B jk ), there is seemingly no chance to get an approximation of such smooth quantities as Oi!3ij involved in the smooth orthogonality constraint (1.48). In order to be able to achieve such an approximation, we need some discrete analogs of the smooth rotation coefficients which would live on edges. For this aim, we turn to the Mobius-geometric description of circular nets from Section 3.1.4, more precisely, to the frame equations (3.25). Introduce vectors Vi = 1jJV(lp-l; then the frame equations become h1jJ)1jJ-l = -eiVi. Expanding these vectors with respect to the basis vectors ek, we have a formula analogous to (1.53):

(5.26)

Vi

v

= 1jJ i1jJ-l = O"iei -

1

"2

L Pkiek + hieoo . kii

The fact that the eoo-component here is equal to hi, is easily demonstrated. Indeed, from (3.25) it follows that TJ - j = hiVi = h i (Ti1jJ)-lei1jJ. Now equation (3.24) allows us to rewrite this equivalently as [eo, (Ti'tP )1jJ-l] = hiei, which proves the claim above. Observe also the normalization condition

(5.27)

0"; =

1

-l L P~i

.

kii

Coefficients Pki are edge variables analogous to smooth rotation coefficients. Indeed, the vectors Vi are defined on the edges of zm parallel to the coordinate axis 'B i . However, these vectors do not immediately reflect the local geometry near these edges; rather, they are obtained by integration of the frame equations (3.25), and thus are of a nonlocal nature. Thus, in the discrete case we have two different analogs of the rotation coefficients: local plaquette variables !3ij for 1 :S i =1= j :S m, defined on the elementary squares of zm parallel to 'B ij , and nonlocal edge variables Pki for 1 :S i :S m, 1 :S k :S N, k =1= i, defined on the edges of zm parallel to 'B i .

5.6. Convergence of circular nets

Evolution equations for tions (3.25):

Vi

Ti Vj

201

are obtained from (3.13) and the frame equa-

= vJi 1ei(Vj + .Bij Vi)ei.

In the derivation one uses the identity Vi(Vj + .BjiVi)Vi = Vj + .BijVi, which easily follows from (3.18). The resulting evolution equations for the edge variables Pkj read: (5.28)

TiPkj

VJi 1 (Pkj+Pki.Bij),

(5.29)

TiPij

VJi 1(-Pij

Here 1 ::; i # j ::; m, 1 ::; k ::; N, and k (3.18) can be now written as

(5.30)

.Bij

+ .Bji =

+ 2(Ji.Bij). # i, j. The

circularity constraint

1

(JiPij

+ (Jjpji - 2 L

PkiPkj,

k-j.i,j

and gives a relation between local plaquette variables .Bij and nonlocal edge variables Pkj. The system consisting of (5.28), (5.29) and (5.30) can be regarded as the discrete Lame system; d. (1.47), (1.48). Turning to the problem of approximation, we start with approximation of a single orthogonal net f : ]Rm -----t ]RN. For the approximating circular nets, we have M = m and all Ei = E. In all the formulas of Section 3.1 and of the present section, we have to replace the lattice functions hi, .Bij, Pkj by Ehi' E.Bij, EPkj, respectively. Observe that formulas (3.16), (5.27) become Vji

= Vij =

(Ji

(1 - E2.Bij.Bji)1/2

= 1 + (')(E2),

E2 ' " 2) 1/2 = ( 1 - "4 ~ Pki = 1 + (')(E2). k-j.i

Under this rescaling, equations (3.12), (3.13), (3.14) and (3.17) can be put into the standard form (5.1) with functions on the right-hand sides approximating, as E -----t 0, the corresponding functions in equations (1. 7)-(1.10) with order (')(E). Nevertheless, Theorem 5.7 still cannot be applied to orthogonal nets. The reason for this is that the full system of differential equations describing orthogonal nets, consisting of equations (1.7)-(1.10) and constraint (1.54), is nonhyperbolic. Its nonhyperbolicity rests on the fact that the constraint (1.54) is not resolved with respect to the derivatives oi.Bij. Note, however, that constraint (1.54) does not take part in the evolution of solutions starting with the data given in the coordinate planes 'B ij : the constraint is satisfied automatically, provided it is fulfilled for the coordinate surfaces ~ij" Therefore, we will obtain a convergence result for orthogonal nets as soon as it will be established for coordinate surfaces.

n

5. Approximation

202

Discretization of a curvature line parametrized surface. Initial data for a smooth curvature line parametrized surface f : 13 12 ----> JR.N are:

(i) two smooth curves f(O);

f

r~i

(i = 1,2), intersecting orthogonally at

(ii) a smooth function ,12 : 13 12 ,12 = ~(aI/h2 - 2 !hd·

---->

JR., whose designated meaning is

a

Let] r~i be the images of the curves f r~i in the Mobius-geometric model Q~ . Let hi = laJI and Vi = h:;laJ be the metric coefficients and unit tangent vectors of the coordinate curves. Choose an initial frame 1}J(0) E :J-C(X) such that ](0) = 1}J-1(0)eo1}J(0), Vi(O) = 'If,-l(O)ei1}J(O) (i = 1,2). Define the frames 1}J : 'Bi ----> :J-C(X) of the curves ] r~i as the solutions of equations (1.51) for i = 1,2 (considered as ordinary differential equations) with the initial value 1}J(0). Rotation coefficients of the curves Jr~i are the functions (3ki : 'Bi ----> JR. defined by the formula (1.53) for i = 1,2. Define the discrete coordinate curves

]£ r~1

by restricting the functions

Jr ~i to the lattice points.

Let hi = I JR. in the expansions £ £ + E1£lie(X). (Jiei - "2E '\""' L Pkiek k#i Finally, let the plaquette function '12 : 1312 ----> JR. be obtained by restricting ,12 to the lattice points (or to the midpoints of the corresponding plaquettes of 1312).

A£(nl.t)-l Vi£ = nl.£ 'f/ Vi 'f/

=

Thus, we get valid Goursat data for a hyperbolic system of first order difference equations for the variables vi, hi, Phi' consisting of (3.12), (3.13), (3.14), (5.28) and (5.29) with distinct i. j E {1,2} and 1 :::; k :::; N, where the following expressions should be inserted:

if,

t E E E (312 = (J1P12-"2

(1"2 'LPk1Pk2-'12 \""' E

£

£

)

,

k>2 The nets ]E : 1312 ----> Q~ defined as solutions of the Goursat problem just described are circular surfaces, since they fulfill the circularity constraint (5.30). They will be called canonical circular surfaces constructed from the above initial data. Theorem 5.17. (Convergence of circular surfaces) There exists r > 0 such that the canonical circular surfaces ]E : 1312 (r) ----> Q~ converge, with

5.6. Convergence of circular nets

203

order ()(E) in C OO (13 12(r)), to the unique curvature line parametrized surface j : 13 12 (r) -> Qij with the initial data n:Bi (i = 1, 2) and ~ (oIlh2 - 02/321) = ')'12· Edge rotation coefficients pL and plaquette rotation coefficients /312, /3~1 of the circular surfaces jt converge to the corresponding rotation coefficients /3ki of the curvature line parametrized surface j. Proof. We begin with showing the convergence of the frames, 'lj;: -> 'lj;, and of the rotation coefficients, Pki -> /3ki , along the discrete curves r:Bi' This follows from two observations. First, Vf(O) = Vi(O) + ~(OiVi)(O) + ()(E2), so that (Ti - 1)'1//(0) = -~ ei'lj;(O) (OiVi) (0) + ()(E2).

r

Second, combining frame equations on two neighboring edges of finds that

(Ti - Ti-1)'lj;t = -ei'lj;t(l- Ti- 1 )v'f = -Eei'lj;t(OiVi) everywhere on theory.

13i.

13i,

one

+ ()(E2)

The claim follows by standard methods of the ODE

Now an application of Theorem 5.7 shows that the functions jt : 1312 -> Qij converge to the functions j : 13 12 -> Qij which solve the Goursat problem for the hyperbolic system of first order differential equations, consisting of (1.44)-(1.47) with distinct i,j E {I, 2} and 1 ::; k ::; N, and

Solutions /3ki satisfy the relation ~ (01/312 - 02/321) = ')'12 and the orthogonality constraint (1.48). D Discretization of an m-dimensional orthogonal net. Given the initial data (0 1,2) for an m-dimensional orthogonal net (see Section 1.4), we can apply the procedure described in the previous paragraph, with an initial frame 'lj; (0) E :J{oo such that

j(O) = 'lj;-l(O)eo'lj;(O),

Vi(O) = 'lj;-l(O)ei'lj;(O)

(1::; i ::; m),

to produce, in a canonical way, the circular surfaces jt

r:B'. t)

and their pla-

quette rotation coefficients /3ij . Thus, we get the data (Ot2) (see Section 3.1) for an E-dependent family of circular nets jt : (Ez)m -> Qij. These nets will be called the canonical circular nets corresponding to the initial data (0 1,2), Theorem 5.18. (Convergence of circular nets) The canonical circular nets jt : 13€(r) -> ]RN converge, as E -> 0, to the unique orthogonal net

204

5. Approximation

j : 'B(r) ~ jRN with the initial data (0 1,2), Convergence is with order O(E) in COO('B(r)). Proof. The data (0~2) yield a well-posed Goursat problem for the hyperbolic system of first order difference equations for the variables jE, vi, hi, f3ij , consisting of (3.12), (3.13), (3.14), (3.17). The convergence of these Goursat data is assured by Theorem 5.17. Now the claim of the theorem follows directly from Theorem 5.7. 0 Discretization of a Ribaucour transformation. Given the initial data (R1,2) for a Ribaucour transform of an orthogonal net (see Section 1.4), define the plaquette rotation coefficients f3 Mi on the "vertical" plaquettes along the edges of the coordinate axes 'Bi by restricting the corresponding functions d)i to lattice points or, alternatively, to midpoints of the corresponding edges of 'Bi: U

E

'Bi,

1

~

i

~

m.

Thus, we get the data (R~2) (see Section 3.1), which, together with (0~2)' allow us to construct in a ~anonical way circular nets FE : (Ez)m X {O, 1}' ~ jRN. They will be called the canonical circular nets corresponding to the initial data (0 1,2), (R1,2).

Theorem 5.19. (Convergence of discrete Ribaucour transformations) The canonical circular nets (r)+ = P(-, 1) : 'BE(r) ~ jRN converge to the unique Ribaucour transform f+ : 'B(r) ~ jRN of f with the initial data (R1,2). Convergence is with order O(E) in COO('B(r)). Proof. Define vlf(O) as the unit vector parallel to 8f(0) = f+(O) - f(O), and set hM(O) = 18f(0)1. These data along with f3 Mi on the coordinate axes, added to the previously found r(O), vi, hi, f3ij for 1 ~ i,j ~ m, form valid Goursat data for the system (3.12), (3.13), (3.14), (3.17). The circularity constraint (3.18) implies that f3iM = -2(vi, vlf ) - EOi on all edges of 'Bi. Perform the substitution

vM = y+O(E),

hM = £+O(E),

f3Mi = EOi+0(E2), (jiM =

-2(Vi, y)+O(E)

in equations (3.13), (3.14), (3.17) with one of the indices equal to M. Taking into account that in this limit one has

ViA} = vM~ = 1 - E(Vi' y)Oi + 0(E 2 ), one sees that the limiting equations coincide with (1.57), (1.58), (1.59). A 0 reference to Theorem 5.7 finishes the proof.

5.7. Convergence of discrete K-surfaces

205

5.7. Convergence of discrete K-surfaces Discretization of a K-surface. Given the initial data (K) for a K-surface (see Section 1.6), we define the initial data (K.6.) (see Section 4.2) for an Edependent family of discrete K-surfaces with EI = E2 = E by restricting nf 'B; to the lattice points, as for general A-surfaces. Thus, we get two intersecting discrete curves in §2. Define discrete M-nets nf : (EZ)2 -> §2 as solutions of the difference equations (4.31) with the initial data (K.6.). Finally, define the discrete K-surfaces (EZ)2 -> ffi.3 with the help of the discrete Lelievre representation (2.69). These will be called the canonical discrete K-surfaces corresponding to the initial data (K).

r :

Theorem 5.20. (Convergence of discrete K-surfaces) Canonical discrete K-surfaces r: :Bf(r) -> ffi.3 converge, as E -> 0, to the unique K-surface f : :B(r) -> ffi.3 with the initial data (K). Convergence is with order CJ(E) in COO(:B( r)). Proof. We have for n = nf: € (n, Tin + T2n) 2 + E(n, (hn aI2 = 1 + (Tin, T2n) = 2 + E(n, 6In + 62n)

+ 62n) + E2(6In, 62n) .

Since (n,6in) = CJ(E), we find that ab

=

1 - ~E2(6In, 62n)

+ CJ(E4).

Comparing this with (1.68), we see that Theorem 5.7 can be applied and yields convergence of the net n€: (EZ)2 -> §2 to the smooth net n: ffi.2 -> §2. Finally, convergence of to f follows exactly as for general A-surfaces. 0

r

Discretization of a Backlund pair. Let the initial data (B) for a Backlund transformation of a given K-surface f, i.e., the point n+(O), be given. Take it as the initial data (B.6.) for the discrete Backlund transformations (r)+ : (EZ)2 -> ffi.3 of the family of discrete K-surfaces constructed in Theorem 5.20.

r

Theorem 5.21. (Convergence of discrete Backlund transformations) Canonical discrete K-surfaces (r)+ : :B€(r) -> ffi.3 converge to the unique Backlund transform f+ : :B(r) -> ffi.3 of the K-surface f with initial data (B). Convergence is with order CJ(E) in COO(:B(r)). Proof. For the Backlund transformation, equations (4.43), (4.44) hold. In the smooth limit we find: (6In, n+ + n) PI = 1 - (Tin, n+) P2 = -

(62n, n+ - n) 1 + (T2n, n+)

206

5. Approximation

Comparing this with (1.41)-(1.42) and applying Theorem 5.7, we prove convergence of the Gauss maps. 0

5.8. Exercises 5.1. Check that each of the following four difference equations approximates the ordinary differential equation ax = f(x) for x : ~ ~ X (where a stands for the ordinary derivative d/dt):

8x = f(x),

8x = f(TX),

~ TX),

8x = f(X

for x : EZ ~ X, where 8x(t) = (x(t theorem where appropriate.

+ E) -

8x = f(x)

~ f(TX)

X(t))/E. Use the implicit function

5.2. Put the Hirota equation for ¢ : (62:)2 ~ ~, 1 sin 4(TIT2¢ - TI¢ - T2¢

E2

1

+ ¢) = "4 sin 4(TIT2¢ + TI¢ + T2¢ + ¢),

and the sine-Gordon equation for ¢ : ~2 ~ ~,

al a2¢ = sin ¢, into the form of hyperbolic first order systems, and check that the former approximates the latter as E ~ O. 5.3. Put the difference equation for discrete Lorentz-harmonic functions n: (EZ)2 ~ §2, TIT2n+n=

(n, TIn + T2n) ( . ) (TIn+T2 n ), 1 + TIn, T2n

and the differential equation for Lorentz-harmonic functions n : ~2 ~ §2, aIa2n= -(aIn,a2n )n,

into the form of hyperbolic first order systems, and check that the former approximates the latter as E ~ O. Why would the approximation claim fail for similar equations in the case of functions with values in IL,N+I,I? 5.4. Prove the following form of the discrete Gronwall lemma: Let Z+ ~ ~+ be three nonnegative sequences satisfying

~,a,

b:

n-I

~(n) :::; an

+ 2: bkb.(k). k=1

Then

n-I

~(n) :::; an

+ 2: akbk

n-I

II

(1 + bj)' k=O j=k+l An interesting (and important) particular case is that of constant coefficients an = /'l, and bn = K.


207

5.9. Bibliographical notes Geometric convergence theorems are available in the literature for problems described by elliptic partial differential equations, such as the Plateau problem in the theory of minimal surfaces; see, for example, Dziuk-Hutchinson (1999). Convergence of metric and geometric properties of general polyhedral surfaces was shown in Hildebrandt-Polthier-Wardetzky (2006) based on the analysis of the "cot an" Laplace operator. For surfaces described by hyperbolic partial differential equations, first approximation results were obtained in Bobenko-Matthes-Suris (2003, 2005). The presentation of this section follows these papers. The complete proof of the main approximation Theorem 5.7 can be found in Matthes (2004). A related purely geometric construction of circular nets approximating general curvature line parametrized surfaces is given in Bobenko-Tsarev (2007).

Chapter 6

Consistency as Integrability

Up to now we have encountered many instances of multidimensional nets which serve as discretizations of smooth geometries traditionally associated with, and described by integrable systems. The idea of consistency (or compatibility) is in the core of the integrable systems theory. One is faced with it already at the very definition of the complete integrability of a Hamiltonian flow in the Liouville-Arnold sense, which means exactly that the flow may be included into a complete family of commuting (compatible) Hamiltonian flows. It is impossible to list all applications or reincarnations of this idea. We mention only some of them relevant for our present account.

• In the theory of solitons nonlinear integrable equations are represented as a compatibility condition of a linear system called the zero curvature representation (also known as Lax, or ZakharovShabat representations). Various analytic methods of investigation of soliton equations (such as the inverse scattering method, algebrogeometric integration, asymptotic analysis, etc.) are based on this representation. • It is a characteristic feature of soliton (integrable) partial differential equations that they appear not separately but are always organized in hierarchies of commuting (compatible) flows.

• Another indispensable feature of integrable systems is that they possess Biicklund-Darboux transformations. These special transformations are often used to generate new solutions from the known ones.

-

209

6. Consistency as Integrability

210

In fact all these properties are interrelated and it is customary to understand the integrability as the presence of OIl(' (or some combination) of these features.

In this chapter we show how the development of discrete differential geometry leads to a new l.Ulderstanding of the very notion of integrability and its properties.

6.1. Continuous integrable systems Consider one of the most celebrated integrable systems having numerous applications in differential geometry as well as in mathematical physics, the sine-Gordon equation (6.1) for a function ¢ : ]R2 ---t]R. Recall the geometric interpretation of the sineGordon equation. Let f : ]R2 ---t ]R3 be a surface parametrized along its asymptotic lines. Surfaces of constant negative Gaussian curvature K = -1 (K-surfaces, for short) in the asymptotic lines parametrization are characterized by the additional requirement that 10tIi does not depend on U2, and lodl does not depend on lJ1. Reparametrizing the asymptotic lines of a K-surface if necessary, one can assume that 10tIi = 102fl = 1. Then the angle ¢ = ¢( u) between the vectors otI and 02f satisfies the sine-Gordon equation (6.1). Integrability of the sine-Gordon equation has many manifestations, two of which will be of special importance for us: the zero curvature representation and the existence of Backlund transformations. To formulate the zero curvature representation of the sine-Gordon equation, consider the matrices

~ ( -A

(6.2)

U

=

(6.3)

V

= :2i

ad)

(

0

-A -010

A-le-iq,

),

A-lei )

o

.

They depend on 'u E ]R2 through the function ¢ and its partial derivatives, and also depend on a (real) parameter A, known in the theory of integrable systems as the spectral parameter. It is usual to think about U, V as functions of U E ]R2 which take values ill the twisted loop algebra

g[A]

= {~ : ]R*

--t

su(2): ~(-A) = (}3~(A)(}:3}.

6.1. Continuous integrable systems

211

Then it is a matter of a straightforward computation to check that 1; is a solution of equation (6.1) if and only if the zero curvature condition EhU - 8 1 V

(6.4)

+ [U, V] = 0

is satisfied identically in A. The name "zero curvature" comes from the fact that (6.4) expresses the flatness of the connection (or, better, the oneparameter family of connections) on ]R2 given by the differential one-form U dU1 + V dU2. This condition assures the solvability of the following system of linear differential equations: (6.5) for a function '11 : ]R2 ~ G[A] with values in the twisted loop group

G[A]

= {S:]R*

~ SU(2) : S(-A)

= 0"3S(A)0"3}.

The existence of the zero curvature representation is considered as one of the main integrability features of the sine-Gordon equation (and the likes). On a general note, it relates a nonlinear equation (6.1) to the system of linear equations (6.5), which are amenable to analysis. In particular, the spectral theory of the first equation in (6.5) lies in the basis of the inverse spectral transformation approach to the solution of certain boundary value problems for the sine-Gordon equation. Also conserved quantities (integrals) of the sine-Gordon equation can be derived directly from its zero curvature representation. Furthermore, the zero curvature representation allows one to reconstruct a K-surface corresponding to a solution 1; of the sine-Gordon equation. Given a solution 1; : ]R2 ~ JR., introduce the matrices (6.2), (6.3) satisfying (6.4). Define the function '11 : JR.2 ~ G[A] as the solution of equations (6.5) with the initial condition '11(0,0; A) = 1. Then the immersion f : ]R2 ~ ]R3 obtained by the Sym formula, (6.6)

f(u) = 2AW-1(u; A)

8W~~; A) 1,\=1 '

under the canonical identification (4.9) of su(2) with ]R3, is an asymptotic lines parametrized K-surface, with the angle 1; between the asymptotic directions. The function '11 is known as the extended frame of f. Moreover, the right-hand side of (6.6) with various A not necessarily equal to 1 delivers a whole family of immersions fA : ]R2 ~ ]R3, all of which turn out to be asymptotic lines parametrized K-surfaces. These surfaces fA constitute the so-called associated family of f. The classical Backlund transformation is the next common feature of all known integrable systems. In the case of the sine-Gordon equation, it is given by the following construction. For a given solution 1; of (6.1), a

212

new solution equations:


(p+ can be found by solving the following system of differential

(6.7) This system is compatible, Eh(81 ¢+) = 8 1 (82 ¢+), provided ¢ is a solution of the sine-Gordon equation, and then ¢+ is also a solution. It is determined by the parameter n and the value ¢+(O,O) at one point. Geometrically, ¢+ is the angle between asymptotic directions of the Backlund transform f+ : IR2 ----) IR3 of a given K-surface f : IR2 ----) IR3 characterized, according to Definition 1.26, as follows: the straight line segments [J (u), f+ (u) 1 are tangent to both surfaces f and f+, and their length is independent of u. It can be checked by a direct computation that equations (6.7) are equivalent to the following matrix differential equations, which are satisfied identically with respect to the spectral parameter A:

(6.8)

81 W = U+W -

wu,

82 W = V+W -

wv,

where the matrix W is given by the formula

(6.9)

W=

(

ei(¢+-¢)/2

-inA

-inA

e- i (¢+-¢)/2

)

On the other hand, (6.8) constitute a solvability condition for the system consisting of (6.5) and similar equations for the matrix function I}i+ = WI}i.

(6.10) One can show that formed surface f+.

I}i+

serves as the extended frame of the Backlund trans-

A remarkable property of Backlund transformations is given by Bianchi's permutability theorem: if ¢(1) is a Backlund transformation of ¢ with parameter nand ¢(2) is a Backlund transformation of ¢ with parameter (3, then there exists a unique solution ¢(12) of the sine-Gordon equation which is simultaneously a Backlund transformation of ¢(1) with parameter (3 and a Backlund transformation of ¢(2) with parameter n; this solution is given by the formula

(6.11) So, integrable systems, for which the sine-Gordon equation is a prototypical example, are characterized by such features as zero curvature representation and Backlund transformations with permutability properties. The origin and the very existence of these features is considered in the classical theory of integrable systems as something mysterious and transcendental.

6.2. Discrete integrable systems

213

6.2. Discrete integrable systems The theory of discrete integrable systems has been developed for some time as part of the general theory of integrable systems. Its aims at the early stages were not very ambitious: just to find difference analogs of integrable differential systems, enjoying the same integrability features. In this introductory section we give an illustration by the example of the integrable discretization of the sine-Gordon equation, known as the Hirota equation: (6.12)

1 sin 4"(T1 T2¢ - T1¢ - T2¢ + ¢)

E2

1

= "4 sin 4"(T1 T2¢ + T1¢ + T2¢ + ¢).

Here ¢ is a real-valued function on (EZ)2, and the shift symbols stand for

Tk¢(U) = ¢(u + Eek). The Hirota equation (6.12) turns out to describe discrete K-surfaces, i.e., discrete A-surfaces f : (EZ)2 ----* IR3 with all edges of the same length E£, so that 18dl = 182 fl = £. Here, of course, 8k f(u) = (f(u + Eek) - f(u))/E. The discrete zero curvature representation of equation (6.12) is formulated in terms of the matrices U, V: (EZ)2 ----* G[A], defined by the formulas ei(Tl'P-¢)/2

(6.13)

U

iEA)

(

£, ;1.)

i(A

e_ il ".2".;)/2

'

2

1

(6.14)

~ e (T2H¢)/2) 2

(

V =

£2;1.)

"- e- ih q,+1>1/2 21.

1

'

2A where the normalizing factors £1 (A) = (1 + E2 A2 /4)1/2 and £2(A) = (1 + E2A- 2/4)1/2 are introduced in order to assure that U, V E G[A]. The matrix equation (6.15) is satisfied identically in A if and only if the function ¢ solves (6.12). Equation (6.15) is called a discrete zero curvature representation of the Hirota equation (6.12). It expresses the flatness of a discrete G[A]-valued connection, given by the matrices U assigned to the directed edges (u, u + Eel) and the matrices V assigned to the directed edges (u, u + Ee2) of the lattice (EZ)2; see Figure 6.1. In its turn, this condition assures the solvability of the following system of linear difference equations:

(6.16) for a function W : (EZ)2

T1 W = Uw, ----*

G[A].

T2W = VW


214

v u Figure 6.1. Discrete flat connection.

As in the continuous case, the discrete zero curvature representation can be used as a starting point for application of the analytical machinery of the inverse spectral methods. It also yields the conserved quantities (integrals) of the Hirota equation. Moreover, it can be used to reconstruct the underlying discrete K-surface, corresponding to a given solution ¢ : (fZ)2 ----+ lR of the Hirota equation, in literally the same fashion as in the smooth case. Given a solution ¢ of equation (6.12), introduce matrices (6.13), (6.14) satisfying (6.15). Define the function 1]/ : (fZ)2 ----+ G[A] as the solution of (6.16) with the initial condition 1]/(0,0; A) = 1. Then the Sym formula (6.6) determines a net j : (fZ)2 ---) lR3 , which is a discrete K-surface with the characteristic angle function ¢ and with the edge length ff, where £ = (1 +1'2/4)-1. Again, the right-hand side of (6.6) for various A not necessarily equal to 1 delivers an associated family 1>.. of discrete K-surfaces. The Backlund transformation for equation (6.12) is given by the following difference analogs of formulas (6.7): (6.17)

. 1 sm4:(T1¢+-¢++T1¢-¢)

I'

20:

sin 4:1(T1 ¢ + + ¢ + - T1 ¢ - ¢ ) ,

(6.18)

Statements analogous to those for the sine-Gordon equation hold. Difference equations (6.17), (6.18) are compatible; that is, T1(T2¢+) = T2(T1¢+), provided ¢ is a solution of (6.12), and then ¢+ is also a solution (determined by the parameter 0: and the value ¢+(O, 0) at one point). Also the geometric meaning of the Backlund transformation is similar to the continuous case: the straight line segments connecting the corresponding points of a discrete K-surface j and its Backlund transform j+ lie in the tangent planes of both surfaces, and their length is independent of U E (EZ)2. A direct computation shows that equations (6.17), (6.18) are equivalent to the matrix equations (6.19)

6.3. Discrete 2D integrable systems on graphs

215

which are satisfied identically in A, with the same matrix W as in (6.9). These equations assure the solvability of the system consisting of (6.16) and similar equations for the matrix function \[1+ defined by (6.10). This latter matrix \[1+ is nothing but the extended frame of the transformed surface. Bianchi's perIllutability theorem is formulated exactly as in the continuous case, and is expressed by the saIlle formula (6.11).

6.3. Discrete 2D integrable systems on graphs Before we turn to the explanation of the crucial idea that the 3D consistency property of 2D equations should be taken as the definition of their integrability, we provide a bit more details on the notion of integrability, corresponding to the traditional view of integrable systems, which is based on discrete zero curvature representations. This latter notion works in a more general context than systems on a regular square lattice 71}, namely it is naturally formulated for systems on graphs. A gmph 9 will mean for us not just a combinatorial object, but will be provided with an additional structure of a strongly regular polytopal cell decomposition of an oriented surface. The set of its vertices will be denoted by V(9), the set of its directed edges, by £7(9), and the set of its faces, by F(9). To any such 9 there canonically corresponds a dual cell decomposition 9*; it is only defined up to isotopy, but can be fixed uniquely with the help of the Voronoi-Delaunay construction. The vertices of 9* are in a one-to-one correspondence with the faces of 9 (actually, they can be chosen as some points inside the corresponding faces; cf. Figure 6.2).

X2

I I

"-

I

""-

.1'3

"-

I

Xl

/Yf, , /

/ /

X"

Figure 6.2. A facE' of 9 and the corrE'sponding vertE'X of 9*.

The variables of a discrete system (fields in the terminology of mathematical physics) will be understood as elements f of some set X (the phase


216

space of a system), assigned either to the vertices or to the edges of 9. (One can imagine also a mixed situation, where part of fields are assigned to the vertices and the others to the edges.) The system itself will be of the following nature. Consider a closed path of directed edges which constitute the boundary of a face of 9:

... , Then, in the case of fields assigned to the vertices, it is supposed that the fields f(xd, ... , f(xn) satisfy a certain condition, of a geometric or an analytic nature, called the equation associated to the face: Q(J(xd,···, f(xn)) = O.

(6.20)

If the fields are assigned to the edges, f(el), ... , f(e n ), then the equation should read correspondingly: (6.21 )

A discrete system is a collection of such equations associated with all faces of 9. One says that such a system admits a discrete zero curvature representation if there is a collection of matrices L(e;.\) E G[.\] from some loop group G[.\], associated with every directed edge e E £(9) (so called transition matrices), with the following properties. For a system with fields on vertices, L(e;.\) depends on the fields f(XI), f(X2) if e = (Xl, X2); for a system with fields on edges, L(e;.\) depends just on the field f(e). The argument.\ of the loops from G[.\] is known in the theory of integrable systems as the spectral parameter. It is required that: • for any directed edge e = (Xl, X2), if -e

L( -e,.\) = (L(e,.\)

(6.22)

= (X2, xd, then

rl;

• for any closed path of directed edges

... , we have (6.23) In the case when the path bounds a face of 9, the discrete zero curvature condition (6.23) must be equivalent to (or at least a consequence of) the equation for the corresponding face. Under conditions (6.22), (6.23) one can define a wave function W : V(9) - t G[.\] on the vertices of 9, by the following requirement: for any

217

6.4. Discrete Laplace type equations

directed edge e = (Xl, X2) E £(9), the values of the wave functions at its ends must be connected via (6.24)

For an arbitrary graph, the analytical consequences of the zero curvature representation for a given collection of equations are not clear. However, in the case of regular graphs, such as those generated by the square lattice Z + iZ c C, or by the regular triangular lattice Z + e27ri / 3 Z C C, such a representation may be used to determine conserved quantities for suitably defined Cauchy problems, as well as to apply powerful analytical methods for finding concrete solutions.

6.4. Discrete Laplace type equations There exist discrete equations on graphs which are not covered by the constructions of Section 6.3.

Definition 6.1. (Discrete Laplace type equations) Let 9 be a graph, with the set of vertices V(9) and the set of edges E(9). Discrete Laplace type equations on the graph 9 for a function f : V(9) - t C read: (6.25)

L

¢(j(xo), f(x); v(xo, x)) =

o.

xE star(xo)

There is one equation for every vertex Xo E V(9); the summation is extended over star(xo), the set of vertices of 9 connected to Xo by an edge (see Figure 6.3); the function ¢ depends on some parameters v : E(9) - t C, assigned to the edges of 9.

The classical (linear) discrete Laplace equations on 9 are a particular case of this definition: (6.26)

L

v(xo, x) (j(x) - f(xo)) = 0,

xE star(xo)

with some weights v : E(9)

-t

lR+ assigned to the (undirected) edges of 9.

The notion of integrability of discrete Laplace type equations is not well established yet. We discuss here a definition which is based on the notion of the discrete zero curvature representation and works under an additional assumption about the graph 9. Namely, like in the previous section, it has to come from a strongly regular polytopal cell decomposition of an oriented surface. We consider, in somewhat more detail, the dual graph (cell decomposition) 9*. Each e E E(9) separates two faces of 9, which in turn correspond to two vertices of 9*. A path between these two vertices is then declared the edge e* E E(9*) dual to e. If one assigns a direction to an edge


218

Xo

Xa

_--~-4':""

IXO

I I

6--

Y4 X4

Figure 6.3. Star of a vertex Xo in the graph

9·

X4

Figure 6.4. Face of 9* dual to a vertex Xo of 9.

e E E(9), then it will be assumed that the dual edge e* E E(9*) is also directed, in a way consistent with the orientation of the underlying surface, namely so that the pair (e, e*) is positively oriented at its crossing point. This orientation convention implies that e** = -e. Finally, the faces of 9* are in a one-to-one correspondence with the vertices of 9: if Xo E V(9), and Xl, ... , Xn E V(9) are its neighbors connected with Xo by the edges el = (Xo, Xl)"'" en = (Xo, Xn) E E(9), then the face of 9* dual to Xo is bounded by the dual edges ei = (YI, Y2), . .. , e~, = (Yn, yI); see Figure 6.4. We will say that a system of discrete Laplace type equations on 9 possesses a discrete zero curvature representation if there is a collection of matrices L(e*; A) E G[A] from some loop group G[A], associated to directed edges e* E E(9*) of the dual graph 9*, such that: • the matrix L(e*; A) depends on the fields f(xo), f(x) at the vertices of the edge e = (xo, x) E E(9), dual to the edge e* E E(9*), and • the flatness conditions (6.22), (6.23) on the dual graph are satisfied. The matrix L(e*; A) is interpreted as a transition matrix along the edge e* E E(9*), that is, a transition across the edge e E E(9). The wave function 'IJI in this situation is defined on the set V(9*) of vertices of the dual graph.

6.5. Quad-graphs Although one can consider 2D integrable systems on very different kinds of graphs on surfaces, there is one kind - quad-graphs -- supporting the most fundamental integrable systems.

219

6.5. Quad-graphs

Definition 6.2. (Quad-graph) A quad-graph'D is a strongly regular polytopaZ cell decomposition of a surface with all quadrilateral faces. Since we are interested mainly in the local theory of integrable systems of quad-graphs, and in order to avoid the discussion of some subtle boundary effects, we shall always suppose that the surface carrying the quad-graph has no boundary. Quad-graphs are privileged because from an arbitrary strongly regular polytopal cell decomposition 9 one can produce a certain quad-graph 'D, called the double of 9. The double 'D is a quad-graph, constructed from 9 and its dual 9* as follows. The set of vertices of the double 'D is V('D) = V(9) u V(9*). Each pair of dual edges, say e = (xo, Xl) E E(9) and e* = (Yl, Y2) E E(9*), defines a quadrilateral (xo, Yl, Xl, Y2). These quadrilaterals constitute the faces of a cell decomposition (quad-graph) 'D. Thus, a star of a vertex Xo E V(9) generates a flower of adjacent quadrilaterals from F('D) around Xo; see Figure 6.5. Let us stress that edges of 'D belong neither to E(9) nor to E(9*).

v--_XI

Figure 6.5. Faces of

1)

around the vertex Xo.

Quad-graphs 'D coming as doubles are bipartite: the set V('D) may be decomposed into two complementary halves, V(TJ) = V(9) UV(9*) ("black" and "white" vertices), such that the ends of each edge from E('D) are of different colors. Equivalently, any closed loop consisting of edges of 'D has an even length. The construction of the double can be reversed. Start with a bipartite quad-graph 'D. For instance, any quad-graph embedded in a plane or in an open disc is automatically bipartite. Any bipartite quad-graph produces two dual polytopal (in general, no more quadrilateral) cell decompositions 9 and


220

9*, with V (9) containing all the "black" vertices of 'D and V (9 *) containing all the "white" ones, and edges of 9 (resp. of 9*) connecting "black" (resp. "white") vertices along the diagonals of each face of 'D. The decomposition of V('D) into V(9) and V(9*) is unique, up to interchanging the roles of 9 and 9*. Notice that if a quad-graph 'D is not bipartite (i.e., if it admits loops consisting of an odd number of edges), then one can easily produce from 'D a new even quad-graph 'D', simply by refining each of the quadrilaterals from F('D) into four smaller ones. Since we are interested mainly in the local theory, we always assume (without mentioning it explicitly) that our quad-graphs are cellular decompositions of an open topological disc. In particular, our quad-graphs 'D are always bipartite, so that 9 and 9* are well defined.

6.6. Three-dimensional consistency An attentive examination of examples in Sections 6.1, 6.2 leads to remarkable observations which relate to the main philosophy of this book. For the continuous sine-Gordon equation the theory seems to consist of several components of a rather different nature: the main object is a partial differential equation, its Backlund transformations are described by a compatible system of two ordinary differential equations, while the superposition formula of Backlund transformations is expressed in purely algebraic terms. In the discrete context situation changes dramatically. All components of the discrete theory have essentially one and the same structure: equation (6.12) which describes discrete K-surfaces, equations (6.17), (6.18) for Backlund transformations of discrete K-surfaces, and equation (6.11) for the superposition principle ofthe latter. Their common structure is captured in the following formula for a function rP : zm ----) lR on an m-dimensionallattice:

(6.27)

. -1 ( rPjk sm

4

+ rPk - rPj - rP ) = -Ok.sm -1 (rPjk - rPk + rPj - rP ) . OJ

4

Here the subscript j stands for the shift in the j-th lattice direction, and parameters OJ are assigned to all edges parallel to the j-th lattice direction. Actually, in the geometric context, we are dealing with the case m = 4. The subscripts 1,2 label the coordinate directions of the discrete surfaces, while the subscripts 3,4 are used as replacements of the Backlund superscripts (1), (2). The relevant values of the parameters are: 01 = f/2, 02 = 2/f, 03 = 0, and 04 = /3. Equations (6.17), (6.11) are exactly of the form (6.27), and equations (6.12), (6.18) are brought into this form upon changing

6.6. Three-dimensional consistency

221

the sign of ¢ on every second hyperplane complementary to the second coordinate direction, i.e., upon the change of variables ¢( u) ----t (-1 )U 2¢( u). This reflects the fact that the underlying geometric properties of discrete K-surfaces and their Backlund transformations are identical and are captured in the definition of multidimensional K-nets, i.e., A-nets (nets in IR3 with planar vertex stars) satisfying the additional requirement that in every elementary quadrilateral the opposite sides have equal length. A discrete K-surface is a K-net with m = 2, iterated Backlund transformations of a discrete K-surface form a K-net with m = 3, while Bianchi's permutability theorem for two Backlund transformations of a discrete K-surface deals with K-nets with m = 4. The variable transformation Ijk

(6.28)

I

I = exp(i¢/2)

puts (6.27) into the form

CYjIj - CYkik CYjik - CYkIj ,

which is also known as the Hirota equation.

Ijk

Iijk

CYi

CYj Ijk

ik

CYi

ik

CYj

CYj Iik

CYk

I

CYk CYj I

CYk

CYkl

CYk

CYj

Ij

/'

~a

CYk Iij 2

CYj

/' I

Figure 6.6. 2D equation.

CYi

Ii

Figure 6.7. 3D consistency.

The Hirota equation (6.28) is a two-dimensional discrete equation, since it relates the variables I at the vertices of any elementary two-dimensional cell (square) of the m-dimensional lattice in such a way that any three variables determine the fourth one uniquely. The possibility to impose this equation everywhere on the m-dimensionallattice hinges on the case m = 3. The corresponding property of three-dimensional consistency should be understood as follows: suppose that four values I, Ii, Ij, ik are given (consult Figure 6.7 for notation). Then equation (6.28) defines Iij, Ijk and Iik' and a further application of this equation gives three a priori different values of Iijk. These three values turn out to automatically coincide for arbitrary


222

initial data. Indeed, a direct computation shows:

(6.29)

r.

k tJ -

+ ai(a~ - aJ)fj/k + aj(ar - a~)fkfi aT)fk + ai(a~ - aJ)fi + aj(ar - a~)fj

ak(o:; - ar)fdj

--=---;c:------;:-:----...."----=--,,.----,----~____,,_,--

ak(aJ -

This coincidence is the meaning of the 3D consistency of the Hirota equation. As a consequence, the Hirota equation can be consistently imposed on all elementary squares of a multidimensional lattice.

6.7. From 3D consistency to zero curvature representations and Backlund transformations Now we are in a position to expose the main idea concerning the understanding of discrete integrable systems, namely that the property of 3D consistency observed in Section 6.6 for the Hirota equation is actually of a fundamental importance and leads directly to the core of the whole theory. We show that other features of integrable systems, such as zero curvature representations and Backlund transformations, are consequences of 3D consistency. The present section is devoted to a realization of this idea for systems on quad-graphs with fields on vertices and with labelled edges. A typical representative of this class of equations is the Hirota system, which we write here once more in the form (6.30)

h2 f

a2!2 ad2 - a2h'

adl -

In the geometric context of K-surfaces we had f = exp(i¢/2) E §1. In the present analytic study we will assume the fields f to be any complex numbers assigned to the vertices of 'I}, while ai are (complex) parameters naturally assigned to the edges of Z2 parallel to 'Bi and constant along the strips in the complementary direction. In a different fashion, one can view ai as fields satisfying the labelling property (6.31)

(6.32) One more example of such a system with vertex variables and edge parameters having the labelling property is given by the cross-ratio equation: (6.33) We already studied this system in Section 4.3 in the context of discrete isothermic surfaces, where the fields f are points of]RN and parameters ai

6.7. From 3D consistency to zero curvature representations

223

are real numbers, the cross-ratio being defined according to the Clifford multiplication in e£(Il~N). Here we will consider a simpler version with fields and parameters being complex numbers. The commutativity of complex multiplication makes the check of the 3D consistency of the cross-ratio equation the matter of a straightforward computation, leading to (6.34)

h23 = (0:1 - 0:2)hh + (0:2 - 0:3)1213 + (0:3 - O:l)hh . (0:2 - 0:1)13 + (0:3 - 0:2)h + (0:1 - 0:3)12

A general system of this class consists of equations (6.35) Here j : Z2 ----) C are complex fields, and O:i are complex parameters on the edges of Z2 parallel to 'B i , satisfying the labelling condition (6.31); see Figure 6.6. Actually, just from the outset we would like to generalize this setup by considering systems on arbitrary quad-graphs instead of Z2. In this case (6.35) should be read as a relation for fields j : V('D) ----) C, with 0: : E('D) ----) C being a labelling of edges of 'D, i.e., a function taking equal values on any pair of opposite edges of any quadrilateral from F('D). In the context of equations on general quad-graphs, there are no distinguished coordinate directions; nevertheless it will be convenient to continue to use notation of (6.35), with the understanding that indices are used locally (within one quadrilateral) and do not stand for shifts into globally defined coordinate directions. So, j, h, h2, 12 can be any cyclic enumeration of the vertices of an elementary quadrilateral. Sometimes we will stress the absence of global coordinate directions by writing (6.35) in a different system of notation, using just a cyclic enumeration of vertices: (6.36) see Figure 6.8.

(3

Figure 6.8. A face of a labelled quad-graph; fields on vertices.


224

For the very possibility to pose equation (6.36) on general quad-graphs, this equation should be uniquely solvable for anyone of its arguments fi E C; therefore the following assumption is natural by considerations in that generality: Linearity. The function Q is a polynomial of degree 1 in each argument fi (multiaffine), with coefficients depending on the parameters a, (3:

(6.37)

Q(h, 12, 13, f4; a, (3) =

al(a, (3)h1213f4

+ ... + a16(a, (3).

For the Hirota equation (6.30) one can take

while for the cross-ratio equation (6.33) with complex-valued arguments one can take

Assume now that equation (6.35) possesses the property of 3D consistency; cf. Figure 6.7. We will demonstrate that this remarkable property automatically leads to two basic structures associated with integrability in the soliton theory: Backlund transformations and zero curvature representation.

Theorem 6.3. (3D consistency yields Backlund transformations) Let equation (6.35) be 3D consistent. Then for any solution f : V('D) --t C of the corresponding system (6.36) on a quad-graph 'D, there is a two-parameter family of solutions f+ : V('D) ----; C of the same system, satisfying (6.38) for all edges (J,Ji) E E('D). Such a solution f+ is called a Backlund transform of f, and is determined by 'its value at one vertex of 'D and by the parameter .A.

Proof. We formally extend the planar quad-graph 'D into the third dimension. For this aim, consider the second copy 'D+ of 'D and add edges connecting each vertex f E V('D) with its copy f+ E V('D+). (We slightly abuse the notation here, by using the same letter f for vertices of the quadgraph and for the fields assigned to these vertices.) On this way we obtain a "3D quad-graph" D, with the set of vertices V(D) = V('D)

u V('D+),

with the set of edges E(D) = E('D) U E('D+) U {(J, f+) : f E V('D)},

6.7. From 3D consistency to zero curvature representations

225

and with the set of faces

F(D) = F(1J) UF(1J+) U {(J,fr,ft,f+): f,fr E V(1J)}. Elementary building blocks of D are combinatorial cubes (j, fro fr2' 12, f+, ft, f0., fi)' as shown in Figure 6.9. The labelling on E(D) is defined in the natural way: each edge (J+, f i+) E E(1J+) carries the same label O'i as its counterpart (J, Ii) E E(1J), while all "vertical" edges (J, f+) carry one and the same label A. Clearly, the content of Figure 6.9 is the same as of Figure 6.7, up to notation. Now, a solution f+ : V(1J+) -----t C on the first floor of D is well defined due to the 3D consistency, and is determined by its value at one vertex of 1J+ and by A. We can assume that f+ is defined 011 V(1J) rather than 011 V(1)+), since these two sets are in a one-to-one 0 correspondence.

fi

0'1

0'2

f0. 0'2

f+

ft

0'1

A

A: I I I

A

A

12 __ --- ~1_

fr2

/

/ / 0'2

f

0'2

0'1

fr

Figure 6.9. Elementary cube of D.

Theorem 6.4. (3D consistency yields zero curvature representation) Let equation (6.35) be 3D consistent. Then the corresponding system (6.36) on a quad-graph 1) admits a zero curvature representation with spectral parameter dependent 2 x 2 matrices: there exist matrices associated to directed edges of 1),

L(e, O'(e); A) : £(1))

(6.39)

-----t

GL(2, A-I, so that equations on the vertical faces of Figure 6.9 read:

This gives the Mobius transformation (6.41) with (6.46) The determinant of this matrix is constant (equal to 1 - AOOi); therefore no further normalization is required. A more usual form of the transition matrices of the zero curvature representation for the complex cross-ratio equation is obtained by the gauge transformation

AU) =

G{),

which leads to the matrices (6.47) These matrices (6.47) are interpreted as matrices of the Mobius transformations acting on the shifted quantities:

To summarize: 3D consistency of 2D quad-equations with complex fields on vertices and with labelled edges implies the existence of Backlund transformations and of the zero curvature representation. This is not a pure existence statement but rather a construction: both attributes can be derived in a systematic way starting with no more information than the equation itself, they are in a sense encoded in the equation provided it is 3D consistent.

6.8. Geometry of boundary value problems for integrable 2D equations There are several important aspects of the problem of embedding of a quadgraph into a regular multidimensional square lattice, related to integrable equations.


228

6.8.1. Initial value problem. We start with the question of correct initial value (Cauchy) problems for discrete 2D equations on quad-graphs. Let P be a path in the quad-graph 'D, i.e., a sequence of edges Cj = (Xj, xj+d E E('D). We denote by E( P) = {Cj} and V (P) = {x j} the set of edges and the set of vertices of the path P, respectively. One says that the Cauchy problem for a path P is well posed if for any set of data f p : V (P) ----) C there exists a unique solution f : V('D) ----) C such that V(P)= fp. It is not difficult to find examples of paths on the square lattice for which the Cauchy problem is well posed. The task we are interested in is to characterize, for a given quad-graph 'D, all paths with this property.

n

-

-

f--- -

-~

-

f-- - --

-

- - f--~

c_ _ _

Figure 6.10. One-corner initial path.

~~

__

u

Figure 6.11. Staircase initial path.

A solution of this problem can be given with the help of the notion of a strip in 'D. Definition 6.5. (Strip) A strip in 'D is a sequence of quadrilateral faces qj E F('D) such that any pair qj-I, qj is adjacent along the edge Cj = qj-I n qj, and Cj, CHl are opposite edges of qj. In other words, a strip is a path in 'D* consisting of edges cj = (qj-l, qj) E E('D*) such that any consecutive pair cj, Cj+l enter and leave the quadrilateral qj along a pair of opposite edges Cj, CHI. The edges Cj are called traverse edges of the strip. So, in a labelled quad-graph 'D any strip may be associated to a label a sitting on all its traverse edges Cj. The strips come to replace coordinate directions in a regular square lattice, and can be considered as a discrete analog of characteristics for hyperbolic systems of partial differential equations with two independent variables. Theorem 6.6. (Well posed Cauchy problems on quad-graphs) Let'D be a finite simply connected quad-graph without self-crossing strips, and let P be a path without self-crossings in 'D. Consider a 3D consistent equation of the type (6.35) on the quad-graph 'D. Then:

6.S. Boundary value problems for integrable 2D equations

229

i) If each strip in 'D intersects P exactly once, then the Cauchy problem for P is well posed. ii) If some strip in 'D intersects P more than once, then the Cauchy problem for P is overdetermined (has in general no solutions).

iii) If some strip in 'D does not intersect P, then the Cauchy problem for P is underdetermined (has in general more than one solution). Proof. We shall only sketch the proof of the claim i). It is based on an embedding T of V ('D) into the unit cube of '!In, where n is the number of edges in P (the number of distinct strips in 'D). Choose any vertex :£0 E V('D), and set T(xo) = E '!In. The image of any other vertex x E V('D) is defined recurrently along a path connecting Xo to x with the help of the following rule:

°

For any two neighbors x, y E V('D), if the edge (x, y) E E('D) belongs to the strip number i E {1, 2, ... , n}, then T(y) = T(x) +ei (mod 2), where ei is the i-th coordinate vector of '!In. The result does not depend on the path connecting x to xo, since any closed path has an even intersection index with any strip; therefore contribution of any strip to T along a closed path vanishes. Edges and faces of 'D correspond to edges and two-faces of the unit cube in '!In. The T-image of the path P is the path (0,0,0, ... ,0), (1,0,0, ... ,0), (1,1,0, ... ,0), ... , (1,1,1, ... ,1). It is clear that for a 3D consistent equation the data along this path define a well-posed Cauchy problem for the unit cube in '!In. In particular, these data uniquely determine the values of the solution on T(V('D)). 0 It should be mentioned that this theorem is not valid for equations without the 3D consistency property. The next theorem is based on the zero curvature representation with a spectral parameter; therefore it is also specific for 3D consistent equations on labelled quad-graphs. We will formulate this theorem for a concrete equation (cross-ratio equation), but actually it applies under much more general circumstances. See, however, Exercise 6.5, illustrating an instance where this theorem is not valid. Theorem 6.7. (Relating data on two Cauchy paths) Consider a generic solution of the cross-ratio equation (6.33) on a simply connected quad-graph 'D. Let each of the two paths P = (xo, Xl, ... , x n ) and P = (xo, Xl, ... ,xn ) in 'D with a common starting point Xo = Xo and a common end point Xn = xn intersect each strip in 'D exactly once. Then the the fields (fo, iI,···, fn) along P determine the fields (fo, 11, ... , 1n) along P uniquely, as soon as the sequences of labels ai = a(Xi-l, Xi) along P and ai = a(Xi-l, Xi) along P are known, that is, without knowing any additional information on the combinatorics of 'D.


230

Proof. The proof is based on the zero curvature representation of the crossratio equation with the transition matrices L given in (6.47). By the hypothesis of the theorem, the sequence (al, ... , an) is a permutation of the sequence (al, ... , an). From the zero curvature condition (6.23) it follows that (6.48) Generally speaking, such an equality does not hold automatically for nonnormalized transition matrices, but in our case det L(fi, fi-1, ai; A) = 1 Aai, which yields the equality of determinants of the both sides of (6.48). Denote the left-hand side of (6.48) by

IIL(fi, ii-I, ai; A).

T(A) =

All entries of this matrix are polynomials in A. We want to show that this matrix can be uniquely refactorized as

T(A) =

II L(fi, h-1' ai; A),

with io = fo, in fn, and with a prescribed permutation (ai) of the parameters (ai) along the path P. We show that there is a unique matrix of the form

L(f1 ,fo, a1; A)

= (

Aa 1(fo

~ id- 1

fo

~ fr)

such that all entries of the matrix T(A)L- l (f1, io, a1; A) are polynomials in A. Since det T(A) = I1(1 - Aad, the points A = ail are exactly those where T(A) is degenerate. For a generic solution, rank T(ail) = 1, so that dim ker T (ai 1) = 1. Define by

11

kerT(all) (recall that io

= fo).

= lR (i1

~ io)

Then the elements of the vector

T(A) (i1

~ io)

are polynomials divisible by 1 - Aal. Now observe that -

L -l(fl ,fo, al; A) = 1 -\a1 (fl

-

~ fO)

-

((fl

-

T

-/0)-1) -

((fl _ °io)-1

which immediately implies that T(A)L -1 (f1,fo, al; A) is a polynomial in A. An inductive application of this procedure yields the desired refactorization of the matrix T(A). It remains to show that for the so found sequence

6.8. Boundary value problems for integrable 2D equations

231

eli), we have ln = fn. But this follows immediately from the fact that the free term of the (12) entry of T().) is equal to fo - fn = In. This finishes the proof. D

10 -

This theorem has rather surprising consequences. Consider a quad-graph obtained from the regular square lattice by replacing some Tn x n rectangle by a finite simply connected quad-graph with the same boundary vertices. The resulting quad-graph is called a regular "quare lattice with a localized defect. We say that a defect is weak if all strips entering the defect leave it in the same direction, possibly in a different order. Figure 6.12 illustrates a weak 3 x 2 defect.

~

Figure 6.12. A weak localized defect in the regular square lattice.

Consider a Cauchy problem for the cross-ratio equation on a regular lattice with a weak defect, with the initial data outside of the defect. Suppose that all horizontal edges outside of the defect carry the same label 0' and all vertical edges outside of the defect carry the same label {3 (so that in Figure 6.12 there should be 0'1 = 0'2 = 0'3). Compare the solution of this problem with the solution of the same Cauchy problem but on the regular square lattice without defects. Surprisingly, as a consequence of Theorem 6.7, the solutions will coincide outside of the defect. One can say that for the cross-ratio equation (and the likes) with a homogeneous labelling the weak defects are transparent. 6.8.2. Extension to a multidimensional lattice. The problem of embedding of a quad-graph 'D into a regular multidimensional cubic lattice has also aspects of a different flavor.


232

Theorem 6.8. (Rhombic embedding) A quad-graph 'D admits an embedding in C with all rhombic faces if and only if the following two conditions are satisfied:

i) No strip crosses itself or is periodic. ii) Two distinct strips cross each other at most once. For a proof of this theorem we refer the reader to Kenyon-Schlenker (2004). We will show that rhombic embeddings are closely related to 3D consistency of equations on 'D. Given a rhombic embedding p : V('D) ---4 C with edges of unit length (which can always be achieved by scaling and will be assumed from now on), one defines the following function on the directed edges of 'D with values in §l = {O E C : 101 = I}: (6.49)

O(x, y) = p(y) - p(x),

\7'(x, y)

E

E('D).

This function can be called a labelling of directed edges, since it satisfies O(-e) = -O(e) for any e E E('D), and the values of 0 on two opposite and equally directed edges of any quadrilateral from F('D) are equal. See Figure 6.13. For any labelling 0 : E('D) ---4 §l of directed edges, the function 0: = 0 2 : E('D) ---4 §l is a labelling of (undirected) edges of'D in our usual sense.

f Figure 6.13. Labelling of directed edges.

Definition 6.9. (Quasicrystallic rhombic embedding) A rhombic embedding p : V('D) ---4 C of a quad-graph 'D is called quasicrystallic if the set of values of the function 0 : E('D) ---4 §l defined by (6.49) is finite, say

e=

{±Ol, ... , ±Od}.

An example of a quasicrystallic (actually periodic) rhombic quad-graph with d = 3 is the so-called dual kagome lattice shown in Figure 6.14. A prototypic example of a nonperiodic quasicrystallic rhombic quadgraph with d = 5 is the famous Penrose tiling shown in Figure 6.15.

6.8. Boundary value problems for integrable 2D equations

233

Figure 6.14. Dual kagome lattice.

Figure 6.15. Penrose rhombic tiling.

It is of a central importance that any quasicrystallic rhombic embedding p can be seen as a sort of projection of a certain two-dimensional sub complex

(combinatorial surface) 01) of a multidimensional regular square lattice 7l,d. The vertices of 01) are given by a map P : V(1)) --t 7l,d constructed as follows. Fix some Xo E V(1)), and set P(xo) = o. The images in 7l,d of all other vertices of 1) are defined recurrently by the property: For any two neighbors x, y E V(1)), if p(y) - p(x) = ±Oi E e, then P(y) - P(x) = ±ei, where ei is the i-th coordinate vector of 7l,d. The edges and faces of 01) correspond to the edges and faces of 1), so that the combinatorics of 01) is that of 1).


234

To exploit possibilities provided by the 3D consistency, we extend the labelling () : £('D) -+ §1 to all edges of'lL,d, assuming that all the edges parallel to (and directed as) ek carry the label ()k. This gives, of course, also the labelling a = ()2 of undirected edges of 'lL,d. Now, any 3D consistent equation can be imposed not only on 01), but on the whole of'lL,d: (6.50)

1

:s; j

-1= k

:s; d.

Here indices stand for the shifts into the coordinate directions. Obviously, for any solution f : 'lL,d -+ C of (6.50), its restriction to V(01)) rv V('D) gives a solution of the corresponding equation on the quad-graph 'D. As for the reverse procedure, i.e., for the extension of an arbitrary solution of (6.36) from 'D to 'lL,d, more thorough considerations are necessary. An elementary step of such an extension consists in finding f at the fourth vertex of an elementary square from the known values at three vertices according to (6.50). Due to 3D consistency this extension is well defined. In particular, one can find f at the eighth vertex of an elementary 3D cube from the known values at seven vertices; see Figure 6.16. This can be alternatively viewed as a flip (elementary transformation) on the set ofrhombically embedded quadgraphs 'D, or on the set of the corresponding surfaces 01) in 'lL,d. Any quadgraph'D (or any corresponding surface 01)) obtainable from the original one by such flips, carries a unique solution of (6.50) which is an extension of the original one.

Figure 6.16. Elementary flip.

Definition 6.10. (Hull) For a given set V C 'lL,d, its hull Je(V) is the minimal set Je C 'lL,d containing V and satisfying the condition: if three vertices of an elementary square belong to Je, then so does the fourth vertex.

'lL,d

One shows by induction that for an arbitrary connected sub complex of with the set of vertices V, its hull is a brick

(6.51)

6.9. 3D consistent equations with noncommutative fields

235

where (6.52) and in the case that nk are unbounded from below or from above on V, we set ak(V) = -00, resp. bk(V) = 00. Combinatorially, all points of the hull JC(V(n1»)) can be reached from by the extension procedure described above. However, there might be obstructions for extending solutions of (6.36) from a combinatorial surface (two-dimensional subcomplex of Zd) to its hull, having nothing to do with 3D consistency. For instance, the surface n shown in Figure 6.17 supports the solutions of (6.36) which cannot be extended to the solutions of (6.50) on the whole of JC(V(n)): the recursive extension will lead to contradictions. The reason for this is nonmonotonicity of n: it contains pairs of points which cannot be connected by a monotone path in n, i.e., by a path in n with all directed edges lying in one octant of Zd. However, such surfaces n do not come from rhombic embeddings, and in the case of n1) there will be no contradictions.

n1)

Figure 6.11. A nonmonotone surface in Z3.

Theorem 6.11. (Extension of solutions from quad-surfaces to Zd) Let the combinatorial surface n1) in Zd come from a quasicrystallic rhombic embedding of a quad-graph 'D, and let its hull be JC(V(n1»)) = IIa,b. An arbitrary solution of a 3D consistent equation (6.36) on n1) can be uniquely extended to a solution of equation (6.50) on IIa,b' The proof of this theorem can be found in Bobenko-Mercat-Suris (2005).

6.9. 3D consistent equations with noncommutative fields The validity of the message formulated in the last paragraph of Section 6.7, saying that 3D consistency of a quad-equation yields a construction of Backlund transformations and of the zero curvature representation, is by no


236

means restricted to the situation for which it was demonstrated (complex fields on vertices). In the present section, we show that it can be extended to equations with fields on vertices taking values in some associative but noncommutative algebra A with unit over a field X, and with edge labels with values in X. The transition matrices of the zero curvature representation are in this case 2 x 2 matrices with entries from A. They act on A according to (6.42), where now the order of the factors is essential. Actually, the proof of Theorem 4.26 given in Section 4.3.7 is an example of a derivation of a zero curvature representation for the cross-ratio equation (6.33) with fields in A = e£(JR N ) and with parameters ai from X = JR, which governs discrete isothermic surfaces in JRN (one has to interpret the arbitrary parameter a3 in equation (4.96) as the spectral parameter >.). The literal generalization of this proof for an arbitrary associative algebra A leads to the following statement.

Theorem 6.12. (Cross-ratio equation in an associative algebra) The cross-ratio equation in an associative algebra A is 3D consistent. It possesses a zero curvature representation with transition matrices (6.47), where the inversion is treated in A. We provide here two more examples of similar results for equations with noncommutative fields. The first will be dealing with the noncommutative Hirota equation. It turns out that the correct way to write such a noncommutative generalization is the following: (6.53)

Theorem 6.13. (Hirota equation in an associative algebra) The noncommutative Hirota equation (6.53) is 3D consistent. It admits a zero curvature representation with the transition matrices (6.54)

Proof. The noncommutative Hirota equation on the face (f, fi, lij, fj) can be written as a formula which gives fij as a linear-fractional transformation of fj: (6.55)

where (6.56)

L(fi,f,ai,aj) =

(

aj I -ad-

-adi ajf-Ifi

).

Here we use the same notation as in the proof of Theorem 4.26 given in Section 4.3.7 for the action of of the group GL(2,A) on A. Thus, equation

6.9. 3D consistent equations with noncommutative fields

(6.53) on the faces as

e 13 , e 23

of the elementary 3D cube

e l23

237

can be written

L(h, j, aI, (3)[h] , L(12, j, a2, (3)[h]·

(6.57) (6.58)

By the shift in the second, resp. the first, coordinate direction we derive the expressions for h23 obtained from equation (6.53) on the faces 72e13, 71 e 23, respectively:

(6.59) (6.60)

h23 h23

L(J12, 12, aI, (3)[123], L(J12, h, a2, (3)[jd·

Substituting (6.57), (6.58) on the right-hand sides of equations (6.60), (6.59), respectively, we represent the equality between the two values of h23 (which we want to demonstrate) in the following matrix form:

In fact, a stronger claim holds, namely (6.61) L(J12, h, a2, (3)L(h, j, aI, (3) = L(J12, 12, aI, (3)L(12, j, a2, (3)' Indeed, the (11) entries on both sides of this matrix identity are equal to a~ + ala2!12j-l. Equating (12) entries on both sides is equivalent to the Hirota equation of the face (J, h,J12, h), and the same holds for the (21) entries. Finally, equating the (22) entries is equivalent to the condition that h2!-1 commutes with 12[1 1 , and this is, of course, true by virtue of (6.53). This proves the 3D consistency of the noncommutative Hirota equation. The claim about the zero curvature representation is nothing but relation (6.61) just proven with a3 replaced by A. 0 We consider here one more equation of this kind: (6.62) with the vertex variables j taking values in A and with the edge labels a from X. In the case of real-valued fields f, this equation is known under the name of the discrete KdV equation; among other things, it expresses the Bianchi-type superposition formula for the Backlund transformations of the Korteweg-de Vries equation. In the case when the fields j are considered to belong to ]RN C A = e£(]RN), the solutions of this equation are special T-nets in ]RN: (6.63)

In the vector form (6.63) this equation is known as the discrete Calapso equation.


238

Theorem 6.14. (Discrete KdV equation in an associative algebra) Equation (6.62) in an associative algebra A with unit is 3D consistent. It possesses a zero curvature representation with the transition matrices (6.64)

Proof. Equations (6.62) on the vertical faces of Figure 6.9 read:

This gives the transition matrices, which can then be used to prove the 3D consistency, in the same manner as in the proof of Theorem 4.26 given in Section 4.3.7 and in the proof of Theorem 6.13. 0 Our last example in this section is of a geometric origin and of a slightly different nature than the previous ones. In Section 4.1, in our study of T-nets in quadrics, we encountered the equation with vertex variables f : 71} ---7 Q = {f E jRN : (j, f) = ",2}: (6.65)

h2 - f

=

a(12 - h),

a

(j,h - h) = ---'-:-----'---''",2 - (jl, h)

2(j,h - h)

Ih-121 2

.

A priori it does not contain any parameters. However, the quantities

(Xi

=

2(j, fi), being functions of the vertex variables f rather than parameters of the equation, possess the labelling property (6.31). Comparing (6.65) with (6.63), we see that the former can be regarded as a particular instance of the latter.

Theorem 6.15. (T-nets in quadrics) Equation (6.65), describing T-nets in quadrics, is 3D consistent. It possesses a zero curvature representation with transition matrices with entries from eC(jRN): (6.66)

Proof. 3D consistency has been proven geometrically in Theorem 4.3. As for the transition matrices, we can take those from (6.64) with (Xi

= 2(j, fi) = - f fi - fif·

Note the geometrical meaning of the spectral parameter: A = 2(j, f+) for the Backlund transformation f+ from which the transition matrices are constructed. 0

239

6.10. Classification of discrete integrable 2D systems. I

6.10. Classification of discrete integrable 2D systems with fields on vertices. I The notion of 3D consistency, being fundamental for definition and study of 2D integrability, proves extremely useful also in various classification problems of the integrable systems theory. Because of its constructive nature, it can be put into the basis of classification within certain Ansiitze. We will present here the solution of a very general problem concerning the 3D consistent systems of (possibly different) quad-equations with complex fields on vertices. Quad-equations will be of the form (6.67) where the field variables Xi E ClP'l are assigned to the vertices of a quadrilateral (ordered cyclically), and Q satisfies only one assumption, namely that of linearity, formulated already in Section 6.7: the function Q is supposed to be an irreducible polynomial of degree 1 in each variable. This implies that equation (6.67) can be solved for any variable, and the solution is a rational function of the other three variables. The problem we would like to solve is that of the 3D consistency of six a priori different quad-equations put on the faces of a coordinate cube: the system of six quad-equations, (6.68)

A(x, Xl, X12, X2) = 0, B(x, X2, X23, X3) = 0, C(x, X3, X23, Xl) = 0,

0, 0, = 0,

A(X3, X13, X123, X23) =

B(Xl' X12, X123, X13) = C(X2' X23, X123, X12)

should admit a unique solution X123 for arbitrary initial data X, Xl, X2, X3; see Figure 6.18. The functions A, . .. ,C are affine linear and a priori are not supposed to be related to each other in any way. In solving a classification problem, one should factor out a possibly large group of transformations that leave the class of objects being classified invariant. In our problem each quad-equation preserves its form under the group (Mob)4 which acts by Mobius transformations on all the vertex fields independently. It will be convenient to denote by p~ the set of polynomials in n variables which are of degree m in each variable, with the following action of Mobius transformations on a polynomial P E p~:

where (6.69)


240

X123

A X3 e:_-------'------'-----. X13

C B - -- ~ - -----,;/ '- - -- -- - f3 , ,

:c .---~---------

/

X2

A

x Figure 6.18. A 3D consistent system of six different quad-equations: A and A are associated to the bottom and to the top faces of the cube, Band E, to the left and to the right faces, and C and C, to the front and to the back ones.

Pi.

Thus, quad-equations (6.67) are characterized by polynomials Q E An important step in the solution of our problem will be classifying such polynomials and finding their normal form modulo the action of (Mob)4. The full problem we are aiming at is classifying and finding normal forms for 3D consistent systems (6.68) modulo the action of (Mob)8 (independent Mobius transformations of all eight vertices of the cube). We will solve these problems under certain nondegeneracy conditions. In formulation of these conditions, as well as in the whole theory, the following operations play an important role: (6.70) (Variables placed as subscripts stand for partial differentiations.) The operation bXi,xj applied to an affine linear polynomial Q(Xl, X2, X3, X4) eliminates the variables Xi, X j, the result being a biquadratic polynomial of the remaining variables Xk,Xl (so that {i,j,k,l} = {1,2,3,4}), which we will denote by hkl(Xk, Xl) = hlk(Xk, xt). Thus, from any Q E the operations bXi,xj produce six biquadratic polynomials hkl E P~, four of them corresponding to the edges of the underlying quadrilateral, and the remaining two corresponding to the diagonals. Note that the operations bXi,xj are covariant with respect to Mobius transformations:

Pi

(6.71) with

~i

given in (6.69).

6.10. Classification of discrete integrable 2D systems. I

241

Definition 6.16. (Nondegenerate biquadratic) A biquadratic polynomial h(x, y) E P~ is called nondegenerate if no polynomial in its equivalence class with respect to Mobius transformations is divisible by a factor x - c or y - c (with c = canst). Thus, a polynomial h(x, y) E P~ is nondegenerate if it is either irreducible or of the form (alxy + f31x + '"'fly + 6d(a2xy + f32x + "(2Y + 62) with ai6i - f3i'Yi i= O. In both cases the equation h = 0 defines Y as a two-valued function on x and vice versa. An example of a degenerate biquadratic is given by h(x, y) = x - y2 (considered as an element of P~), since under the inversion x f---> l/x it turns into x(l - xy2).

Definition 6.17. (Quad-equation of type Q) A multiaffine function Q E Pi, or the corresponding quad-equation Q = 0, is said to be of type Q if its four accompanying edge biquadratics h jk E P~ are nondegenerate, and it is said to be of type H otherwise. It turns out that multiaffine equations of type Q admit an exhaustive classification modulo (Mob)4, with only four normal forms.

Theorem 6.18. (Classification of equations of type Q) Any multiaffine equation Q(Xl' X2, X3, X4) = 0 of type Q is equivalent, modulo Mobius transformations from (Mob)4 acting on each of the variables independently, to one of the equations in the following list, called the list Q: (Q4)

sn( a)(xlx2 +X3X4) - sn(f3) (XIX4 + X2X3) - sn( a - (3)(XIX3 +X2X4) + sn(a - (3) sn(a) sn(f3)(l + k2xIX2X3X4) = 0,

(Q3)

sin( a )(XIX2+X3X4) -sin(f3) (XIX4 +X2X3) -sin( a- (3)(XIX3 +X2X4) +6 sin(a - (3) sin(a) sin(f3) = 0,

(Q2)

a(xlx2 + X3X4) - f3(XIX4 + X2X3) - (a - (3)(XIX3 + X2X4) +af3(a-f3)(xl+X2+X3+x4)-af3(a-f3)(a2-af3+f32) = 0,

(Q1)

a(xlx2 + X3X4) - f3(XIX4 +6a;3( a - (3) = o.

+ X2X3)

- (a - (3)(XIX3

+ X2X4)

In equation (Q4) the notation sn(a) = sn(a; k) is used for the Jacobi elliptic function with modulus k. The parameter 6 in equations (Q3), (Ql) can be scaled away, so that one can assume without loss of generality that 6 = 0 or 6 = 1. It is important to observe that there were no a priori built-in parameters a, 13 in the polynomial Q E Pi; they appear in the course of classification. They turn out to be naturally assigned to the edges of the quadrilateral (Xl, X2, X3, X4). Equation (Q4) is the most general one of the list; it is parametrized by two points of an elliptic curve. Equations (Q1)-(Q3) are obtained from (Q4) upon degenerations of an elliptic curve into rational


242

curves. One could be tempted to reduce the list Q to one item (Q4). However, the limit procedures necessary for that are outside of our group of admissible (Mobius) transformations, and, on the other hand, in many situations the "degenerate" equations (Q1)-(Q3) are of interest by themselves (for instance, the simplest equation of the list, (Q1) with 6 = 0, is nothing but the complex cross-ratio equation). This resembles the situation with the list of the six Painleve equations and the coalescences between them. It remains to find out how the equations of Theorem 6.18 can be assembled into 3D consistent systems on a cube.

Theorem 6.19. (3D consistent systems of type Q) Each equation of the list Q is 3D consistent. Conversely, any 3D consistent system (6.68) with all six equations of type Q is equivalent, modulo Mobius transformations from (M ijb)8 acting on each variable independently, to the system

(6.72) where (ij k) stands for any of the three cyclic permutations of (123), and Q(Xl, X2, X3, X4; ex, (3) is one of the polynomials (Q1)-(Q4)· In the next section, we sketch the main ideas behind the proof of this remarkable result.

6.11. Proof of the classification theorem 6.11.1. 3D consistent systems, biquadratics and tetrahedron property. Biquadratic polynomials h ij for a given Q E :P~ are closely related to the so-called singular solutions of the basic equation (6.67). Generically, the equation Q(XI, X2, X3, X4) = 0 can be solved with respect to any variable: if Q = p(Xj, Xb Xl)Xi + q(Xj, Xk, Xl) then Xi = -q/p for generic values of Xj, xk, Xl. However, Xi is not determined if the point (Xj, Xk, Xl) lies on the curve Si in (CJP'1)3 defined by (6.73)

Si:

p(Xj, Xk, Xl) = q(Xj, Xk, Xl) = O.

Since p = QXi and q = Q - XiP = Q - XiQx;, equations (6.73) are equivalent to Q(XI,X2,X3,X4) = Qx;(XI,X2,X3,X4) = o. Definition 6.20. (Singular solution) A solution (Xl,X2,X3,X4) of equation (6.67) is called singular with respect to Xi if it satisfies also the equation Qx; (Xl, X2, X3, X4) = O. The set of solutions singular with respect to Xi zs parametrized by the curve (6.73) called the singular curve for Xi. The projection of the curve Si onto the coordinate plane (k, l) is exactly the biquadratic hkl = pqXj - PXjq = Qx;QXj - QQX.i,Xj = O.

6.11. Proof of the classification theorem

243

Lemma 6.21. (Singular solutions and biquadratics) If a solution (Xl, X2, X3, X4) of equation (6.67) is singular with respect to Xi, then h jk = h jl = hkl = 0 on this solution. Conversely, if hkl = 0 for some solution, then this solution is singular with respect to either Xi or Xj. Proof. Since hkl = QXiQXj - QQxi,Xj' the equation hkl of the equation Q = 0 is equivalent to QXiQXj = O.

= 0 on the solutions 0

In the following theorem we will deal with biquadratic polynomials corresponding to various multiaffine ones; we will denote the biquadratics by the same letters as their parent quad-equations, with the superscripts for the remaining variables, so that, e.g., AO,1 = 6X2,X12A is the result of eliminating X2,X12 from A(X,XI,XI2,X2). Theorem 6.22. (Tetrahedron property and biquadratics for 3D consistent systems) Consider a 3D consistent system (6.68) with all six functions A, ... ,C being of type Q. Then:

• The system (6.68) possesses the tetrahedron property: the value of Xl23 as a function of the initial data x, Xl, X2, X3 does not depend on X; see Figure 6.19 . • For any edge of the cube, the two biquadratic polynomials corresponding to this edge, coming from the two faces sharing this edge, coincide up to a constant factor. Proof. The values of Xl23 obtained from the equations A = 0, B = 0 and C = 0, respectively, result from elimination of X12, Xl3 and X23, which can be expressed by the equations 2 I

I

3

I

F(x, Xl, X2, X3, XI23)

= AX13,X23BC 2 3

I

I

AX23BCX13 - AX13Bx23C + ABx23CX13

= 0,

I

G(x, Xl, X2, X3, XI23)

= BX12,X13CA - BX13CAx12 - BX12CX13A + BCX13Ax12 = 0, 2 I

3

I

H(x, Xl, X2, X3,

I X123)

= CX12,X23AB - CX12ABx23 - CX23Ax12B + CAX12 B X23 = O. Here the numbers over the arguments of the polynomials F, G, H indicate their degrees in the corresponding variables. These degrees are in the projective sense, that is in agreement with the action of Mobius transformations, and can be read off the right-hand sides. Due to 3D consistency, the expressions for Xl23 as functions of X,XI,X2,X3 found from these three equations,


244

coincide. Therefore the polynomials F, G, H must factorize as: F

2

= J(x,x3)K,

G

2

= g(x,xdK, 1

1

1

H

2

= h(X,X2)K,

1

K = K(X,X1,X2,X3,X123), where the polynomial K yields the common value of X123 as a function of x, Xl, X2, X3. Here J, g, h are some polynomials of degree 2 in the second argument. The degrees of J, g, hand K in X remain to be determined. We do this by analyzing singular solutions. Let the initial data x, Xl, X2 be free variables, and let X3 be chosen to satisfy the equation J(x, X3) = O. Then F == 0, and thus the system B = C = A = 0 does not determine the value of Xl23. Moreover, the equation B = 0 can be solved with respect to X23 since otherwise the initial data must be constrained by the equation BO,2(x, X2) = O. Analogously, the equation C = 0 is solvable with respect to Xl3. Therefore, the uncertainty appears from the singularity of equation A = 0 with respect to Xl23. Hence, the relation A3,23(X3, X23) = 0 is valid. In view of the assumption of the theorem, X23 is a (two-valued) function of X3 and does not depend on X2. This means that the equation B = 0 is singular with respect to X2 and therefore BO,3(x, X3) = O. Analogously, CO,3(x, X3) = O. Thus, we have proven that if X3 = 'P(x) is a zero of the polynomial J then it is also a zero of the polynomials BO,3, CO,3. If one of these three polynomials is irreducible, then this already implies that they coincide up to a constant factor. If the polynomials are reducible, then we could have J = a 2, BO,3 = ab, CO,3 = ac, where a, b, care multiaffine in x, X3. In any case, deg x J = 2, and this is sufficient to complete the proof. Indeed, this implies deg x K = 0, so the tetrahedron property is valid, and the first statement of the theorem is proven. In turn, the tetrahedron property can be used to prove the relation (6.74)

The variables in this relation separate:

B O,3 CO,3

AO,l BO,2 - CO,l . AO,2 '

so that BO,3/CO,3 may only depend on x. Due to nondegeneracy of the biquadratics BO,3, CO,3, this ratio is constant, which proves the second statement of the theorem.

245


So, it remains to prove equation (6.74). For this goal, rewrite the system

(6.68) in the form

= a(x,xI,X2), X23 = b(X,X2,X3), Xl3 = C(X,XI,X3), Xl23 = a(x3,xI3,X23) = b(XI,XI2,XI3) = C(X2,XI2,X23). Assuming the tetrahedron property, i.e., Xl23 = d(XI, X2, X3), we find by Xl2

differentiation: dXl dX2 dX3

= aX13 CXl , = bXl2 a X2 , = CX23 b X3 ,

dX2 =

aX23 b X2 ,

dX3 = dXl =

bXl3 CX3 '

aXl3 c X + aX23 b x , 0= bx12 a x + bXl3 C X ' 0= CX23 b x + cXl2 a X '

0=

cX12 a Xl ,

These equations readily imply the relation aX2bx3cXl

+ a Xl b X2 c X3

= O.

It is equivalent to (6.74) in view of the identity

a X2 /a Xl

X23,. _ _ _ _ _~8

=

AD,I/AD,2.

0

Xl23

I

_- - - r

X3

(q.."-'----.-----"CI3 : "

I "

)'.-

".l.'''' X 2).'9:", -,- -

x

Xl2

. _ - - - - - - 0 · Xl

Figure 6.19. Tetrahedron property.

The astonishing tetrahedron property, possessed by all 3D consistent systems of type Q, is illustrated in Figure 6,19. It means that the fields Xl, X2, X3, Xl23 sitting at the vertices of the white tetrahedron are connected by a certain multiaffine relation K(XI, X2, X3, X123) = O. Of course, for symmetry reasons, a multiaffine relation L(x, X12, X23, X13) = 0 also holds for the fields at the vertices of the black tetrahedron. 6.11.2. Analysis: descending from multiaffine Q to quartic r. In the further analysis, one more operation similar to (6.70) will be useful, namely

(6,75)

8Xk : P~ ~

pi,

8Xk (h) =

h;k -

2hh xkXk '

The operation 8Xk applied to a biquadratic polynomial h(XI, X2) actually computes its discriminant with respect to the variable Xk which gets eliminated, the result being a quartic polynomial which we will denote by rl (xz)


246

(where {k, l} = {1, 2}). Thus, from any h E P~, the operations 6Xk produce two quartic polynomials rl E The operation Xk is covariant with respect to Mobius transformations:

Pi.

6

(6.76)

The following statement is proved by a straightforward computation.

Lemma 6.23. (Commutativity of discriminants) For any multi affine polynomial Q(XI, X2, X3, X4) E

Pi,

6Xk (6 x;,xj(Q)) = 6xj (6 x;,Xk(Q)),

(6.77)

so that the following diagram is commutative:

r4(x4)

6x1 (6.78)

6X3

1

hI4(XI' X4)

r3(x3)

------>

16x1 ,x2 6X2 ,X3 f-----

1

6x2

6X1 ,X4

Q(XI, X2, X3, X4)

----+

h 23 (X2, X3)

1 6x3 ,x4

6x41 rl (xI)

6X4

h 34 (X3, X4)

+--

6X2

1 6x3 6X1

hI2(XI' X2)

+--

r2(x2)

------>

In fact, this diagram can be completed by the polynomials h 13 , h24 corresponding to the diagonals (so that the graph of the tetrahedron appears), but we will not need them. Further on we will make an extensive use of relative invariants of polynothese mials under Mobius transformations. For quartic polynomials r E relative invariants are well known and can be defined as the coefficients of the Weierstrass normal form r = 4 x3 - g2X - g3. For a given polynomial r(x) = r4x4 + r3x3 + r2x2 + rlX + ro they are given by

Pi

g2(r) = 418 (2rrIV - 2r'r'" + (r")2) =

1~ (12ror4 -

g3(r) = _1_(12rr"r IV - 9(r')2r IV - 6r(r"'f

=

3456 1 2 432 (72r or2r4 - 27rl r4

Under the Mobius change of x = the constant factors:

+ 9r lr2 r 3 -

Xl

3rlr3

+ r~),

+ 6r'r"r'" 2

2(r")3)

3

27ror3 - 2r2)·

these quantities are just multiplied by

247

6.11. Proof of the c1assiEcation theorem

For a biquadratic polynomial h E P~ , (6.79) h(x, y) = h22x2y2+h2IX2y+h2ox2+hI2xy2+hllxy+hlOX+ho2y2+hOly+hoo, the relative invariants are defined as

Notice that i3 can be defined also by the formula

Under the Mobius change of x = Xl and y = X2,

ik(M[h]) = ~~ ~~ik(h),

k = 2,3.

The following properties of the operations 6x ,y, 6x are proven straightforwardly. Lemma 6.24. (Opposite biquadratics and all four quartics have equal invariants) For any multiaffine polynomial Q(XI, X2, X3, X4) E Pl set: h I2 (XI,X2) =6X3 ,X4(Q) andh 34 (x3,x4) =6X1 ,X2(Q). For any biquadratic polynomial h(XI' X2) E P~ set: rl(xI) = 6X2 (h) and r2(x2) = 6X1 (h). Then (6.80)

ik(hI2) = idh34),

(6.81)

gk(r2) = gk(rl),

k = 2,3, k = 2,3.

In other words, in the diagram (6.78), the pairs of biquadratic polynomials on the opposite edges have the same invariants i2, i3, and all four quartic polynomials ri have the same invariants g2, g3. These results suggest the following approach to the classification of multiaffine equations Q = 0 modulo Mobius transformations. Suppose that, for a given Q E Pl, the four quartic polynomials ri(xi) associated to the vertices of the quadrilateral in the diagram (6.78) are known. Then one can use Mobius transformations to bring these polynomials into a canonical form. After that, one can reconstruct the edge biquadratics h ij from the pairs of vertex polynomials ri, rj. Finally, one can reconstruct the multiaffine Q from the edge biquadratics. 6.11.3. Synthesis: ascending from quartic r to biquadratic h. According to formulas (6.71), (6.76),

6X1 (6Xj ,Xk (M[Q])) = ~]~~~fM[6xl(6xj,xk(Q))l = C~i2Mh]'


248

where C = ~I ~~~~~~. Since the polynomial Q is defined up to an arbitrary constant factor, we may assume that Mobius changes of variables in the equation Q = induce transformations

°

ri

f-+

~i2 Mhl

of the polynomials r i. This allows us to bring each r i into one of the following six canonical forms: (6.82) r=(x 2 -1)(k 2x 2 -1),

r=x2-1,

r=x2,

r=x,

r=1,

r=O,

according to the six possibilities for the root distribution of r: four simple roots, two simple roots and one double, two pairs of double roots, one simple root and one triple, one quadruple root, or, finally, r vanishes identically. Note that in the first canonical form it is always assumed that k 2 1= 0,1, so that the second and third forms are not considered as particular cases of the first form. Not every pair of such polynomials is admissible as a pair of polynomials at two adjacent vertices, since the relative invariants of the polynomials of such a pair must coincide according to (6.81). We identify all admissible pairs, and then solve the problem of reconstruction of the biquadratic polynomial (6.79) by the pair of its discriminants (6.83)

which is equivalent to a system of 10 (bilinear) equations for 9 unknown coefficients of the polynomial h.

Lemma 6.25. (Reconstructing biquadratic from two discriminants) Nondegenerate biquadratic polynomials with a given pair of discriminants (rl(x),r2(y)) in the canonical form (6.82) exist if and only ifrl(x) = r(x) and r2 (y) = r(y) with one and the same canonical form r. These polynomials h can be brought into the following normal forms, possibly after Mobius transformations of x, y not affecting r:

(q4)

r(x)

=

(x 2 -1)(k 2x 2 -1):

h

=

1

2a(x2+y2-2Axy-a2-k2a2x2y2),

where A2 = r(a);

(q3)

r(x) = 8 - x 2 : where 8

h =

1

. ( ) (x 2 + y2 - 2 cos (a) xy) -

2 sm a

= 0, 1;

(q2)

r (x) = x :

(q1)

r(x)

=8:

1

h = 4a (x - y) 2 h

=

a

-

a3

2" (x + y) + 4 ;

(x - y)2 8a 2a - "2'

where

8 = 0,1.

8 sin(a) 2

,


249

In the cases (q4), (q3)8=1 and (q2) any biquadratic h with a given pair of discriminants (r(x), r(y)) is automatically of the form given in the lemma; in the cases (q3)8=O and (q1) an additional Mobius transformation might be necessary to bring h to this form (for instance, in the case (q1)8=O, that is, r(x) = r(y) = 0, any biquadratic h = (KXY + AX + flY + v)2 has this pair of discriminants, and any Mobius transformation of x, y preserves this form of h). One clearly sees the origin of the elliptic curve in the case (q4): the solution of the problem of finding a biquadratic h(x, y) with the pair of discriminants (r(x), r(y)) in the case r(x) = (x 2 -1)(k 2x 2 -1) is parametrized by a point (a, A) of the corresponding elliptic curve. Introducing the uniformizing variable 0: by a = sn( 0:), so that A = sn' (0:) = cn( 0: )dn( 0:), we can write the corresponding biquadratic (q4) in the form (6.84) h(x, y; 0:) = 2 s~( 0:) (x 2 +y2 - 2 cn( o:)dn( o:)xy - sn2 ( 0:)(1 + k 2x 2y2)).

One can recognize this polynomial as the addition theorem for the elliptic function sn(x; k); more precisely, h(x, y; 0:) = 0 if and only if X = sn(~; k) and y = sn(7]; k) with ~ - 7] = ±o:. 6.11.4. Synthesis: ascending from biquadratics h ij to multiaffine Q. The next step is the reconstruction of the multiaffine polynomials from the biquadratic ones. In doing this, the following facts are useful (they are proven by a direct computation). Lemma 6.26. (Reconstructing multiaffine equation from edge biquadratics) For any multiaffine polynomial Q E Pl, with the notation h ij = bXk,Xl (Q) E P~, the following identities hold:

4i3(h12)h14

(6.85)

where

(6.86)

f

=

= det

23 h 34 h 23 h 34 - h X3 X3X3 X3

12 h;;

12 h Xl 12 h X1X2

12 h X2X2

12 X 2 h X1X2

C

f fX2

),

f X2X2

34 . + h 23 h X3X3 '

12 h 34 _ h 14 h 23 + h 23 h 34 _ h 23 h 34 h'Xl Xl X3 X3 h 12 h 34 - h 14 h 23

Identity (6.85) shows that h14 can be expressed through the other three biquadratic polynomials (provided i3(h12) i= 0). Differentiating (6.86) with respect to X2 or X4 leads to a relation of the form Q2 = F[h 12 , h 23 , h 34 , h14], where F is a rational expression in terms of hij and their derivatives. Therefore, if the biquadratic polynomials on three edges (out of four) are known, then Q can be found explicitly. Of course, it is seen from Lemma 6.26


250

that not every set of three biquadratic polynomials comes as h ij from some

QE

Pl.

Proof of Theorem 6.18. We demonstrate the reconstruction procedure in the most interesting case (Q4). Let the polynomials h 12 , h2:3, h 34 and h14 be of the form (q4), with parameters denoted by (a, A), (b, B), (ii, A) and (b,.8), respectively, all of them lying on the elliptic curve A 2 = r( a). The relative invariants i2, i3 of hP and h 34 must coincide because of (6.80), and it is easy to check that this condition allows only the following possible values for (ii, A):

(a, A),

(-a, -A),

1 ka 2 (a, -A),

1

ka 2 (-a, A).

According to (6.71), a Mobius change of variables in the equation Q = 0 yields JX/r,xll\;I[Q])

=

~k~IM[JXhl;I(Q)]

=

C~il~jlM[hij],

where C = ~1~2~3~4. Since Q is only defined up to a multiplicative constant, we may assume that a Mobius change of variables induces transformations h ij ~ ~il ~jl M[h ij ] of the biquadratic polynomials h ij . In particular, if

h 34 = h(X3' X4; -a, -A)

or

11 31 = h( X3, Xl; kIa' -

then the corresponding Mobius transformation, X3 will change h 34 to

-h( -X3, X4; -a, -A),

resp.

~

k~2)'

-X3 or X3

~

1/(kx3),

2 ( 1 1 A )' - kX3h kX3' X4; ka' - ka 2

both of which coincide with h(X3, X4; a, A) due to the symmetries of the polynomial (q4). Thus, performing a suitable Mobius transformation of the variable X3 (which does not affect the polynomial r(x3)), we may assume without loss of generality that (ii, A) = (a, A). After that, the polynomial h14 is uniquely found according to formula (6.85), and it turns out that the equality (b,.8) = (b, B) is fulfilled automatically. Thus, the change of one variable allows us to achieve the equality of the parameters corresponding to the opposite edges of the square. A direct computation using formula (6.86) yields the equation

a(XIX2 + X3X4) + b(XIX4 + X2X3) - C(XIX3 + X2X4) - abc(l + k 2 XIX2.r3:r'1) = 0, where c = (aB +bA)/(l- k 2a 2b2). The uniformizing substitution a = sn(a), b = - sn(;3) , so that A = sn'(a), B = sn'(;3), and therefore c = sn(a - (3), brings it to the form (Q4).


251

Also in the other cases (Q1)-(Q3), suitable Mobius changes of the variables X2, X3, X4 allow us to bring the polynomials into the form hl2 = h(XI,X2ia), h 23 = h(X2,x3i(3), h 34 = h(x3,x4ia). A direct computation with formula (6.85) proves that this yields hl4 = h(XI' X4i (3). Then the multiaffine Q is found by using (6.86). 0 6.11.5. Putting equations Q = 0 on the cube. Proof of Theorem 6.19. Given a 3D consistent system (6.68) with all equations of type Q, one can use the Mobius transformations from (Mob)8 to bring all six equations into the canonical form from the list Q. Since by Theorem 6.22 biquadratics coming to an edge from two adjacent faces must coincide up to a constant factor, all six equations have to be of one and the same type (Q1)-(Q4). Moreover, the parameters k 2 in the case (Q4) and § in the cases (Q3), (Q1) have to be the same on each face of the cube. Therefore, the equations on all faces may differ only by the values of a and (3. Consider the equations corresponding to three faces adjacent to one vertex, say to x:

A(X,Xl,X12,X2) = Q(X,XI,XI2,X2ia,P) = 0, B(x, X2, X23, X3) = Q(x, X2, X23, X3i (3,;Y) = 0, C(x, X3, X13, xt) = Q(x, X3, x13, Xli 'Y, a) = O. We will show that one can write these three equations as (6.87) For the polynomials (Q1)-(Q4) from the list Q we have:

(6.88) (6.89)

h I2 (Xl, X2) h 14 (XI,X4)

= §x3,x4Q(Xl, X2, X3, X4i a, (3) = ",(a, (3)h(XI, X2i a), =

§x2,x3Q(XI,X2,X3,x4ia,(3) = ",((3,a)h(xI,x4i(3),

with the biquadratics h(x, Yi a) listed in Theorem 6.25 as (q1)-(q4). Thus, we find:

AO,l(X, Xl) = ",(a, P)h(x, Xli a),

02

-

-

BO,2(X, X2) = ",((3, ;Y)h(x, x2i (3),

A ' (x, X2) = ",((3, a)h(x, x2i (3), BO,3(x, X3) = ",(;y, (3)h(x, x3; ;Y),

CO,3(x, X3) = ",(,,(, a)h(x, x2i 'Y),

CO,I(x, Xl) = ",(a, 'Y)h(x, Xl; a).

According to the second statement of Theorem 6.22 and to formula (6.74), the following relations must hold: (6.90) h(X,XI;a) _ ( _) h( X,XI;a ) - rn a, a , (6.91)


252

For the most complicated case (Q4), to which we will restrict ourselves in this proof, the biquadratic (q4) is given in (6.84), and a direct computation gives K,(a,{3) = 2sn(a) sn(f3) sn(a - {3). Equations (6.90) yield that & may only take the values ±a, which correspond to m(a, &) = ±1, and analogously for {3,'Y. Equation (6.91) with the above-mentioned values of K,(a,{3) yields m(a, &)m({3, (J)m("(, 1') = 1. Thus, up to a change of enumeration, two cases are possible: or

&

= a,

{3 = -{3,

'Y

= -"(.

In the first case the equations A = 0, B = 0, C = 0 have the desired form (6.87) with al = a, a2 = {3, a3 = 'Y. In the second case it is enough to observe that the equation B = 0 is not affected by the replacement of parameters ({3,1') with (-{3, -i') = (-{3, 'Y), which again leads to the desired form (6.87) with al = a, a2 = -{3, a3 = 'Y. Continuing to argue in a similar manner for faces adjacent to other vertices, one shows that the signs of edge parameters can always be adjusted on the whole cube as in system 0 (6.72).

6.12. Classification of discrete integrable 2D systems with fields on vertices. II In the previous two sections, we classified quad-equations Q = 0 of type Q, that is, those with all nondegenerate edge biquadratics, and showed their 3D consistency. However, quad-equations of type H, i.e., those with (some of) the edge biquadratics being degenerate, are by no means less interesting or less important. It is enough to mention that the very prominent Hirota equation is of type H (which is the reason for the choice of the latter notation). A classification of multiaffine equations of type H seems to be a rather complicated and tiresome task. Nevertheless, postulating some additional properties, a classification can be achieved. Our assumptions for the quad-equation Q = 0 will be as follows: ~ Linearity. The left-hand side of the equation (6.92) is a polynomial of degree 1 in each variable, depending on two parameters assigned to the edges. ~

Symmetry. The function Q has the symmetry properties

(6.93)

EQ(XI,X4,X3,x2;{3,a),

E

(6.94)

aQ(x2,x3,x4,xI;{3,a),

a = ±1.

~

= ±1

Tetrahedron property. The value X123, existing due to 3D consistency, depends on Xl, X2 and X3, but not on x.

6.12. Classification of discrete integrable 2D systems. II

253

The symmetry properties are natural to require, because to enable us to pose our equations on arbitrary quad-graphs, the equations should not depend on the enumeration of vertices. Note that the normal forms of the list Q possess these symmetries. The tetrahedron property is admittedly a less natural classification assumption, but it holds for the vast majority of known interesting examples, including all the equations of the list Q and the Hirota equation itself; see formula (6.32). We consider here the problem of 3D consistency for equation (6.92) in the sense of the system (6.72), with one and the same polynomial Q. Due to the symmetry assumption, the natural transformation group, which can be used to put the equation in the normal form, is essentially smaller than in Section 6.10; namely, all vertex fields should be acted on by one and the same Mobius transformation. Theorem 6.27. (Classification of symmetric equations with tetrahedron property) Any 3D consistent quad-graph equation (6.92) possessing the linearity, symmetry, and tetrahedron properties is equivalent, modulo Mobius transformations acting simultaneously on all variables Xi and modulo point transformations of the parameters a, (3, to one of the equations of the following lists. List Q from Theorem 6.18: (Q4)

sn( a) (XIX2 +X3X4) - sn({3)(xIX4 +X2X3) - sn( a - (3)(XIX3 + X2X4) + sn( a - (3) sn( a) sn({3) (1 + k2xIX2X3X4) = 0,

(Q3)

sin( a) (XIX2+X3X4) -sin({3) (XIX4 +X2X3) -sin( a- (3)(XIX3+X2X4) +0 sin(a - (3) sin(a) sin({3) = 0,

(Q2)

a(xlx2 + X3X4) - (3(XIX4 + X2X3) - (a - (3)(XIX3 + X2X4) +a{3(a-{3)(xI +X2+X3+x4)-a{3(a-{3)(a2-a{3+{32) = 0,

(Q1)

a(xlx2 + X3X4) - (3(XIX4 +oa{3( a - (3) = 0;

+ X2X3)

- (a - (3)(XIX3

+ X2X4)

list H:

+ X3X4)

+ X2X3) + 0(a 2 - (32) = 0, X4) + ({3 - a)(XI + X2 + X3 + X4) + {32 -

(H3)

a(xlx2

(H2)

(Xl - X3)(X2 -

(H1)

(Xl - X3)(X2 - X4)

- (3(XIX4

+ (3 -

a

a 2 = 0,

= 0;

and list A: (A2)

sin(a)(xlx4 + X2X3) - sin({3) (XIX2 + X3X4) - sin(a - (3)(1 + XIX2X3X4) = 0,

(A1)

a(xlx2 + X3X4) - (3(XIX4 -oa{3(a - (3) = 0.

+ X2X3) + (a -

(3)(XIX3

+ X2X4)

Remarks. 1) The parameter 0 in equations (Q3), (Q1), (H3), (A1) can be scaled away, so one can assume without loss of generality that 0 = or 0 = 1.

°


254

2) If one extends the transformation group of equations by allowing Mobius transformations to act on the variables Xl, X3 differently than on x2, X4 (white and black subgraphs of a bipartite quad-graph), then equation (A2) turns into (Q3)8=O by the change (X2, X4) I---> (1/X2, 1/X4), and equation (A1) turns into (Q1) by the change (X2, X4) I---> (-X2, -X4). SO, really independent equations are given by the lists Q and H. 3) Equation (H3) is the most general in the list H, since (H1) and (H2) can be considered as its limiting cases. Note that (H1) is the discrete KdV equation and (H3)8=O is a version of the Hirota equation with the symmetry properties (6.93), (6.94). The general scheme of the proof of Theorem 6.27 is the same as in Section 6.11. We start with the "analysis" part. Due to the symmetry assumption, all edge biquadratics for the polynomial Q(Xl, X2, X3, X4; a, (3) are given by one and the same biquadratic polynomial g(x, y; a, (3), so that

h 12 (Xl, X2) = c5 X3 ,X4 (Q) = g(Xl, X2; a, (3), h 14 (Xl,X4) = c5 X2 ,X3(Q) = g(Xl,x4;(3,a). Moreover, the polynomial 9 is symmetric: g(x, y; a, (3) = g(y, x; a, (3). Lemma 6.28. (Descending from multiaffine Q to quartic r) The biquadratic g(x, y; a, (3) admits a representation (6.95)

g(X, y; a, (3) = k(a, (3)h(x, y; a),

where the factor k is antisymmetric, k((3, a) = -k( a, (3), and the coefficients of the polynomial h(x, y; a) depend on a single parameter a in such a way that the discriminant r(x) = c5 y (h) does not depend on a at all. Proof. In the proof of Theorem 6.22 we used the previously demonstrated tetrahedron property to derive formula (6.74). In the present setup the tetrahedron property has been postulated, thus we can still use formula (6.74), which, due to the symmetry assumptions, takes the following form:

g(x, Xl; aI, a2)g(x, X2; a2, a3)g(x, x3; a3, ad

= -g(x, Xl; aI, a3)g(x, X2; a2, adg(x, X3; a3, a2). This relation implies that the fraction g(x,xl;al,a2)/g(x,xl;al,a3) does not depend on Xl, and due to the symmetry it does not depend on X either. We see that the symmetry assumptions has been used in this argument to replace the nondegeneracy of biquadratics which has been required in Theorem 6.22 to come to the same conclusion. We find:

g(x, Xl; aI, a2) g(x, xl; aI, a3)

K:(al, a2) K:(al,a3)'

6.12. Classification of discrete integrable 2D systems. II

255

where the function /(x, y; a, (3) = lJI(x, y; a - (3), (6.105) (Ql)8=O: Additive three-leg form with ¢(x, y; a, (3) = 'lj;(x, y; a - (3), 'lI'{Xo, Xl; a)

(6.106)

a

= --Xo - Xl

(H3): Multiplicative three-leg form with (6.107)

iJ>(x, y; a, (3) =

(3x - ay ax-

(3' y

lJI(XO, Xl; a) = XOXI

+ ba.

(H2): Multiplicative three-leg form with (6.108)

iJ>(x,y;a,(3)

=

x-y+a-(3 x-y-a+

(3'

lJI(xo,xI;a)=xo+xI+a.

(HI): Additive three-leg form with (6.109)

¢(x, y; a, (3)

a-(3

= -- , x-y

'l/}(xo, Xl; a)

=

Xo

+ Xl.

Proof. The proof of this theorem is obtained by a direct computation; see Exercise 6.15. However, this does not give any insight in how these three-leg forms could be found. A general way to derive three-leg forms is the subject 0 of Exercise 6.16. Remark. It should be mentioned that the existence of a three-leg form allows us to derive (and, in some sense, to explain) the tetrahedron property of Section 6.11. Indeed, consider three faces adjacent to the vertex Xl23 in Figure 6.19, namely the quadrilaterals (Xl, X12, X123, X1.'3) , (X2' X23, X123, XI2), and (X3, X13, X123, X23). A summation (resp. multiplication) of the three-leg forms (centered at X123) of equations corresponding to these three faces leads to the additive equation (6.110)

¢(XI23, xl; a2, a3)

+ ¢(X123, X2; a3, al) + ¢(XI23, X3; aI, a2) =

0,

respectively to the multiplicative equation (6.111)

This is the equation which relates the fields at the vertices of the "white" tetrahedron in Figure 6.19. Note that it can be interpreted as a discrete Laplace type equation coming from a spatial flower with three petals. The functions ¢(x,y;a,(3) (resp. iJ>(x,y;a,(3)) corresponding to the "long" legs, yield additive (resp. multiplicative) Laplace type equations on arbitrary planar graphs. Studying the list of Theorem 6.32, one sees that there are only six "long" legs functions. Three of them are rational in x, y;


260

each of the corresponding Laplace type systems admits two different extensions to a quad-graph system: one from the list Q, where the form of the "short" legs coincides with the form of the "long" ones, (Q3)8=O, (Q1)8=1, and (Q1)8=O, and the other from the list H, with different "short" legs, (H3), (H2), and (H1), respectively. The other three functions are rational in Y only, and require a uniformizing change of the variable x. The corresponding Laplace type systems admit only one extension to a quad-graph system, (Q4), (Q3)8=1, and (Q2). Theorem 6.33. (Integrability of Laplace type equations) Additive Laplace type equations (6.98) with the "long legs" functions ¢(x, Yi Cl:, (3), resp. multiplicative Laplace type equations (6.100) with the "long legs" functions (x, Yi Cl:, (3) from Theorem 6.32, are integrable in the sense of Section 6.4. Proof. Observe that the functions ¢, resp. , always contain the parameters Cl:, (3 in the combination Cl: - (3. This means that the edge parameters of the extension of the Laplace type system to a quad-graph system are only defined up to an additive constant A. Let the equations of the extended quad-graph system read

Q(xo, Yb Xk, Yk+li Cl:k

+ A, Cl:k+l + A)

=

o.

Rewrite this equation as a Mobius transformation, (6.112) Then the (normalized, if necessary) matrix L(xo, Xb Cl:k, Cl:k+li A) is the transition matrix across the edge (xo, Xk) E E(9), or along the edge (Yk, Yk+d E E(9*), in the zero curvature representation of the Laplace type system on 9. 0 Example. Consider the additive Laplace type system on 9, corresponding to the equation (Q1)8=O (cross-ratio equation): Cl:k - Cl:k+l =

'"""'

(6.113)

L....; xkE

star(xo)

O.

Xo - Xk

The extension to the quad-equation reads:

(xo - Yk)(Xk - Yk+l) Cl:k + A (Yk - Xk)(Yk+l - xo) Cl:k+l + A' which can be written as the Mobius transformation (6.112) with the matrix L(XO,Xk,Cl:k,Cl:k+li A) (6.114)

( \ + n,k)l + Cl:k A

LX.

Cl:k+l

Xo - Xk

(Xk 1

-XOXk) -Xo

261

6.14. Fields on edges: Yang-Baxter maps

These matrices give a zero curvature representation of the Laplace type system (6.113) on an arbitrary surface graph 9.

6.14. Fields on edges: Yang-Baxter maps We now turn to the study of another large class of 2D systems on quadgraphs with fields assigned to the edges. In this situation it is natural to assume that each elementary quadrilateral carries a map F : X x X -> X x X, with X being the set where the fields x, Y take values, so that F(x, y) = (X2' yd; see Figure 6.22. The concept of 3D consistency of such maps may be interpreted in several ways, depending on the initial value problem one would like to pose on the elementary 3D cube.

x

Figure 6.22. Map encoded by an elementary quadrilateral with fields on edges.

One way is to choose the initial data x, y, z on three edges of an elementary cube adjacent to one vertex. One computes first

F(x, y) = (X2' Yl),

F(y, z) = (Y3, Z2),

F(z, x) = (Zl' X3),

and then

F(X3, Y3) = (X23, Y13),

F(Yl, Zl) = (Y13, Z12),

F(Z2' X2) = (Z12' X23),

so that there are two a priori different answers for any of the fields X23, Y13, Z12 with two indices; see Figure 6.23. Definition 6.34. (3D consistent map) A map F : X x X -> X x X is called 3D consistent if the two answers for each of the fields (X23, Y13, Z12) in Figure 6.23 coincide for any initial data (x, y, z).

An important example of a 3D consistent map, coming from discrete differential geometry, is discussed in Exercises 6.19, 6.20. Definition 6.34 has one notational inconvenience: since each initial edge is used on the first step by two different maps, it is not possible to express the property of 3D consistency in terms of compositions of maps. This can be overcome by a different choice of initial data, namely by choosing them on a path consisting of three edges of three different coordinate directions.


262

Y3

Y13 I

X3

Z2 1 I I

Z

I X }- ____2_

Y/ /

/

X

Figure 6.23. 3D consistency of 2D systems with fields on edges.

This leads to the notion of Yang-Baxter maps (traditionally denoted by R rather than by F). Definition 6.35. (Yang-Baxter map) A map R : X x X ---+ X x X is called a Yang-Baxter map if it satisfies the Yang-Baxter relation (6.115) where each ~j X 3 1----+ X 3 acts as the map R on the factors i, j of the Cartesian product X 3 , and acts identically on the third factor.

Equation (6.115) is understood as follows. The fields x, yare supposed to be assigned to the edges parallel to the I-st and the 2-nd coordinate axes, respectively. Additionally, consider the fields z assigned to the edges parallel to the 3-rd coordinate axis. Initial data are the fields x, y, z on a path consisting of three edges of different coordinate directions; see Figure 6.24. The left-hand side of this figure corresponds to the composition of maps on the left-hand side of equation (6.115), which are visualized as maps along the three front faces of the cube:

(Here and below we slightly abuse the notation by omitting the arguments on which our maps act identically.) Similarly, the right-hand side of the figure corresponds to the chain of maps on the right-hand side of (6.115), which are visualized as maps along the three back faces of the cube:

So, equation (6.115) assures that the two ways of obtaining from the initial data (x, y, z) lead to the same results.

(X23, Y13, Z12)


263

X23

;1:2:3

Yl3

Y13

Rl~

Y3

X3

~

Z2

Z12 R13

I

\'i'l I

Z Z12

Z

YI~_ X2 _ _

//Rl~

Y

R13

X

Y

X Figure 6.24. Yang-Baxter relation.

The notion of the zero curvature representation makes perfect sense for Yang- Baxter maps, and can be expressed as

L(x, >")L(y, >..) = L(Yl, >")L(X2' >..).

(6.116)

There is a construction of zero curvature representations for Yang-Baxter maps with no more input information than the maps themselves, close in spirit to Theorem 6.4. Consider a parameter-dependent Yang-Baxter map R(o, (3), with parameters CI', (3 E C assigned to the same edges of the quadrilateral in Figure 6.22 as the fields x, y, opposite edges carrying the same parameters. Although this can be considered as a particular case of the general notion, by introducing :X = X x C and R(:E, 0; y, (3) = R(o, (3)(x, y), it is convenient for us to keep the parameter separately. Thus, in Figure 6.24 all edges parallel to the :1: (resp. y, z) axis carry the parameter 0: (resp. (3, 1'), and the corresponding version of the Yang-Baxter relation reads: (6.117) Theorem 6.36. (Zero curvature representation for Yang-Baxter maps) Suppose that there is an effective action of thc linear group G = G L( N, q on the set X (i. e.. A E G acts identically on X only if A = I), and that the Yang-Baxter map R(o:, (3) has the following special form:

(6.118)

X2

=

B(y, (3, o:)[x] ,

Yl

=

A(x, o,fJ)[y].

Here A, B : X x ex C ---t G are some matrix-valued functions on X depending on parameters ° and (3, and A[x] denotes the action of A EGan x E X. Then, whenever (X2' yd = R(o, (3)(.T, y), we have (6.119)

A(x,o:,>")A(y,(3,>")

A(Yl, (3, >")A(X2' 0:, >..),

(6.120)

B(y,(3,>")B(x, 0:, >..)

B(X2' 0, >")B(YI' (3, >..).


264

In other words, both A(x, a,).) and B-1(x, a,).) (or BT(x, a, ).)) form zero curvature representations for R. Proof. Look at the values of Z12 produced by the two sides of the YangBaxter relation (6.117): the left-hand side gives Z12 = A(x, a, ,)A(y,,8, ,)[z], while the right-hand side gives Z12 = A(Yl,,8,,)A(x2,a,,)[z]. Now since we assume that the action of G is effective, we immediately arrive at the relation

A(x,a,,)A(y,,8,,) = A(Yl,,8,,)A(x2,a,,), which holds whenever (X2, yd = R(a, ,8)(x, y). This coincides with (6.119), an arbitrary parameter, playing the role of the spectral parameter ).. Similarly, one could look at the values of X23 produced by the two sides of (6.117): the left-hand side gives X23 = B(Y3,,8, a)B(z2", a)[x], while the right-hand side gives X23 = B(z", a)B(y,,8, a)[x]. Effectiveness of the action of G again implies:

B(Y3,,8, a)B(z2", a) = B(z", a)B(y,,8, a), whenever (Y3, Z2) = R(,8, ,)(y, z). This coincides with (6.120); here the role of the spectral parameter). is played by an arbitrary parameter a. 0 In order to cover all known examples, the scheme of Theorem 6.36 must be extended in the following way. We say that A(x, a,).) gives a projective zero curvature representation for the Yang-Baxter map R if the relation (6.116) holds up to multiplication by a scalar matrix cI, where c may depend on all the variables in the relation. Assume that the action of G = GL(N, q on X is projective, i.e., scalar matrices and only they act trivially. Then the previous considerations show that the matrices A(x, a,).) and B-1(x, a,).) give projective zero curvature representations for the corresponding YangBaxter maps (6.118). In practice, the natural choices of matrices A, B in (6.118) actually lead to proper zero curvature representations, as the following examples show. Example 1: Adler map. Here X =

_ a-,8 x = y - --, x+y

(6.121)

C]!Dl

and the map has the form

_

,8-a x+y

y = x - --.

Then fj

= x _,8 - a = x 2 + xy - (,8 - a) = A(x,a,,8)[y], x+y

x+y

where

A(x,a,).)=

(

X

1

x2

+ xa

-). )

'

and the group G = GL(2, q acts projectively on C]!Dl by Mobius transformations. In this example B(x, a,).) = A(x, a, ).), so the matrices BT = AT provide us with an alternative zero curvature representation.


265

Example 2: Interaction of matrix solitons. Our next example comes from mathematical physics. The matrix Korteweg-de Vries equation Ut + 3UUx + 3Ux U + Uxxx = 0 admits one-soliton solutions of the form U(x, t) = 2a 2P sech2 (ax - 4a 3 t), where a is the parameter measuring the soliton velocity, and the matrix amplitude P must be a projector: p2 = P. Projectors of rank 1 have the form P = ~r? / (~, 17). It turns out that the change of the matrix amplitudes P of two solitons with velocities a1 and a2 after their interaction is described by the following Yang-Baxter map:

R(a1' a2) : (6,171; 6, 172)

->

(~l' ih; ~2' T72),

(6.122) (6.123) In this example X is the set of projectors P of rank 1 which is the variety CpN-1 x CpN-1, and a projective action of the group G = GL(N, q on X is induced by A[(~, 17)] = (A~, A'I]). It is easy to see that formulas (6.123) can be written as

with the matrices ~17T

2a A-a

A(~,17,a,'\)

= 1+ - \ - . -(-)' ~,17

Thus, the matrices A(~, 17, a,'\) give a projective zero curvature representation for the interaction map, but it is not difficult to see that this is actually a genuine zero curvature representation. As in Example 1, B(~, 17, a,'\) = A(~, 17, a, '\).

Example 3: Yang-Baxter maps ansmg from geometric crystals. Let X = C n , and define R : X x X -> X x X by the formulas (6.124)

xJ --

X

p

_J_ J

-

p'

Yj

j-1

Pj-1 = Yj -po '

j = 1, ...

,n,

J

where (6.125)

Pj =

t (IT a=l

k=}

Xj+k

IT

YJ+k)

k=a+}

(in this formula subscripts j + k are taken (mod n) ). Clearly, the map (6.124) keeps the subsets Xn x Xf3 C X x X, where n

Xn =

{(X}, ...

,xn ) EX:

II k=}

Xk

= a},


266

invariant. It can be shown that the restriction of R to Xn x Xt3 may be written in the form (6.118). For this, the following trick is used. Embed Xn x Xt3 into ClP'n-l x ClP'n-l via J(:c,y) = (z(x),w(y)), where

z(x) = (1 : ZI

: ... :

zn-d,

w(y) =

j

Zj

=

II

(WI: ... : W n-l :

1),

n

Xk ,

Wj

=

k=l

II

Yk·

k=j+1

Then it is easy to see that in the coordinates (z, 111) the map R is written as

z = B(y, (3, ex)[z] ,

'Ii! = A(x, ex, (3)[w] ,

with certain matrices B, A from G = GL(n, q, where the standard projective action of GL(n, q on ClP'n-1 is used. Moreover, a simple calculation shows that the inverse matrices are cyclic two-diagonal:

(6.126)

(6.127)

B- 1 (y, (3, ex)

A-I(x, ex, (3)

Yl

-1

0 0

Y2

0 -1

0

Y3

0 -ex

0 0

0 0

Xl

0

-1 0

X2

0 0

-1

X3

0 0

0 0

0 0

0 0 0

0 0 0

Yn-l

-1

0

Yn

0 0 0

-(3 0 0

.Tn -1

0

-1

Xn

To be more precise, the matrices A. B are defined only up to multiplication by scalar matrices. These scalar matrices are chosen in (6.126), (6.127) in such a way that the dependence of the matrices B- 1 , A-Ion their "own" parameters ((3, resp. ex) drops out, so that the only parameter remaining in the zero curvature representation is the spectral one. In other words, the zero curvature representation does not depend on the subset Xn x Xt3 to which we restricted the map. It can be checked that this is actually a genuine (not only projective) zero curvature representation. Note also that in this example the matrices BT coincide with A (so they cannot be used to produce an alternative zero curvature representation for R).

6.15. Classification of Yang-Baxter maps Consider Yang-Baxter maps R : X x X ----+ X x X, (.T,y) f---+ ('U,v) in the following special framework. Suppose that X is an irreducible algebraic variety, and R is a birational automorphism of X x X. Thus, the birational

6.15. Classification of Yang-Baxter maps

267

map R- I : X x X --> X x X, (u,v) ~ (x,y) is defined. This is depicted in the left square in Figure 6.25. Furthermore, a nondegeneracy condition is imposed on R: rational maps u(',y) : X - t X and v(x,') : X --> X must be well defined for generic x, resp. y. In other words, birational maps If : X x X --> X x X, (x, v) ~ (u, y) and If-I: X x X - t X X X, (u, y) ~ (x, v), called companion maps to R, must be defined. This requirement is visualized in the right square in Figure 6.25. Birational maps R satisfying this condition are called quadrirational. A formal definition of a slightly more general notion (where different spaces are allowed for the arguments x and y) looks as follows.

Definition 6.37. (Quadrirational map) Let Xl, X2 be two irreducible algebraic varieties over C. A rational map F : Xl X X2 --> Xl X X 2 , identified with its graph, an algebraic variety r F C Xl X X2 X Xl X X 2 , is called quadrirational if for any fixed pair (x, y) E Xl X X 2 , except possibly some closed subvarieties of codimension ~ 1, the variety r F intersects each of the sets {x} x {y} X Xl X X 2 , Xl X X2 X {x} x {y}, Xl x {y} X {x} X X2 , and {x} X X2 X XI X {y} exactly once, i. e., if r F is a graph of four rational maps F,F-I,P,P-I: Xl X X2 --> Xl X X 2 .

u

u

Y;(,V x

x

Figure 6.25. A map F on X x X, its inverse and its companions.

It is possible to classify all quadrirational maps in the case Xl = X2 = ClP'I; we give a short presentation of the corresponding results. Birational isomorphisms of ClP'1 x ClP'l are necessarily of the form

(6.128)

F. u = a(y)x + b(y) . c(y)x + d(y) ,

A(x)y + B(x) v - ----'-'-----'- C(x)y + D(x) ,

where a(y), .. . , d(y) are polynomials in y, and A(x), .. . , D(x) are polynomials in x. For quadrirational maps, the degrees of all these polynomials are :S 2. Depending on the highest degree of the coefficients of each fraction in (6.128), we say that the map is [1:1], [1:2], [2:1], or [2:2]. The richest and most interesting subclass is [2:2]. For the maps of this subclass the polynomials .6.(x) = A(x)D(x) - B(x)C(x) and 8(y) = a(y)d(y) - b(y)c(y) are of


268

degree four. A quartic polynomial belongs to one of the following five types, depending on the distribution of its roots: I: four simple roots,

II: two simple and one double root, III: two double roots, IV: one simple and one triple root, V: one quadruple root. It turns out that a necessary condition for a map of the subclass [2:2] to be quadrirational is that the polynomials ~(x) and 8(y) are simultaneously of one of the types I-V. Sufficient conditions are more complicated and can be expressed in terms of singularities of the map F, i.e., those points (~, 'T1) E «][»1 x CIP'1 where both the numerator and the denominator of at least one of the fractions in (6.128) vanish:

(6.129) or A(~)'T1

(6.130)

+ B(O = 0,

C(O'T1 + D(~) =

o.

For instance, if both polynomials ~(x) and 8(y) are of type I, then the necessary and sufficient condition for the quadrirationality of the map (6.128) is that the roots Xi, Yi (i = 1, ... ,4) of ~(x), 8(y) can be ordered so that both equations (6.129), (6.130) be satisfied for (~, 'T1) = (Xi, Yi), i = 1, ... ,4; in other words, the four singularities of both fractions in (6.128) be at the points (Xi, Yi). One can find normal forms for all quadrirational [2:2] maps with respect to the action of the natural transformation group, which in this case is the group (Mob)4 of Mobius transformations acting independently on each of the fields X, y, u, v.

Theorem 6.38. (Classification of quadrirational maps on CIP'1 x C1P'1) Any quadrirational (2:2) map on CIP'1 xCIP'1 is equivalent, under some change of variables acting by Mobius transformations on each field X, y, u, v independently, to exactly one of the following five normal forms: Fr:

u = oyP,

v = (3xP,

Frr:

y u=-P, 0

v=

Fm:

y u= -P, 0

X

73 P, X

v=

73 P,

P =

P= P=

(1 - (3)x + (3 - 0 + (0 - l)y (3(1 - o)x + (0 - (3)yx + 0((3 - l)y'

ox - (3y + (3 - 0 , x-y ox - (3y , x-y


F rv : u Fv:

=

yP,

u = y + P,

v

xP,

P

v=x

+ P,

=

with suitable constants

0:,

=

1 + (3 -

269

0: ,

x-v

p=o:-(3 ,

x-v

(3.

Each one of the maps F r , ... ,Fv is an involution and coincides with its companion maps, so that all four arrows in Figure 6.25 are described by the same formulas. Note also that these maps come with the intrinsically built-in parameters 0:, (3. Neither their existence nor a concrete dependence on parameters is presupposed in Theorem 6.38. A geometric interpretation of these parameters can be given in terms of singularities of the map; it turns out that the parameter 0: is naturally assigned to the edges x, u, while (3 is naturally assigned to the edges y, v. For instance, for the map Fr the parameter 0: is nothing but the cross-ratio of the four roots Xi of the polynomial ~(x), and similarly (3 is the cross-ratio of the four roots Yi of the polynomial 8(y). The most remarkable fact about the maps F r , ... ,Fv is their 3D consistency. For 'J = I, II, III, IV or V, denote the corresponding map Fy of Theorem 6.38 by Fy(o:, (3), indicating the parameters explicitly. Moreover, for any 0:1, 0:2, 0:3 E C, denote by Fij = F J-( O:i, O:j) the corresponding maps acting nontrivially on the i- th and the j- th factors of (ClP l )3. Theorem 6.39. (Normal forms of quadrirational maps on ClP l x ClP l are 3D consistent) For any 'J = I, II, III, IV or V, the system of maps Fij is 3D consistent, and also satisfies the Yang-Baxter relation with parameters (6.117). Proof. The proof can be obtained by a direct computation (Exercise 6.22). It will also follow from Theorem 6.40 below, after we provide a geometric interpretation of the maps F'J'. D

Actually, 3D consistency of quadrirational maps on ClP l x ClP l holds not only for the normal forms F'J' but also under much more general circumstances. The only condition for quadrirational [2:2] maps consists in matching singularities along all edges of the cube. Similar statements hold also for quadrirational [1:1] and [1:2] maps, so that in the case Xl = X2 = ClP l the properties of being quadrirational and of being 3D consistent are related very closely. The maps F'J' of Theorem 6.38 admit a very nice geometric interpretation. Consider a pair of nondegenerate conics Ql, Q2 on the plane ClP2 , so that both Qi are irreducible algebraic varieties isomorphic to ClPl. Take

270


X E Ql, Y E Q2, and let £ = (XY) be the line through X, Y (well-defined if X =1= Y). Generically, the line £ intersects Ql at one further point U =1= X, and intersects Q2 at one further point V =1= Y. This defines the map

(6.131)

:t(X, Y)

= (U, V);

see Figure 6.26 for the ]R2 picture. The map :t is quadrirational, it is an involution and moreover coincides with both its companions. This follows immediately from the fact that, knowing one root of a quadratic equation, the second is a rational function of the input data. Intersection points X E Ql n Q2 correspond to the singular points (X, X) of the map :t.

Figure 6.26. A quadrirational map on a pair of conics.

Generically, two conics intersect at four points; however, degeneracies can happen. There are five possible types I - V of intersection of two conics: I: II: III: IV: V:

four simple intersection points; two simple intersection points and one point of tangency; two points of tangency; one simple intersection point and one second order tangency point; one point of third order tangency.

All conics sharing a quadruple of points build a linear family, or a pencil of conics. There are five types I-V of pencils of conics. Using rational parametrizations of the conics:

Cpl :3 x

I--f

X(x) E Ql

C

Cp2,

resp.

Cpl:3 y

I--f

Y(y) E Q2

C

Cp2,

it is easy to see that :t pulls back to the map F : (x, y) I--f ('U, v) which is quadrirational on Cpl x Cpl. One shows by a direct computation that the maps F for the above five situations are exactly the five maps listed in Theorem 6.38. Now, we obtain the following geometric interpretation of the statement of Theorem 6.39.


271

Figure 6.27. 3D consistency on a linear pencil of conics.

Theorem 6.40. (3D consistent maps on a pencil of conics) Let Qi, i = 1,2,3, be three nondegenerate members of a linear pencil of conics. Let X E Ql, Y E Q2 and Z E Q3 be arbitrary points on these conics. Define the maps 'Jij as in (6.131), corresponding to the pair of conics (Qi,Qj). Set (X2' YI) = 'J12(X, Y), (X3, Zl) = 'J13(X, Z), and (Y3, Z2) = 'J23(Y, Z). Then

X 23 = (X 3Y3) n (X 2Z 2) E Ql, (6.132)

Y13

= (X3Y3) n (Y1ZI)

E

Q2,

Z12 = (Y1Z1) n (X 2Z 2) E Q3.

In other words, the maps 'Jij are 3D consistent.

Proof. We will work with equations of lines and conics on Cp2 in homogeneous coordinates, and use the same notation for geometric objects and homogeneous polynomials vanishing on these objects. Construct the lines a = (YZ), b = (XZ) and c = (XY), respectively. Let

X 2 = (c n Qd \ X,

Y1 = (c n Q2) \ Y,

X3 = (b n QI) \ X,

Y3

= (a n Q2) \ y.

Next, construct the line C = (X 3 Y3 ), and let

X 23 = (CnQd \X3,

Y13 = (CnQ2) \ Y3·

Finally, construct the lines A = (Y1 Y13 ) and B = (X 2X 23 ). We have four points X, X 2, X3 and X 23 on the conic Ql, and two pairs of lines (C, c) and (B, b) through two pairs of these points each. Therefore, there exists J-ll E Cpl such that the conic Ql has the equation Ql = 0 with

Ql = J-l1Bb + Cc.


272

Similarly, the conic Q2 has the equation Q2 = 0 with Q2

= /-l2Aa + Cc.

Consider the conic Q1 - Q2 = /-llBb - /-l2Aa = O.

It belongs to the linear pencil of conics spanned by Q1 and Q2. Furthermore, the point Z = a n b lies on this conic. Therefore, the conic Q1 - Q2 must coincide with Q3, which has therefore the equation Q3 = 0 with

Q3 = /-ll Bb - /-l2 Aa.

Moreover, the two points Z2 = an Band Zl = b n A also lie on Q3. Since Z2 E B, we have B = (X2Z2). Similarly, since Zl E A, we have A = (Y1Zd. Finally, we find that the point Z12 = An B = (Y1Zd n (X 2Z 2) also lies on Q3, which is equivalent to (6.132). 0

6.16. Discrete integrable 3D systems The major part of this chapter has been devoted to the very rich theory of integrability of 2D equations, the root of which has been identified in their 3D consistency. In this last section we turn our attention to integrability of 3D systems, now understood as 4D consistency. The most striking feature is that the number of integrable systems drops dramatically with the growth of dimension: we know of only half a dozen of discrete 3D systems with the property of 4D consistency. All of them are of a geometric origin and in fact appeared already in Chapters 2, 3, and 4. We are going to briefly discuss their general algebraic features. In the 3D context, there are a priori many kinds of systems, according to where the fields are assigned: to the vertices, to the edges, or to the elementary squares of the cubic lattice. 6.16.1. Fields on 2-faces. Consider first the situation when the fields (assumed to take values in some space X) are assigned to the the elementary squares. Denote by a, b, c the fields attached to the 2-faces parallel to the coordinate planes 12, 13, 23, respectively. The system under consideration is a map F : X3 f---+ X3, which we write as F(a,b,c) = (T3a,T2b,T1C) = (a3, b2, cd. One can think of the fields a, b, c as sitting on the bottom, front, and left faces of a cube, and a3, b2 , q, on the top, back, and right faces. This is visualized in Figure 6.28. The concept of 4D consistency of such a map assumes that one can extend it to a four-dimensional square lattice. Thus, in addition to the fields a, b, c, there are fields d, e, f, attached to the 2-faces parallel to the coordinate planes 14, 24, 34, respectively. Initial data a, b, c, d, e, f are the

6.16. Discrete integrable 3D systems

273

c

b I

b2

I

/

/

}-----

a

/

Figure 6.28. 3D system on an elementary cube: a map with fields on 2-faces.

fields on six 2-faces of a 4D cube adjacent to one vertex. They allow one to apply the map F on four 3-faces of a 4D cube (the inner, bottom, front, and left ones in Figure 6.29): H23 : (a, b, c)

f---->

(a3, b2, cd,

H24 : (a, d, e)

f---->

(a4, d2, ed,

F 134 : (b, d, j)

f---->

(b 4, d3, h),

F234: (c,e,j)

f---->

(c4,e3,12).

Here Fijk denotes the map F acting on a 3-face of the coordinate directions 'B ijk . Now one can apply the map F on the other four 3-faces (the outer, top, back, and right ones):

F123 : (a4, b4, C4)

f---->

(a34, b24 , C14),

H24: (a3,d3,e3)

f---->

(a34,d23,e13),

F134 : (b2, d2, h)

f---->

(b24, d23, 112)'

F234 :

f---->

(C14, e13,

(Cl, el,

h)

h2).

Thus, one obtains two answers for each of the six fields a34, b24, C14, d23 , e13, h2, and the map F is 4D consistent if these pairs of answers identically coincide for all six fields and for all initial data. We mention here two examples of systems of the kind just discussed, both of geometric origin. The first is the discrete Darboux system which describes Q-nets in the affine setting; see Section 2.1.3. For this system, each 2-face of the coordinate direction 'B ij , i < j, carries a field consisting of a pair of real numbers hij"ji). The map is given by the formulas (6.133)

Tklij =

lij + liklkj , 1 -'jk,kj

k=l=i,j.

Theorem 6.41. (4D consistency of the discrete Darboux system) The discrete Darboux system (6.133) is 4D consistent.

The second example is the star-triangle map which describes T-nets; see Section 2.3.8. For this system, each 2-face of the coordinate direction 'Bij, i < j, carries just one real number aij, and the convention aij = -aji holds.

6. Cow;istency as Integrability

274

f

c

:b a "-----t . . . . . . . . . . . . . . .

c Cl..J

d

Figure 6.29. Initial data and results of two-stage application of a 4D consistent map with fields on 2-faces.

The map is given by the formulas (6.134)

Tkaij

aij

=aijajk

+ (ljkaki + (lkiaij

k

i- i. j.

A symmetric appearance of this formula is due to the above convention. If one would like to consider aij with i < j only, there would appear some minus signs in the denominator. Thus. in the index-free notation for the fields a = a12, b = a13, e = a23, used at the beginning of this section, the star-triangle map is written as (6.135)

F(a,b.e) = (a3,b 2,ed = (

a . b , e ). ab + be - ca ab + be - ea ab + be - ca

Theorem 6.42. (4D consistency of the star-triangle map) The startriangle map (6.135) is 4D consistent. As in the case of 2D systems (see Section 6.14). our definition of consistency cannot be written in terms of composition of maps. since each piece of the initial data is used simultaneously in two different maps. It turns out to be possible to change the initial value problem on a 4D cube in such a way that the resulting consistency condition can be formulated in terms of compositions. It is not difficult to realize that for this the initial data should be prescribed on six 2-faces (of all six two-dimensional coordinate directions) which form a surface topologically equivalent to a disk. Such a surface is depicted on the left in Figure 6.:30. One can apply to this initial surface two different sequences of flips of the kind depicted in Figure 6.28, both leading to the surface on the right in Figure 6.30. One sequence starts with flipping the inner 3-face. and then


275

a34

d e

q4 b

b24

C

f

a

h2

- - - - - - ...

e13 d 23

Figure 6.30. Initial data surface for a map satisfying the fUllctional tetrahedron equation, and the result of its four-fold flipping.

the top, front, and right ones. Denote the maps corresponding to these flips by 3 ij k; they are "companion maps" for the original F, i.e., they arise from F by regarding it along various diagonals of the basic cube. There appears a composition of maps:

3 123 : (a, b, c) e-t (a3, b2, cd, 3 134 : (b 2, d 2, f) e-t (hl, d 23 , h),

3 12 4 : (a3, d, e)

e-t

3 234 : (cl,el,h)

(a3.J,

e-t

d2 , ed,

(cl,t,e13,f12).

Another sequence starts with flipping the left 3-face, and proceeds with the back, bottom, and outer ones:

3 234 : (c,e,f) 3 124 : (a,d3,e3)

e-t

e-t

(c.l,e3,h),

(a4,d 23 ,e13),

3 134 : (b, d, h)

e-t

3 123 : (a4, b4, C4)

(b 4, d3, !I2),

e-t

(a34' b24 , C14).

The requirement that the two chains of maps lead to identical results can be thus encoded in the formula (6.136)

Definition 6.43. (Functional tetrahedron equation) A map 3 : X3 --> X3 'is said to satisfy the functional tetrahedron equation if (6.136) holds, where each 3 ijk is a map on X6(a, b, c, d, e, f) acting as 3 on the factors of the Cartesian product X6 corresponding to the variables sitting on the faces parallel to the planes ij, ik, j k, and acting tr-ivially on the other three factor-so Thus, we see that the concept of functional tetrahedron equation essentially coincides with the concept of 4D consistency of 3D systems with fields on 2-faces, the main difference lying in how the initial value problem is posed for the system at hand. It can be demonstrated (see Exercise 6.26)


276

that the 4D consistency of the star-triangle map (6.135) is translated into the following result.

Theorem 6.44. (Star-triangle solution of the functional tetrahedron equation) The map (6.137)

S(a, b, c) = (a3, b2, cd = (

ab b ,a + c - abc, be b) a+c-ac a+c-ac

satisfies the functional tetrahedron equation (6.136). The map (6.137) is related to the map (6.135) via conjugation by b 1---* 1/ b. One of the integrability features of 4D consistent maps (or, equivalently, of maps satisfying the functional tetrahedron equation) is a 3D analog of the zero curvature representation. For the map (6.137) it is discussed in Exercise 6.27. 6.16.2. Fields on vertices. Another version of 3D systems deals with fields assigned to vertices. In this case each elementary cube carries just one equation (6.138) relating the fields x E X in its eight vertices. Such an equation should be solvable for any of its eight arguments in terms of the other seven. This is shown in Figure 6.31.

X23 __- - - - - - {

Xl23

X3 e----i-----{.X13

I

X2 •

- - - - - - -

Xl2

/ / /

X - - - - - - - - { . Xl

Figure 6.31. 3D system on an elementary cube: an equation with fields on vertices.

The 4D consistency of such a system is defined as follows. Initial data on a 4D cube are 11 fields x, Xi (1 ::::: i ::::: 4), Xij (1 ::::: i < j ::::: 4). This data allow one to uniquely determine, by virtue of (6.138), all the fields Xijk (1 ::::: i < j < k ::::: 4). Then one has four different possibilities to find X1234, corresponding to the four 3-faces adjacent to the vertex Xl234 of the 4D cube; see Figure 6.32. If all four values coincide for any initial data, then


277

equation (6.138) is 4D consistent. For such systems, one can consistently impose equations (6.138) on all three-dimensional cubes of the lattice Z4.

X1234 1

X3

"

X134

1

"

1

~13

X2 ' 1

X

•

X12

--

," ',' X

.. ' • __ ~, __________ - ~ .. ' , , ' X24 "

X124

X14',

Figure 6.32. 4D consistency of a 3D system with fields on vertices.

The only examples of 4D consistent equations with one scalar field attached to each vertex we know are related to the star-triangle relation and appear through different factorizations of the face fields aij' Given a (complex-valued) solution aij of equation (6.134), the relations (6.139)

'Tkaij

'Tiajk

'Tjaki

aij

ajk

aki

yield the existence of a function (6.140)

Z :

zm -+ .) = L(Y1, >')L(x, >.) with the matrices L(x,>')=>'l+x. Can you derive this zero curvature representation, following the ideas of Theorems 6.4, 6.36? 6.21. Specializing the map (6.149), or, equivalently, (6.150) to the case when all fields belong to the algebra lHI of quaternions, show that the real parts of the quaternions x, X2 are equal, as well as the real parts of the quaternions y, Y1, and that the imaginary parts of the four quaternions build a (nonplanar) quadrilateral in ]R3 with opposite sides of equal length (Chebyshev quadrilateral). In other words, there exist (t, (3 E ]R and f, h, 12, f12 E ~lHI = su(2) c:::'. ]R3 such that

+ (h - J), = a1 + (112 - h),

x = (t1 X2

+ (12 - J), = (31 + (112 - h),

y = (31

Yl

with (6.151)

Ih2 -

121 =

Ih

-.n

Ih2 - hi

=

112 - fl·

Moreover, we have the relation (6.152)

h2 - h - 12

+f

=

1 (112 - J) x 2(a - (3)

(12 - h),

which fixes the proportionality coefficient between the vectors h2- ft and (112 - J) x (12 - h); these vectors are parallel due to

(112 - h - 12

12+ f

+ f, h2 - 1) = (112 - h - 12 + f, 12 - h) = 0,

which is equivalent to relations (6.151). The 3D consistency of map (6.149) yields the 3D consistency of equation (6.152) with ]R3- valued fields on vertices, if a, (3 is considered as a real-valued edge labelling.

6.17. Exercises

283

6.22. Check (by hand or with the help of a computer algebra system) that the maps from Theorem 6.38 satisfy the Yang-Baxter relation. 6.23. Construct zero curvature representations for the Yang-Baxter maps from Theorem 6.38, based on Theorem 6.36. 6.24. Consider a pencil of conics having a triple tangency point at the point (WI: W2 : W3) = (0 : 1 : 0) (in homogeneous coordinates on ClP'2). Conics of this pencil and their rational parametrization are given (in nonhomogeneous coordinates) by the formulas

Q(a): W2 -

wf - a

= 0,

X(x)

= (WI (x), W2(x)) =

Check that if QI = Q(a), Q2 = Q((3), then the map given in coordinates by Fv of Theorem 6.38.

(x, x 2 + a).

:.r defined

in (6.131) is

6.25. Consider a pencil of conics through four points 0= (0,0), (0,1), (1,0), (1, 1) E ClP'2, where nonhomogeneous coordinates (WI, W2) on the affine part C 2 of ClP'2 are used (any four points on ClP'2, no three of which lie on a straight line, can be brought into these four by a projective transformation). Conics of this pencil are described by the equation

Q(a): W2(W2 - 1) = aW1(W1 - 1). A rational parametrization of such a conic is given, e.g., by

x-a x(x-a)) X(x) = (WI(X), W2(x)) = ( 2 ' 2 . X -a x-a Here the parameter x has the interpretation of the slope of the line (OX). The values of x for the four points of the base locus of the pencil on Q(a) are x = a, 00, 0 and 1. Show that if Q1 = Q(a), Q2 = Q((3), then the map :.r defined in (6.131) is given in coordinates by H of Theorem 6.38. 6.26. The geometric content of the discrete Moutard equation Xij - x = aij (x j - Xi) is the parallelism of the lines (XXij) and (XiX j ). Therefore, there are in principle four ways to introduce the field aij as the proportionality coefficient between the two vectors under consideration: ±aij and ±1/aij. Prove that one can introduce the fields aij for six two-dimensional coordinate directions in £:;4 so that all four maps Sijk in (6.136) be given by the formulas (6.137). 6.27.* Check the following 3D analog of the zero curvature representation for the map (6.137):

L23(C)L13(b)LI2(a) = L12(a3)L13(b2)L23(cd, where

L12(a) =

(l~aa

I-a a

o

L13(b)

= (

-b 0

0

1 1+b 0

1-o b) , b


284

and L 23(C)

~G

o

-c l+c

Can you derive this representation? 6.28. Consider the system of linear equations

X2 - X = a(xI - x),

(6.153)

X3 - X = b(XI - x)

for a scalar-valued function x on Z3. Equations in (6.153), as well as the coefficients a, b, are naturally assigned to triangles; see Figure 6.33. Show

a3 . _---

x

X2

- -- -- --- - - - - -- -Xl2

Figure 6.33. Equations on triangles.

that the compatibility of these equations is assured as soon as the coefficients a, b satisfy the following equations: (6.154)

(a3 - l)(b - 1) = (b 2 - l)(a - 1),

which should be understood as a map (a, aI, b, bd f----+ (a3, b2)' Valid initial data for such a map can be prescribed on a surface shown in Figure 6.34.

Figure 6.34. Initial conditions for the system (6.154).

6.17. Exercises

285

Show that, due to the second equation in (6.154), there exists a scalar function 1 on Z3 such that a = hi h, b = hi h, and that this function solves the equation

h - 12 + 12 - h + h - h = o.

(6.155)

112

123

113

The function x on Z3 solves the multiratio equation, also known as the Schwarzian discrete KP equation: (Xl -

(6.156)

XI2)(X2 - X23)(X3 - X13) _

-1

(X12 - X2)(X23 - X3)(XI3 - xd -

.

Give a geometric interpretation of equations (6.153), (6.156) (hint: these equations encode a Menelaus configuration). Can you find a linear system similar to (6.153) which would generate the so-called bilinear octahedron equation (or bilinear Hirota equation, or discrete KP equation): (6.157) It is natural to call equations of the type (6.155), (6.156), (6.157), which do not involve the fields at the vertices X and X123 of an elementary cube, octahedron equations, as opposed to the general cube equations (6.138).

6.29. Octahedron equations (6.155), (6.156), (6.157) have a sort of 4D consistency property. One imposes such an equation for three 3D coordinate directions (ij4): X12

(6.158)

=

1(Xl,X2,X4,XI4,X24),

X13 = g(Xl, X3, X4, X14, X34), X23

=

h(x2, X3, X4, X24, X34).

Compared to the usua14D consistency of cube equations, the vertices x and Xij4 do not appear in this system, and only three equations are considered. Check that the following holds: equations (6.159)

Xl23

=

l(g, h, X34,

fI, h) = g(J, h, X24,j, h) =

h(J, g, X14,

j, fI),

are satisfied identically with respect to 11 independent variables (chosen as initial data):

In equations (6.159) the "hat" denotes the shift in the 4-th direction:

j = 74(J) =

1(xI4, X24,

X44,

X144, X244),

etc.

Verify also that for each of the above systems, an equation of the form (6.160)


286

holds. For instance, for the multiratio equation (6.156), (X14 -

X12)(X24 -

X23)(X34 -

X13) =

(X12 -

X24)(X23 -

X34)(X13 -

X14)

-1 .

6.18. Bibliographical notes Sections 6.1, 6.2: Continuous and discrete integrable systems. The theory of integrable systems (called also the theory of solitons) is a vast field in mathematical physics with huge literature. The focus of different publications in this area varies from algebraic geometry, enumerative topology, statistical physics, quantum groups and knot theory to applications in nonlinear optics, hydrodynamics and cosmology. We mention here a selection of mathematical monographs (in chronological order): Toda (1978), Novikov-Manakov-Pitaevskii-Zakharov (1980), Ablowitz-Segur (1981), Calogero-Degasperis (1982), Newell (1985), Faddeev-Takhtajan (1986), Ablowitz-Clarkson (1991), Dubrovin (1991), Matveev-Salle (1991), Hirota (1992), Korepin-Bogoliubov-Izergin (1992), Belokolos-Bobenko-Enol'skii-Its-Matveev (1994), Hitchin-Segal-Ward (1999), Kupershmidt (2000), Rogers-Schief (2002), Babelon-Bernard-Talon (2003), Reyman-SemenovTian-Shansky (2003), Suris (2003), Dubrovin-Krichever-Novikov (2004), Fokas-Its-Kapaev-Novokshenov (2006). Concerning the basic concrete example of these sections, the sine-Gordon equation: the Backlund transformation was found by Backlund (1884); the permutability theorem is due to Bianchi (1892). The zero curvature representation is due to Ablowitz-Kaup-Newell-Segur (1973) and Takhtajan (1974). The immersion formula for surfaces with constant negative Gaussian curvature in terms of the frame is in Sym (1985). The discretization (6.12) of the sine-Gordon equation along with its Backlund transformation is due to Hirota (1977b). The geometric meaning was uncovered in BobenkoPinkall (1996a). Sections 6.3, 6.4, 6.5: Integrable systems on graphs. Our presentation of the general theory of integrable systems on graphs follows BobenkoSuris (2002a). Examples of integrable systems on the regular triangular lattices were considered in Adler (2000), Bobenko-Hoffmann-Suris (2002) and Bobenko-Hoffmann (2003). The fundamental role of quad-graphs for discrete integrability was understood in Bobenko-Suris (2002a). A different framework for integrable systems on graphs was developed by Novikov with collaborators. In particular, the Laplace transformations on graphs were studied in Dynnikov-Novikov (1997), the theory of discrete Schrodinger operators on graphs was developed in Novikov (1999a, b ), and the scattering theory on trees is due to Krichever- Novikov (1999).


287

Sections 6.6, 6.7: From 3D consistency to zero curvature representations and Backlund transformations. The idea of consistency (or compatibility) is at the core of the theory of integrable systems. It appears already in the very definition of complete integrability of a Hamiltonian flow in the Liouville-Arnold sense, which says that the flow may be included into a complete family of commuting (compatible) Hamiltonian flows; see Arnold (1989). In the discrete context the (d + 1)-dimensional consistency of d-dimensional equations was observed many times. In the case d = 1 it was used as a possible definition of integrability of maps in Veselov (1991). A clear formulation in the case d = 2 was given in Nijhoff-Walker (2001). A decisive step was made in Bobenko-Suris (2002a) and independently in Nijhoff (2002): it was shown that the existence of a zero curvature representation follows for two-dimensional systems from the three-dimensional consistency. Section 6.8: Geometry of boundary value problems for integrable 2D equations. The discussion of the Cauchy problem on quad-graphs in Subsection 6.8.1 follows Adler-Veselov (2004). Embedding of quad-graphs into cubic lattices as a purely combinatorial problem was studied in a more general setting of arbitrary cubic complexes in Dolbilin-Stan'ko-Shtogrin (1986, 1994) and Shtan'ko-Shtogrin (1992). Theorem 6.8 is due to KenyonSchlenker (2004). The notion of the quasicrystallic rhombic embeddings and the extension to multi-dimensional lattices in Subsection 6.8.2 is due to Bobenko-:l\1ercat-Suris (2005). Note that intersections of 01) with bricks correspond to combinatorially convex subsets of 'D, as defined in Mercat (2004). Section 6.9: 3D consistent equations with noncommutative fields. The notion of 3D consistency in the noncommutative setup was introduced in Bobenko-Suris (2002b), where also the derivation of the zero curvature representation was given. Further examples due to Adler and Sokolov can be found in Adler-Bobenko-Suris (2007). The discrete Calapso equation (6.63) together with its zero curvature representation appeared in Schief (2001). There is a big literature on noncommutative integrable systems. One of the fundamental results in the theory of quantum integrable systems with discrete space-time is the quantization of the Hirota system by FaddeevVolkov (1994). A systematic exposition of noncommutative integrable systems is given in Kupershmidt (2000). References on discrete noncommutative systems include Matveev (2000), Nimmo (2006), Schief (2007). Sections 6.10, 6.11, 6.12: Classification of discrete integrable 2D systems with fields on vertices. The classification of discrete integrable 2D systems based on the notion of 3D consistency was given in

288


Adler-Bobenko-Suris (2003, 2007). The first of this papers deals with equations possessing the cubical symmetry and the tetrahedron property (Theorem 6.27). In Sections 6.10, 6.11 we present the classification of the second paper made under much weaker assumptions (Theorems 6.18, 6.19). Equations (H3)o=o and (HI) are perhaps the oldest in the lists; they can be found in the work of Hirota (1977a,b). Equations (Ql) and (Q3)o=o go back to Quispel-Nijhoff-Capel-Van der Linden (1984). Equation (Q4) was found in Adler (1998) (in the Weierstrass normalization of an elliptic curve). This equation in the Jacobi normalization is due to Hietarinta (2005). Equations (Q2), (Q3)o=1, (H2) and (H3)o=1 appeared explicitly for the first time in Adler-Bobenko-Suris (2003). The master equation (Q4) was investigated in Adler-Suris (2004), where its relation to various 2D integrable systems was revealed. Special solutions to this equation were found in Atkinson-Hietarinta-Nijhoff (2007). Symmetries of quad-equations from our lists were studied in Papageorgiou-TongasVeselov (2006) and Rasin-Hydon (2007). A 3D consistent equation without the tetrahedron property was found in Hietarinta (2004). This equation was shown to be linearizable by RamaniJoshi-Grammaticos-Tamizhmani (2006). Its geometric interpretation is given in Adler (2006). Section 6.13: Integrable discrete Laplace type equations. The relation of discrete (hyperbolic) systems on quad-graphs to Laplace type (elliptic) equations was discovered in Bobenko-Suris (2002). Examples of Laplace type equations on graphs previously appeared in Adler (2001). The threeleg forms of integrable quad-equations were found in Adler-Bobenko-Suris (2003) (with a formula for (Q4) in the Weierstrass normalization). In AdlerSuris (2004) the three-leg form of (Q4) was used to derive elliptic Toda systems on graphs. Section 6.14: Yang-Baxter maps were introduced in Drinfeld (1992) under the name of set-theoretical solutions of the Yang-Baxter equation. In Veselov (2003) the term "Yang-Baxter maps" was proposed instead of "set-theoretical solutions", and various notions of integrability were studied. In particular, commuting monodromy maps were constructed and zero curvature representations were discussed. A general construction of zero curvature representations (Theorem 6.36) was given subsequently in SurisVeselov (2003). A good survey on the topic is by Veselov (2007).

The map of Example 1 first appeared in Adler (1993). Example 2 is treated in Goncharenko-Veselov (2004) along with more general Yang-Baxter maps on Grassmannians. Example 3 is investigated in Noumi-Yamada (1998) and in Etingof (2003).


289

Section 6.15: Classification of Yang-Baxter maps. Quadrirational Yang-Baxter maps were introduced and classified in Adler-Bobenko-Suris (2004). On pencils of conics used in this classification one can read, for example, in Berger (1987). Section 6.16: Discrete integrable 3D systems. Various algebraic structures relevant for integrability of higher-dimensional discrete systems appeared in the literature. The role played in 2D by the zero curvature representation goes in 3D to the so-called local Yang-Baxter equation introduced in l'vlaillet-Nijhoff (1989). Several 3D systems possessing this structure were found in Kashaev (1996). The functional tetrahedron equation was introduced in Kashaev-Korepanov-Sergeev (1998) as one of the versions of the 4D consistency. Note that their notation is different from the one in formula (6.136): their indices 1 ::; i,j, k ::; 6 of Sijk numerate two-dimensional coordinate planes. This paper contains also a list of solutions of this equation possessing local Yang-Baxter representations with a certain Ansatz for the participating tensors. The discrete Darboux system was derived in Bogdanov-Konopelchenko (1995). The fact that the star-triangle map satisfies the functional tetrahedron equation was observed in Kashaev (1996). In discrete differential geometry the star-triangle map appeared in Konopelchenko-Schief (2002a). The discrete BKP equation goes back to Miwa (1982). Its double crossratio form is due to Nimmo-Schief (1997). Its 4D consistency was observed in Adler-Bobenko-Suris (2003). Theorem 6.47 is due to Tsarev-Wolf (2007). The first works on quantization of discrete differential geometry appeared recently. Quantum versions of the discrete Darboux system and its reduction for circular nets were investigated in Sergeev (2007) and BazhanovMangazeev-Sergeev (2008). A quantization of circle patterns is proposed in Bazhanov-Mangazeev-Sergeev (2007).

Section 6.17: Exercises. Ex. 6.8: This result is due to Adler-Sokolov; see Adler-Bobenko-Suris (2007). Ex. 6.9, 6.10, 6.11: See Adler-Bobenko-Suris (2003). Ex. 6.12: See Hietarinta (2004). Ex. 6.13: See Adler-Bobenko-Suris (2007). Ex. 6.14: See Adler-Bobenko-Suris (2007) and Atkinson (2008). Ex. 6.16: Unpublished result by Adler. Ex. 6.18: See Adler-Suris (2004). Ex. 6.19: In this generality the result seems to be new.

290


Ex. 6.20, 6.21: See Hoffmann (2008), Schief (2007), and Pinkall-Springborn-Weifimann (2007). Ex. 6.25: See Adler-Bobenko-Suris (2004). The map FI appeared also in a different context in Tongas-Tsoubelis-Xenitidis (2001). Ex. 6.27: See Kashaev-Korepanov-Sergeev (1998). Ex. 6.28: A related material can be found in Konopelchenko-Schief (2005). Ex. 6.29: From a work in progress with Adler.

Chapter 7

Discrete Complex Analysis. Linear Theory

7.1. Basic notions of discrete linear complex analysis Many constructions in discrete complex analysis are parallel to discrete differential geometry in the space of real dimension 2. Recall that a harmonic function U : ~2 c:::: C ---+ ~ is characterized by the relation 02U 02u b.u = ox2 + oy2 = o. A conjugate harmonic function v : ~2 c:::: C Riemann equations ov ou ov ox oy ox' Equivalently, j = u + iv : ~2 Cauchy-Riemann equation

c::::

C

---+

---+ ~

is defined by the Cauchy-

ou oy·

C is holomorphic, i.e., satisfies the

oj . oj oy = ~ ox· The real and the imaginary parts of a holomorphic function are harmonic, and any real-valued harmonic function can be considered as a real part of a holomorphic function. A standard classical way to discretize these notions is the following. A function u : 7!} ---+ ~ is called discrete harmonic if it satisfies the discrete Laplace equation

(b.u)m ,n =

Um+l ,n

+ Um-l ,n + Um,n+l + Um,n-l

-

4um,n = O.

-

291

7. Discrete Linear Complex Analysis

292

A natural domain of a conjugate discrete harmonic function v :

UZ 2 )*

---+ lR is the dual lattice; see Figure 7.1. The defining discrete Cauchy-Riemann

C(-- __ -yv I I I I

I I I I

U

0--- ---0 Figure 7.1. Regular square lattice and its dual.

equations read: Vm+l/2,n+I/2 - v m+I/2,n-I/2

Um+l,n - um,n,

Vm+l/2,n+I/2 - Vm-I/2,n-l/2

-(Um,n+1 - um,n),

with the natural indexing of the dual lattice; cf. Figure 7.2.

The corre-

9 VI I

t-----;-:

Uo ••

Vo 0- - - - - -0

--------(e*))

= 1.

e*Estar(yo;9*)

These conditions should be compared with conditions characterizing the angles 4> : E(9) U E(9*) ~ (0, n) of a rhombic embedding of a quad-graph '.D, which consist of (7.11) and

4>(e*) = 2n,

4>(e) = 2n,

(7.13) eEstar(xo;9)

e* Estar(y();S*)

for all Xo E V(9) and all Yo E V(9*). Thus, the integrability condition (7.12) says that the system of angles 4> : E(9) U E(9*) ~ (0, n) comes from

7.3. Integrable discrete Cauchy-Riemann equations

299

a realization of the quad-graph 'D as a rhombic ramified embedding in C. Flowers of such an embedding can wind around its vertices more than once. Another formulation of the integrability conditions is given in terms of the edges of the rhombic realizations.

Theorem 7.6. (Integrable Cauchy-Riemann equations in terms of rhombic edges) Integrability condition (7.9) for the weight function 1/ : E(9) U E(9*) -----; lR+ is equivalent to the following: there exists a labelling of directed edges of'D, (): E('D) -----; §l, such that, in the notation of Figure 7.4,

(7.14)

l/(Xo, Xl) =

1 ( 1/

Yo, YI

. ()o

) = ~ ()

0

- ()l

+ (). I

Under this condition, the 3D consistency of the discrete Cauchy-Riemann equations is assured by the following values of the weights 1/ on the diagonals of the vertical faces of D:

(7.15)

1/

( +) - . () - >.. x, y - ~ () + >..'

where () = ()(x, y), and>" E C is an arbitrary number which is interpreted as the label assigned to all vertical edges of D: >.. = ()(x, x+) = ()(y, y+).

So, integrable discrete Cauchy-Riemann equations can be written in a form with parameters assigned to directed edges of 'D:

(7.16)

f(yt) - f(yo) f(xI) - f(xo)

()l - ()o

()l

+ ()o'

where

and p : V(9) -----; C is a rhombic realization of the quad-graph 'D. Since ()l - ()o

()l

+ ()o

p(yt) - p(yo) p(xt) - p(xo) ,

we see that for a discrete holomorphic function f : V(9) -----; C, the quotient of diagonals of the f - image of any quadrilateral (xo, Yo, Xl, yt) E F ('D) is equal to the quotient of diagonals of the corresponding rhombus. A standard construction of zero curvature representation for 3D consistent equations, given in Theorem 6.4, leads in the present case to the following result.

Theorem 7.7. (Zero curvature representation of discrete CauchyRiemann equations) The discrete Cauchy-Riemann equations (7.16) admit a zero curvature representation with spectral parameter dependent 2 x 2

300


matrices along (x, y) E E ('D) given by (7.17)

L(y, x, ex;

A) = (A + () o

-2()(f(x) + f(y))) , A-()

where () = p(y) - p(x). Linearity of the discrete Cauchy-Riemann equations is reflected in the triangular structure of the transition matrices. Also, all constructions of Section 6.8 can be applied to integrable discrete Cauchy-Riemann equations. In particular, for weights coming from a quasicrystallic rhombic embedding of the quad-graph 'D, with labels 8 = {±()I, ... , ±()d}, discrete holomorphic functions can be extended from the corresponding surface D,]) C Zd to its hull, preserving discrete holomorphy. Here we have in mind the following natural definition: Definition 7.8. (Discrete holomorphic functions on Zd) A function f : Zd - t C is called discrete holomorphic if it satisfies, on each elementary square of Zd, the equation

f(n f(n

(7.18)

+ ej + ek) - f(n) + ej) - f(n + ek)

For discrete holomorphic functions in Zd, the transition matrices along the edges (n, n + ek) of Zd are given by

(7.19)

Lk(n;

A) = (A + ()k o

IJ,

-2()k(f(n + ek) A - ()k

+ f(n)))

.

All results of this section hold also in the case of generic complex weights which leads to () E C and to parallelogram realizations of 'D.

7.4. Discrete exponential functions An important class of discrete holomorphic functions is built by discrete exponential functions. We define them for an arbitrary rhombic embedding p : V('D) - t C. Fix a point Xo E V('D). For any other point x E V('D), choose some path {Cj }j=1 C E('D) connecting Xo to x, so that Cj = (Xj-I, Xj) and Xn = x. Let the slope of the j-th edge be ()j = p(Xj) - p(xj-d E §1. Then n z+(). e(x; z) = z_/ .

II

j=1

J

Clearly, this definition depends on the choice of the point Xo E V('D), but not on the path connecting Xo to x.

7.4. Discrete exponential functions

301

An extension of the discrete exponential function from n'D to the whole of Zd is given by the following simple formula: (7.20)

e(n;z)

=

+ fh)nk . rrd (Zz-fh k=l

The discrete Cauchy-Riemann equations for the discrete exponential function are easily checked: they are equivalent to a siq'lple identity

( Z + Bj z - Bj

.

z + Bk _ l)j(Z + Bj z - Bk z - Bj

_

z + BI) = Bj Z - ~k Bj

+ Bk -

.

Bk

At a given n E Zd, the discrete exponential function is rational with respect t? the parameter z, with poles at the ~. ElB l , ... , EdBd, where Ek = sIgn nk. / Equivalently, one can identify the discrete exponential function by its initial values on the axes:

z+Bk)n e(nek;z)= ( Z-Bk .

(7.21)

Another characterization says that e ( .; z) is the Backlund transformation of the zero solution of discrete Cauchy-Riemann equations on Zd, with the "vertical" parameter z. We now show that the discrete exponential functions form a basis in some natural class of functions (growing not faster than exponentially).

Theorem 7.9. (Discrete exponentials form a basis of discrete holomorphic functions) Let f be a discrete holomorphic function on V(1)) rv V(n'D), satisfying

'In E V(n'D),

(7.22)

with some C E R Extend it to a discrete holomorphic function on the hull 9{(V(n'D)). There exists a function g defined on the disjoint union of small neighborhoods around the points ±Bk E C and holomorphic on each of these neighborhoods, such that

(7.23)

1. f(n) - f(O) = -2

rg(.\)e(n; .\)d.\,

1n ir

where r is a collection of 2d small loops, each running counterclockwise around one of the points ±Bk.

Proof. The proof is constructive and consists of three steps. (i) Extend f from V(n'D) to 9{(V(n'D)); inequality (7.22) propagates in the extension process, if the constant C is chosen large enough.


302

(ii) Introduce the restrictions I~k) of I : 9-C(V(f22»))

-----t

C to the coor-

dinate axes:

(iii) Set g(A) = 'L%=1 (gk(A) + g-k(A)), where the functions g±k(A) vanish everywhere except in small neighborhoods of the points ±Ok, respectively, and are given there by convergent series (7.24)

1 ((k) gk(A) = 2A 11 - 1(0)

~(A-Ok)n (k) +~ A + Ok (fn+1

(k») - In-I) ,

and a similar formula for g-k(A). Formula (7.23) is then easily verified by computing the residues at A = ±Ok (see Exercise 7.5).

o It is important to observe that the data I~k), necessary for the construction of g(A), are not among the values of I on V(1)) rv V(f22») known initially, but are encoded in the extension process.

7.5. Discrete logarithmic function We now define the discrete logarithmic function on a rhombic quad-graph 1). Fix some point Xo E V(1)), and set

(7.25)

£(x) =

2~i

Ir 10;i

A) e(x; A)dA,

'\Ix

EV(1)).

Here the integration path r is the same as in Theorem 7.9, and fixing Xo is necessary for the definition of the discrete exponential function on 1). To make (7.25) a valid definition, one must specify a branch of 10g(A) in a neighborhood of each point ±Ok. This choice depends on x, and is done as follows. Assume, without loss of generality, that the circular order of the points ±Ok on the positively oriented unit circle §,1 is the following: 01, ... , 0d, -01, ... , -Od. We set Ok+d = -Ok for k = 1, ... , d, and then define Or for all r E Z by 2d-periodicity. For each r E Z, assign to Or = exp(i[r) E §,1 a certain value of the argument "Ir E JR.: choose a value "II of the argument of 01 arbitrarily, and then extend it according to the rule

"Ir+1-"Ir

E

(0,7f),

'\Ir E Z.

Clearly, "Ir+d = "Ir + 7f, and therefore also "Ir+2d = "Ir + 27f. It will be convenient to consider the points Or, supplied with the arguments "ITl as belonging to the Riemann surface A of the logarithmic function (a branched covering of the complex A-plane).

7.5. Discrete logarithmic function

303

For each Tn E Z, define the "sector" Urn on the plane C carrying the quad-graph 1> as the set of all points of V(1)) which can be reached from .TO along paths with all edges from {Om, ... , Om+d- d. Two sectors Unq and Um2 have a nonempty intersection if and only if ITnI - Tn21 < d. The union U = U:=-= Urn is a branched covering of the quad-graph 1>, and it serves as the domain of the discrete logarithmic function. The definition (7.25) of the latter should be read as follows: for x E E A. The integration path r consists of d small loops on A around these points, and arg('\) = S'log('\) takes values in a small open neighborhood (in JR) of the interval

Urn, the poles of e(x;'\) are exactly the points Om, ... ,Om+d-l

(7.26) of length less than 7f. If Tn increases by 2d, the interval (7.26) is shifted by 27f. As a consequence, the function e is discrete holomorphic, and its restriction to the set V(9) of "black" points is discrete harmonic everywhere on U except at the point .TO: (7.27) Thus, the functions gk in the integral representation (7.23) of an arbitrary discrete holomorphic function, defined originally in disjoint neighborhoods of the points Qr, in the case of the discrete logarithmic function are actually restrictions of a single analytic function log('\) / (2,\) to these neighborhoods. This allows one to deform the integration path r into a connected contour lying on a single leaf of the Riemann surface of the logarithm, and then use standard methods of complex analysis to obtain asymptotic expressions for the discrete logarithmic function. In particular, one can show that at the "black" points of V(9), (7.28)

€(x) '"" log Ix

- xol,

.T --) 00.

Properties (7.27), (7.28) characterize the discrete Green's function on Thus:

9.

Theorem 7.10. (Discrete Green's function) The discrete logarithmic function on 1>. restricted to the set of vertices V(9) of the "black" graph 9, coincides with discrete Green's function on 9. Now we extend the discrete logarithmic function to Zd, which will allow us to gain significant additional information about it. In addition to the unit vectors ek E Zd (corresponding to Ok E §l), we introduce their opposites ek+d = -ek, k E [1, d] (corresponding to Ok+d = -Ok), and define e r for all


304

r E Z by 2d-periodicity. Then m+d-l

8m =

(7.29)

EB

Ze r C Zd

r=m

is a d-dimensional octant containing exactly the part of n1) which is the P-image of the sector Um C 'D. Clearly, only 2d different octants appear among the 8 m (out of 2d possible d-dimensional octants). Define 8m as the octant ~m equipp!d with the interval (7.26) of values for 'Slog(Br ). By definition, 8 m1 and 8 m2 intersect if the underlying octants 8 m1 and 8 m2 have a nonempty intersection spanned by the common coordinate semi axes Ze r , and the 'S log( Br) for these common semiaxes match. It is easy to see that 8 m1 and 8 m2 intersect if and only if iml - m2i < d. The union 8 = U~=-OO 8m is a branched covering of the set U~=l 8 m C Zd. ~

~

Definition 7.11. (Disc~ete logarithmic function on Zd) The discrete logarithmic function on 8 is given by the formula (7.30)

£( ) = _1 n 27ri

( . A)dA irr10g(A) 2A e n, ,

'in E 8,

where for n E 8 m the integration path r consists of d loops around the points Bm, . .. , Bm+d- 1 on A, and'S 10g(A) on r is chosen in a small open neighborhood of the interval (7.26).

The discrete logarithmic function on 'D ca~ be described as the restriction of the discrete logarithmic function on 8 to a branched covering of n1) rv 'D. This holds for an arbitrary quasicrystallic quad-graph 'D whose set of edge slopes coincides with e = {±Bl, ... , ±Bd}. Now we are in a position to give an alternative definition of the discrete logarithmic function. Clearly, it is completely characterized by its values ~ner), r E [m, m + d -1], on the coordinate semiaxes of an arbitrary octant 8 m . Let us stress once more that the points ner do not lie, in general, on the original quad-surface n1). Theorem 7.12. (Values of discrete logarithmic function on coordinate axes) The values ~~) =~£(ner), r E [m,m + d - 1], of the discrete logarithmic function on 8 m C 8 are given by: (7.31)

£~) =

{

2(1+~+."'+ n~l)' 10g(Br)=Z"fr,

Here the values 10g(Br )

= i'"Yr

n even, n odd.

are chosen in the interval (7.26).

Proof. Comparing formula (7.30) with (7.24), we see that the values e~) can be obtained from the expansion of 10g(A) in a neighborhood of A = Br

7.5. Discrete logarithmic function

305

into the power series with respect to the powers of (A - Or) / (A + Or). This expansion reads:

~l-(-1)n(A-O )n n A+O:'

10g(A)=10g(Or)+~

Thus, we come to a simple difference equation n(e(r) _ e(r) ) n+l n-l

(7.32)

= 1 _ (_l)n ,

with the initial conditions (7.33)

£~r) = £(0) = 0,

o

which yield (7.31).

Observe that values (7.31) at even (resp. odd) points imitate the behavior of the real (resp. imaginary) part of the function 10g(A) along the half-lines arg(A) = arg(Or)' This can be easily extended to the whole of S. Restricted to the black points n E S (those with nl + ... + nd even), the discrete logarithmic function models the real part of the log~rithm. In particular, it is real-valued and does not branch: its values on 3 m depend on m (mod 2d) only. In other words, it is a well-defined function on 3 m . On the contrary, the discrete logarithmic function restricted to the white points n E S (those with nl + ... + nd odd) takes purely imaginary values, and increases by 27fi as m increases by 2d. Hence, this restricted function models the imaginary part of the logarithm. It turns out that recurrence relations (7.32) are characteristic for an important class of solutions of the discrete Cauchy-Riemann equations, namely for the isomonodromic solutions. In order to introduce this class, recall that discrete holomorphic functions in Zd possess a zero curvature representation with transition matrices (7.19). The moving frame we, A) : Zd --> GL(2, C) [A] is defined by prescribing some w(O; A), and by extending it recurrently according to the formula

(7.34) Finally, define the matrices A(-; A) : Zd

(7.35)

A(n; A) =

-->

gl(2, C)[A] by

dW~~; A) w- 1 (n; A).

These matrices satisfy a recurrence relation, which is obtained by differentiating (7.34),

(7.36)

A(n + ek; A) =

dLk(n; A) -1 -1 dA Lk (n; A) + Lk(n; A)A(n; A)Lk (n; A),

and therefore they are determined uniquely upon fixing some A(O; A).


306

Definition 7.13. (Isomonodromy) A discrete holomorphic function f : Zd -+ C is called isomonodromic if, for some choice of A(O; .\), the matrices A(n;.\) are meromorphic in .\, with poles whose positions and orders do not depend on n E Zd. This term originates in the theory of integrable nonlinear differential equations, where it is used for solutions with a similar analytic characterization.

It is clear how to extend Definition 7.13 to functions on the covering 5. In the following statement, we restrict ourselves to the octant 51 = (Z+)d for notational simplicity.

Theorem 7.14. (Discrete logarithmic function is isomonodromic) For a proper choice of A(O; .\), the matrices A(n;.\) at any point n E (Z+)d have simple poles only: . _ A(O)(n) A(n,.\) .\

(7.37)

~ (B(l)(n)

+L

.\

1=1

with

(~

(7.39)

n, (~ C(l)(n)

~

'"

C(l)(n))

+ .\ _ ()

I

,

I

(-I)n;: +"'),

(7.38)

(7.40)

+

()

-(f(n)

(~

f(n

+;(n -

e,))) .

+ e,; H(n))

.

At any point n E 5, the following constraint holds: d

(7.41)

Lnl(e(n+q) -£(n-e/))

=

1- (-lt 1 +··+ nd •

/=1

Proof. The proper choice of A(O;.\) mentioned in the Theorem, can be read off formula (7.38): A(O;.\)

=

1(0 1)

>:

0 0 .

The proof consists of two parts. (i) First, one proves the claim for the points of the coordinate semiaxes. For any r = 1, ... , d, construct the matrices A(ne r ;.\) along

7.6. Exercises

307

the r-th coordinate semi-axis via formula (7.36) with transition matrices (7.19). This formula shows that the singularities of A(ner ; A) are poles at A = 0 and at A = ±()r, and that the pole A = 0 remains simple for all n > O. By a direct computation and induction, one shows that it is exactly the recurrence relation (7.32) for fAr) = f(ne r ) which assures that the poles A = ±()r remain simple for all n > O. Thus, (7.37) holds on the r-th coordinate semiaxis, with B(l)(ne r ) = C(l) (ne r ) = 0 for I =1= r.

(ii) The second part of the proof is conceptual, and is based upon the multidimensional consistency only. Proceed by induction, filling out the hull of the coordinate semiaxes: each new point is of the form n + ej + ek, j =1= k, with three points n, n + ej, and n + ek known from the previous steps, where the statements of the proposition are assumed to hold. Suppose that (7.37) holds at n + ej, n + ek. The new matrix A(n + ej + ek; A) is obtained by two alternative formulas, (7.42)

A(n+ej+ek;A)=

dLk(n + ej; A) -1 dA Lk (n+ej;A)

+Lk(n + ej; A)A(n + ej; A)L};l(n + ej; A),

and the other with k and j interchanged. Equation (7.42) shows that all poles of A(n + ej + ek; A) remain simple, with the possible exception of A = ±()k, whose orders might increase by 1. The same statement holds with k replaced by j. Therefore, all poles remain simple, and (7.37) holds at n + ej + ek. Formulas (7.38)-(7.40) and constraint (7.41) follow by direct computations based on (7.42). D

7.6. Exercises 7.1. Let 'D be a bipartite quad-graph, with black vertices Xj and white vertices Yj. Let J-l : E('D) -> C be a function such that, for any elementary quadrilateral (xo, Yo, Xl, yd E F('D), J-l(Xo, YO)J-l(Xl, yd = J-l(Xo, Yl)J-l(Xl, yo).

Show that there exists a function () : V('D) -> C such that for every edge (x, y) E E('D) we have iJ-l(x, y) = ()(y)/()(x). If J-l is real-valued, then one can assume that () takes real values at black points and imaginary values at white points.

7.2. Prove by induction that the entries of the matrix


308

are given by

A=

~(II(1+ivk)+ II(l-ivk)), k

k

B=

~i(II(1+ivk)- II(1-ivk))' k

k

7.3. Check that the function f : 7i} ----+ C given by f(m, n) = (m(h satisfies the discrete Cauchy-Riemann equation

f(m f(m

+ 1, n + 1) - f(m, n) + 1, n) - f(m, n + 1)

+ n(h)2

(h + (h 01 - O2'

Generalize this function ("discrete z2,,) for 7l,d and for arbitrary quad-graphs 1).

7.4. Find the "discrete z3", i.e., the function f : 7l,2 ----+ C which is polynomial in m, n of degree 3, with cubic terms (mOl +n02)3, and satisfying the discrete Cauchy-Riemann equations.

7.5. Prove that for the functions 9k(>') from (7.24),

>. + Ok)n _ (k) Res,X=li k ( >. _ Ok 9k(>') - fn - f(O). 7.6. Estimate the difference £~k) -logn for the values given in (7.31), for n even. 7.7. Bibliographical notes Section 7.1: Basic notions of discrete linear complex analysis. The standard discretization of harmonic and holomorphic functions on the regular square grid goes back to Ferrand (1944) and Duffin (1956). This discretization of the Cauchy-Riemann equations apparently preserves the majority of important structural features. A pioneering step in the direction of further generalization of the notions of discrete harmonic and discrete holomorphic functions was undertaken by Duffin (1968), where the combinatorics of 7l,2 was given up in favor of arbitrary planar graphs with rhombic faces. A far reaching generalization of these ideas was given by Mercat (2001), who extended the theory to discrete Riemann surfaces. Section 7.2: Moutard transformation for discrete Cauchy-Riemann equations. For general Moutard transformations see the bibliographical note to Section 2.3 and Exercise 2.27. A further discussion of the Darboux transformation for discrete Laplace operators induced by the Moutard transformation for discrete Cauchy-Riemann equations can be found in DoliwaGrinevich-Nieszporski-Santini (2007). Section 7.3: Integrable discrete Cauchy-Riemann equations. Condition (7.13) on the system of angles ¢ : E(9) U E(9*) ----+ (0,71') characterizing rhombic embedding was given in Kenyon-Schlenker (2004). Theorems


309

7.5, 7.6 characterizing 3D consistent (integrable) Cauchy-Riemann equations and their zero curvature representation from Theorem 7.7 are from Bobenko-Mercat-Suris (2005). Section 7.4: Discrete exponential functions. A discrete exponential function on 7!} was defined and studied in Ferrand (1944) and Duffin (1956). It was generalized for quad-graphs 'D in Mercat (2001) and Kenyon (2002). The question whether discrete exponential functions form a basis in the space of discrete holomorphic functions on 'D (Theorem 7.9) was posed in Kenyon (2002) and answered in Bobenko-Mercat-Suris (2005). Section 7.5: Discrete logarithmic function. The discrete logarithmic function on a rhombic quad-graph 'D was introduced in Kenyon (2002). Also the asymptotics (7.28) as well as Theorem 7.10 were proven in that paper. All other results in this section, starting with the extension of the discrete logarithmic function to Zd, are from Bobenko-Mercat-Suris (2005). For the theory of isomonodromic solutions of differential equations and its application to integrable systems see Fokas-Its-Kapaev-Novokshenov (2006). Isomonodromic constraint (7.41) was found in Nijhoff-RamaniGrammaticos-Ohta (2001), with no relation to the discrete logarithmic function.

Chapter 8

Discrete Complex Analysis. Integrable Circle Patterns

8.1. Circle patterns The idea that circle packings and, more generally, circle patterns serve as a discrete counterpart of analytic functions is by now well established. We give here a presentation of several results in this area, which treat the interrelations between circle patterns and integrable systems.

Definition 8.1. (Circle pattern) Let 9 be an arbitrary cell decompos'ition of an open or closed disk in C. A map Z : V(9) ---> C defines a circle pattern with combinatorics of 9 if the following condition is satisfied. Let y E F(9) '" V(9*) be an arbitrary face of 9, and let Xl, X2, . .. , Xn be its consecutive vertices. Then the points Z(XI), Z(X2), ... , z(xn) E C lie on a circle, and their circular order is just the listed one. We denote this circle by C(y), thus putting it into a correspondence with the face y, or, equivalently, with the respective vertex of the dual cell decomposition 9*.

As a consequence of this condition, if two faces yo, YI E F(9) have a common edge (xo. :rd, then the circles C(Yo) and C(yI) intersect in the points z(:r:t), Z(X2)' In other words, the edges from E(9) correspond to pairs of neighboring (intersecting) circles of the pattern. Similarly, if several faces Yl, Y2, ... , Ym E F (9) meet in one point Xo E V (9), then the corresponding circles C (Yl ), C (Y2), ... , C (Ym) also have a cornman intersection point Z (xo). A finite piece of a circle pattern is shown in Figure 8.1.

-

311

312

8. Integrable Circle Patterns

Figure 8.1. Circle pattern.

Given a circle pattern with combinatorics of 9, we can extend the function z to the vertices of the dual graph, setting

z(y) = center of the circle C(y),

Y E F(9) c::: V(9*).

After this extension, the map z is defined on all of VeD) = V(9) U V(9*), where']) is the double of 9. Consider a face of the double. Its z-image is a quadrilateral of the kite form, whose vertices correspond to the intersection points and the centers of two neighboring circles Co, C 1 of the pattern. Denote the radii of Co, C 1 by TO, Tl, respectively. Let xo, Xl correspond to the intersection points, and let Yo, Y1 correspond to the centers of the circles. Give the circles Co, C 1 a positive orientation (induced by the orientation of the underlying q, and let ¢ E (0, 7f) stand for the intersection angle of these oriented circles. This angle ¢ is equal to the kite angles at the "black" vertices z(xo), Z(X1); see Figure 8.2, where the complementary angle ¢* = 7f - ¢ is also shown. It will be convenient to assign the intersection angle ¢ = ¢( e) to the "black" edge e = (xo, xd E E(9), and to assign the complementary angle ¢* = ¢(e*) to the dual "white" edge e* = (Yo, yt) E E(9*). Thus, the function ¢ : E(9) U E(9*) -----7 (0,7f) satisfies (7.11). The geometry of Figure 8.2 yields following relations. First of all, the cross-ratio of the four points corresponding to the vertices of a quadrilateral face of']) is expressed through the intersection angle of the circles Co, C 1 :

(8.1)

q(z(xo), z(yo), z(xd, Z(Y1)) = exp(2i¢*).

8.2. Integrable cross-ratio and Hirota systems

313

Co

Figure 8.2. Two intersecting circles.

Furthermore, running around a "black" vertex of 'D (a common intersection point of several circles of the pattern), we see that the sum of the consecutive kite angles vanishes (mod 21f), hence:

rr

(8.2)

exp(i¢(e)) = 1,

Vxo

E

V(9).

eEstar(xo;9)

Finally, let 1/;01 be the angle of the kite (z(xo), z(Yo), Z(Xl), z(yI)) at the "white" vertex z(yo), i.e., the angle between the half-lines from the center z(yo) of the circle Co to the intersection points z(xo), z(xI) with its circle C 1 . It is not difficult to calculate this angle:

'oi,) TO+Tlexp(i¢*) exp (to/Ol = . TO + Tl exp( -i¢*)

(8.3)

Running around the "white" vertex of 'D, we come to the relation

(8.4)

rr rn

" )=1

TO

+

Tj

exp(i¢j)

-----,----"---,- = 1 TO + T)" exp( -i¢)~) ,

VyO

E

V(9*),

where the product is extended over all edges ej = (Yo, Yj) E star(yo; 9*), and ¢j = ¢(ej), while Tj are the radii of the circles Cj = C(Yj).

8.2. Integrable cross-ratio and Hirota systems Our main interest is in the circle patterns with prescribed combinatorics and with prescribed intersection angles for all pairs of neighboring angles. According to formula (8.1), prescribing all intersection angles amounts to prescribing cross-ratios for all quadrilateral faces of the quad-graph 'D. Thus, we come to the study of cross-ratio equations on arbitrary quad-graphs.


314

Let there be given a function Q condition

(8.5)

E(9)

Q(e*) = l/Q(e),

u E(9*)

----t

C satisfying the

\Ie E E(9).

Definition 8.2. (Cross-ratio system) The cross-ratio system on 1> corresponding to the function Q consists of the following equations for a function z: V(1)) ----t C, one for any quadrilateral face (:ro,yo,:rl,yd of1>:

(8.6)

q(z(xo), z(Yo), z(xt), Z(Yl))

= Q(xo, xd = l/Q(yo, yd·

An important distinction from the discrete Cauchy-Riemann equations is that the cross-ratio equations actually do not depend on the orientation of quadrilaterals. We have already encountered 3D consistent cross-ratio systems on Zd in Section 6.7 (see equation (6.33)), in the version with labelled edges. A natural generalization to the case of arbitrary quad-graphs is this: Yl

Xo

Yo Figure 8.3. Quadrilateral, with a labelling of undirected edges.

Definition 8.3. (Integrable cross-ratio system) A cross-ratio system is called integrable if there exists a labelling 0: : E(1)) ----t C of undirected edges of 1) such that the function Q admits the follow'ing factorization (in the notation of Figure 8.3): (8.7) Clearly, integrable cross-ratio systems are 3D consistent (see Theorem 4.26), admit Backlund transformations, and possess zero curvature r-epr-esentation with the transition matrices (6.47). It is not difficult to give an equivalent reformulation of the integrability condition (8.7). Theorem 8.4. (Integrability condition of a cross-ratio system) A cross-ratio system with the function Q : E(9) U E(9*) ----t C is integrable if

315

8.2. Integrable cross-ratio and Hirota systems

and only if for all Xo are fulfilled: (8.8)

E

V(9) and for all Yo

II

E

II

Q(e) = 1,

eEstar(xo;9)

V(9*) the following conditions Q(e*)

= 1.

e*Estar(yo;9*)

For a labelling of undirected edges a : E('D) --t C, we can find a labelling {} : E('D) --t C of directed edges such that a = (}2. The function P : V('D) --t C defined by p(y) - p(x) = {}(x, y) gives, according to (8.8), a parallelogram realization (ramified embedding) of the quad-graph 'D. The cross-ratio equations are written as (}2

(8.9)

q(z(xo), z(yo), z(xd, Z(Yl)) =

(}g = q(p(xo),p(Yo),p(Xd,p(Yl)); 1

in other words, for any quadrilateral (xo, Yo, Xl, Yl) E F('D), the cross-ratio of the vertices of its image under the map z is equal to the cross-ratio of the vertices of the corresponding parallelogram. In particular, one always has the trivial solution z(x) == p(x) for all x E V('D). A very useful transformation of the cross-ratio system is given by the following construction.

Definition 8.5. (Hirota system) For a given labelling of directed edges {} : E('D) --t C, the Nirota system consists of the following equations for the function W : V('D) --t C, one for every quadrilateral face (xo, Yo, Xl, Yl) E F('D):

(8.10) (}ow(xo)w(yo)

+ (}lW(YO)W(Xl)

- {}ow(xdw(Yd - (}lW(Yl)W(XO)

= o.

Note that the Hirota equation coincides with equation (6.30) of Section 6.7 (by the way, this shows that also in that previous version it was natural to assign parameters to directed edges). In terms of the parallelogram realization p : V('D) --t C of the quad-graph 'D corresponding to the labelling {}, equation (8.10) reads: (8.11)

+ W(YO)W(Xl) (p(xd - p(yo)) P(Xl)) + W(Yl)W(XO) (p(xo) - p(Yl))

W(xo)W(yo) (p(yo) - p(xo)) +w(xl)w(yd(p(yd -

= O.

Obviously, a transformation W f---> cw on V(9) and W f---> c-lw on V(9*) with a constant c E C, hereafter called a black-white scaling, maps solutions of the Hirota system into solutions. A relation between the cross-ratio and the Hirota system is based on the following observation:

Theorem 8.6. (Relation between cross-ratio and Hirota systems) Let W : V('D) --t C be a solution of the Hirota system. Then the relation (8.12)

z(y) - z(x) = (}(x,y)w(x)w(y) = W(X)W(y) (p(y) - p(x))


316

for all directed edges (x, y) E E('D) defines a unique (up to an additive constant) function z : V(1)) ----+ C which is a solution of the cross-ratio system (8.9). Conversely, for any solution z of the cross-ratio system (8.9), relation (8.12) defines a unique (up to a black-white scaling) function w : V(1)) ----+ C; this function w solves the Hirota system (8.10).

In particular, the trivial solution z(x) = p(x) of the cross-ratio system corresponds to the trivial solution of the Hirota system, w(x) == 1 for all x E V(1)). By a direct computation one can establish the following fundamental property.

Theorem 8.7. (Integrability of Hirota system) The Hirota system (8.10) is 3D consistent. As a usual consequence, the Hirota system admits Backlund transformations and possesses zero curvature representation with transition matrices along the edge (x, y) E E(1)) given by (8.13)

L(y, x, (); >.) = (

1 ->.()/w(x)

-()w(y) ) w(y)/w(x)

,

where () = p(y) - p(x).

8.3. Integrable circle patterns Returning to circle patterns, let {z(x) : x E V(9)} be the intersection points of the circles of a pattern, and let {z(y) : y E V(9*)} be their centers. Due to (8.1), the function z : V(1)) ----+ C satisfies a cross-ratio system with Q : E(9) U E(9*) ----+ §1 defined as Q(e) = exp(2i¢(e)). Because of (8.2), the first of the integrability conditions (8.8) is fulfilled for an arbitrary circle pattern. Therefore, integrability of the cross-ratio system for circle patterns with prescribed intersection angles ¢ : E(9*) ----+ (0,7f) is equivalent to (8.14)

II

exp(2i¢(e*))

= 1,

Vyo E V(9*).

e* Estar(yo; 9*)

This is equivalent to the existence of the edge labelling a : E(1)) that, in the notation of Figure 8.2, (8.15)

.

exp(2~¢*)

----+

C such

ao

= -. al

Moreover, one can assume that the labelling a takes values in §l. Our definition of integrable circle patterns will require somewhat more than integrability of the corresponding cross-ratio system.

8.3. Integrable circle patterns

317

Definition 8.8. (Integrable circle pattern) A circle pattern with prescribed intersection angles ¢ : E(9*) ---> (0,7r) is called integrable if

II

(8.16)

exp(i¢(e*))

= 1,

\:Iyo E V(9*),

e* Estar(yo; 9*)

i. e., if for any circle of the pattern the sum of its intersection angles with all neighboring circles vanishes (mod 27r).

This requirement is equivalent to a somewhat sharper factorization than (8.15), namely, to the existence of a labelling of the directed edges () : £('D) ---> §l such that, in the notation of Figure 8.2, (8.17)

exp( i¢) =

()l ()o

.A.*) = -

exp (Z'I'

()o

()l .

(Of course, the last condition yields (8.15) with 0: = ()2.) The parallelogram realization p : V(1)) ---> CC corresponding to the labelling () E §l is actually a rhombic one.

Theorem 8.9. (Isoradial integrability criterion) Combinatorial data 9 and intersection angles ¢ : E(9) ---> (0, 7r) belong to an integrable circle pattern if and only if they admit an isoradial realization. In this case, the dual combinatorial data 9* and intersection angles ¢ : E(9*) ---> (0, 7r) admit a realization as an isoradial circle pattern, as well. Proof. The rhombic realization p : V(1)) ---> CC of the quad-graph 1) corresponds to a circle pattern with the same combinatorics and the same intersection angles as the original one and with all radii equal to 1, and, 0 simultaneously, to an analogous dual circle pattern. Consider a rhombic realization p : V(1)) ---> CC of 1>. Solutions z : ---> CC of the corresponding integrable cross-ratio system which come from integrable circle patterns are characterized by the property that the z-image of any quadrilateral (xo, Yo, Xl, YI) from F(1)) is a kite with the prescribed angle ¢ at the black vertices z(xo), Z(Xl) (cf. Figure 8.2). It turns out that the description of this class of kite solutions admits a more convenient analytic characterization in terms of the corresponding solutions w : V('D) ---> CC of the Hirota system defined by (8.12).

V(1))

Theorem 8.10. (Circle pattern solutions of Hirota system) The solution z of the cross-ratio system corresponds to a circle pattern if and only if the solution w of the Hirota system, corresponding to z via (8.12), satisfies the condition (8.18)

w(X) E

§1,

w(y) E lR+,

\:Ix E V(9), y E V(9*).


318

The values w(y) E lR+ have then the interpretation of the radii of the circles C(y), while the (arguments of the) values w(x) E §l measure the rotation of the tangents to the circles intersecting at z (x) with respect to the isoradial realization of the pattern. Proof. As is easily seen, the kite conditions are equivalent to

lw(xo)l = 1 and Iw(xI}1

o

This yields (8.18), possibly upon a black-white scaling.

The conditions (8.18) form an admissible reduction of the Hirota system with () E §l, in the following sense: if any three of the four points w(xo), w(yo), w(xI}, W(Yl) satisfy the condition (8.18), then so does the fourth one. This is immediately seen, if one rewrites the Hirota equation (8.10) in one of the two equivalent forms: (8.19)

w(xI} w(xo)

w(yI) w(yo)

(}lW(Yl) - (}ow(yo) (}lW(YO) - (}ow(yI)

(}ow(xo) (}ow(xI)

+ (}lW(XI) + (}l w(xo)"

As a consequence of this remark, we obtain Backlund transformations for integrable circle patterns.

Theorem 8.11. (Backlund transformations of integrable circle patterns) Let all () E §l, and let p : V ('1» --- O

2a

This function is called the discrete logarithmic function; it should not be confused with the namesake function £(n) in the linear theory (Section 7.5). From (8.42) the following characterization is found: the discreteJogarithmic function L is the solution of the discrete cross-ratio system on S defined by


324

the values on the coordinate semiaxes L};) = L(ne r ), r E [m,m which solve the recurrence relation (Ln+l - Ln)(Ln - Ln-d 1 (8.43) n = Ln+1 - Ln- 1 2 with the initial conditions

Lg·) = L(O) =

(8.44)

+d -

1],

00,

where log er is chosen in the interval (7.26). Explicit expressions: (8.45)

(r)

L2n = loger

~1 1 + ~ k + 2n '

(r)

_

L2n+1 -loger

~1 + ~ k·

k=l

k=l

L

Theorem 8.17. (Circle pattern logarithm is isomonodromic The discrete logarithm is isomonodromic and satisfies, at any point n E S, the following constraint:

L nj (L(n + ej) d

(8.46)

L(n))(L(n) - L(n - ej)) = ~. L(n + ej) - L(n - ej) 2

j=l

By restriction to quad-surfaces On, we come to the discrete logarithmic function on arbitrary quasicrystallic quad-graphs 'D. By construction, they all correspond to circle patterns. A conjecture that these circle patterns are embedded seems plausible (see Figure 8.5).

Figure 8.5. Discrete logarithm circle patterns with combinatorics of the regular square and hexagonal lattices.

8.5. Linearization Let e : E('D) ---t C be an edge labelling, and let p : V('D) ---t C be the corresponding parallelogram realization of'D defined by p(y)-p(x) = e(x, y). Consider the trivial solutions zo(x) = p(x),

wo(x) = 1,

Vx E V('D)

8.5. Linearization

325

of the cross-ratio system (8.9) and the corresponding Hirota system (8.11). Suppose that Zo : V('D) ~ C belongs to a differentiable one-parameter family of solutions ZE : V('D) ~ C, E E (-EO, EO), of the same cross-ratio system, and denote by WE : V('D) ~ C the corresponding solutions of the Hirota system. Denote (8.47)

g -dZ-E I - dE E=O'

f =

(W;l dWE) dE

. E=O

Theorem 8.18. (Discrete derivative for discrete holomorphic functions) Both functions f,g : V('D) ~ C solve discrete Cauchy-Riemann equations (7.16). Proof. By differentiating (8.12), we obtain a relation between the functions f, 9 : V('D) ~ C:

(8.48)

g(y) - g(x) = (J(x)

+ f(y)) (p(y) - p(x)),

\i(x, y)

E

E('D).

The proof of the theorem is based on this relation solely. Indeed, the exactness condition for the form on the right-hand side on an elementary quadrilateral reads

(J(xo) + f(yo)) (p(yo) - p(xo)) + (J(yo) + f(Xl)) (p(Xl) - p(yo)) +(J(Xl) + f(Yl)) (p(Yl) - P(Xl)) + (J(Yl) + f(xo)) (p(xo) - p(Yl)) which is equivalent to (7.16) for the function condition for f, that is,

= 0,

f. Similarly, the exactness

(J(xo) + f(yo)) - (J(yo) + f(Xl)) + (J(xI) + f(Yl)) - (J(yI) + f(xo)) = 0, yields

g(yo) - g(xo) _ g(xI) - g(yo) p(yo) - p(xo) p(xI) - p(yo)

+ g(Yl) - g(xI) _ g(xo) - g(yI) p(yI) - p(xI)

= 0.

p(xo) - p(Yl)

Under the conditionp(yo)-p(xo) = p(Xl)-p(Yl), this is equivalent to (7.16) for g. D Remark. This proof shows that, given a discrete holomorphic function f : V('D) ~ C, relation (8.48) correctly defines a unique, up to an additive constant, function 9 : V('D) ~ C, which is also discrete holomorphic. Conversely, for any 9 satisfying the discrete Cauchy-Riemann equations (7.16), relation (8.48) defines a function f uniquely (up to an additive black-white constant); this function f also solves the discrete Cauchy-Riemann equations (7.16). Actually, formula (8.48) expresses that the discrete holomorphic function f is the discrete derivative of g, and so 9 is obtained from f by discrete integration. Summarizing, we have the following statement.


326

Theorem 8.19. (Linearization of circle patterns) a) A tangent space to the set of solutions of an integrable cross-ratio system, at a point corresponding to a rhombic embedding of a quad-graph, consists of discrete holomorphic functions on this embedding. This holds in both descriptions of the above set: in terms of variables z satisfying the crossratio equations, and in terms of variables w satisfying the Hirota equations. The corresponding two descriptions of the tangent space are related via the discrete derivative (resp. antiderivative) of discrete holomorphic functions. b) A tangent space to the set of integrable circle patterns of a given combinatorics, at a point corresponding to an isoradial pattern, consists of discrete holomorphic functions on the rhombic embedding of the corresponding quad-graph, which take real values at white vertices and pure imaginary values at black ones. This holds in the description of circle patterns in terms of circle radii and rotation angles at intersection points (Hirota system). A spectacular example of this linearization property is delivered by the isomonodromic discrete logarithm studied in Section 7.5 and isomonodromic z2a circle patterns of Section 8.4.

Theorem 8.20. (Linearization of w 2a - 1 circle patterns is the discrete logarithm) The tangent vector to the space of integrable circle patterns along the curve consisting of patterns w 2a - 1 , at the isoradial point corresponding to a = 1/2, is the discrete logarithmic function f defined in Section 7.5. Proof. We have to prove that the discrete logarithm f and the discrete power function w 2a - 1 are related by

f(n) =

(~~ w 2a - 1 (n)) 2 da

. a=1/2

Due to Theorem 8.18, it is enough to prove this for the initial data on the coordinate semiaxes. But this follows by differentiating with respect to a the initial values (8.31) at the point a = 1/2, where all w = 1: the result coincides with (7.31). 0

8.6. Exercises 8.1. Check that formulas (8.30), (8.31) give solutions to the corresponding difference equations (8.26), (8.28).

8.2. Prove asymptotic relations (8.32), (8.33). 8.3. Fill in the details of the proofs of Theorems 8.15, 8.16.

8.7. Bibliographical notel:i

327

8.4. For every solution z : 'lL d ---t C of the cross-ratio system (8.21), define the dual solution z* : 'lL d ---t C by z*(n+ej)-z*(n)= ( . z n

(]2 + ej) - z (n)

The dual solution is defined uniquely up to translation, and this freedom can be fixed by prescribing z*(O). Show that for a E (0,1) the dual solution to the discrete z2a, normalized to vanish at n = 0, coincides with the discrete z2(I-al.

8.5. Show that the limit a ---t 1 in Definition 8.12 leads to the discrete z2 as a solution of the cross-ratio equation, satisfying the recurrence relations (8.26) with a = 1 on the coordinate semiaxes, and with the initial data 2

z(O) = 0,

z(ej)

= 0,

z(2ej)

= ()j)

z(ej

+ ek) =

In particular, one sector of the discrete z2, defined ()l = 1, B2 = i, is characterized by the initial data z(O,O) = z(1, 0) = z(O, 1) = 0,

z(2,O) = 1,

Oil

()J - ()~ 2(10g()j -log()k) . ('lL+) 2 , in the case of

z(O,2) = -1,

8.6. Show that the dual solution to the discrete L.

z2

z(1, 1)

=

.2

2- • 1r

is the discrete logarithm

8.7. Show that for the cross-ratio system on ('lL+)2 with Bl = 1, B2 = 'i, the dual solution to z(m, '11.) = l/(m + in) is given by

z*(m,n)

=

1 3((m +in)3 - ('In -in)).

This can be regarded as the discrete z3.

8.7. Bibliographical notes Section 8.1: Circle patterns. The idea that circle packings and, more generally, circle patterns serve as a discrete counterpart of analytic functions is by now well established; see the monograph by Stephenson (2005). The origin of this idea is connected with the approach by Thurston (1985) to the Riemann mapping theorem via circle packings. Since then the theory bifurcated to several areas.

One of them is dealing mainly with approximation problems. The most popular are hexagonal packings, for which the convergence to the Riemann mapping was established in Rodin-Sullivan (1987). In He-Schramm (1998) it was shown that this convergence actually holds in the class Coo, that is, all higher derivatives are approximated. Sirnilar results are available also for

328


circle patterns with combinatorics of the square grid introduced in Schramm (1997), and even for more general circle patterns; see Bucking (2007). Another area concentrates around the uniformization theorem of KoebeAndreev-Thurston, and is dealing with circle packing realizations of cell complexes of prescribed combinatorics, rigidity properties, constructing hyperbolic 3-manifolds, etc.; see Thurston (1997), He (1999), Stephenson (2005). A variational description of circle packings was initiated by Colin de Verdiere (1991). Further progress is due to Bragger (1992), Rivin (1994), and Bobenko-Springborn (2004). The extremals of the functional used in the last paper are described by equation (8.4). An application of this approach in discrete differential geometry is the construction of discrete minimal surfaces through circle patterns in Bobenko-Hoffmann-Springborn (2006). The main topic of this chapter is interrelations of circle patterns with integrable systems. See the notes to Section 8.3.

Section 8.2: Integrable cross-ratio and Hirota systems. In this generality (for arbitrary quad-graphs) this material is due to Bobenko-Suris (2002a). On 71} the relation between the cross-ratio and Hirota systems is considered in Capel-Nijhoff (1995). Our presentation follows BobenkoMercat-Suris (2005). Section 8.3: Integrable circle patterns. Orthogonal circle patterns with combinatorics of the square grid were studied in Schramm (1997). Hexagonal circle patterns with fixed intersection angles were investigated in BobenkoHoffmann (2003), and with the multiratio property, in Bobenko-HoffmannSuris (2002). The general theory presented here is formulated in BobenkoMercat-Suris (2005). Section 8.4: za and log z circle patterns. The circle patterns za on the square lattice were introduced in Bobenko (1999) and studied in BobenkoPinkall (1999) and Agafonov-Bobenko (2000). The conjecture that these patterns are embedded, i.e., the interiors of different kites are disjoint, was formulated in the first of these papers. The study was extended to the regular hexagonal grid in Bobenko-Hoffmann (2003). The fact that the circle patterns za are immersed, i.e., the neighboring kites do not overlap, was proven in Agafonov-Bobenko (2000) for the square grid and in AgafonovBobenko (2003) for the hexagonal grid combinatorics. The embeddedness was proven in Agafonov (2003) for the case of the square grid combinatorics. The isomonodromic constraint (8.37) was obtained first for a = 1/2 in Nijhoff (1996), with no geometric interpretation. For the Hirota system, the isomonodromic constraint (8.41) was derived in Nijhoff-RamaniGrammaticos-Ohta (2001), also with no relation to geometry. Our presentation here follows Bobenko-Mercat-Suris (2005).


329

Section 8.5: Linearization. The operation of discrete integration for discrete holomorphic functions was considered in Duffin (1956, 1968), and Mercat (2001). Linearization of circle patterns was studied in BobenkoMercat-Suris (2005); in particular, the derivation of Green's function from the za circle pattern is taken from this paper. Section 8.6: Exercises. Ex. 8.3: See Bobenko-Mercat-Suris (2005). Ex. 8.5: See Agafonov-Bobenko (2000). Ex. 8.6: See Agafonov-Bobenko (2000) in the case of the regular square grid. Ex. 8.7: See Bobenko-Pinkall (1999).

Chapter 9

Foundations

For the reader's convenience we give here a brief introduction to projective geometry and the geometries of Lie, Mobius, Laguerre and Plucker. We also include a number of classical incidence theorems relevant to discrete differential geometry. For extensive presentations of these classical results we recommend, in particular, the textbooks: Blaschke (1954) and Pedoe (1970) on projective and Plucker line geometry, Blaschke (1929) on sphere geometries, Cecil (1992) on Lie geometry, and Hertrich-Jeromin (2003) on Mobius geometry.

9.1. Projective geometry Projective geometry studies properties of geometric objects which remain invariant under the group of projective transformations, which is generated by Euclidean motions, homotheties, and central projections. A suitable analytical framework for doing projective geometry is given by the notion of homogeneous coordinates. The main space of real projective geometry is

which is the set of equivalence classes of equivalence relation: X rv Y

{:}

X = AY,

]RN+1 \

X,Y E ]RN+l \

{O} modulo the following {O},

A E ]R*.

On a general note, building the set of equivalence classes with respect to the relation rv is called a projectivization. Thus, points of JP>]RN are projectivizations of I-dimensional vector subspaces of ]RN+1.

-

331

332

9. Foundations

The equivalence class of x = (Xl, ... , XN, xN+d E JR.N+l \ {O} is denoted by [x] = [Xl: ... : XN : XN+I]. The space JR.N+l is called the space of homogeneous coordinates on JR.lP'N. One says that X E JR. N+I \ {O} is a lift of [x] to the space of homogeneous coordinates, or a representative of [x] in the space of homogeneous coordinates. The usual space JR.N can be identified with the subset of equivalence classes of elements of JR.N+I with XN+I # 0:

This subset is called an affine part of lP'JR. N . The complement of an affine part, i.e., the set of equivalence classes [Xl : ... : XN : 0], is called the hyperplane at infinity, and its elements are called infinitely remote points. Of course, XN+I plays a distinguished role in this construction. One can single out a coordinate other than XN+I and will then obtain different affine parts. The N + 1 affine parts obtained in this way build an atlas of lP'JR. N as a real manifold, consisting of N + 1 charts. More generally, projective hyperplanes in lP'JR. N are projectivizations of hyperplanes, that is, of vector N-spaces in JR.N+I. Any hyperplane U can be described by an equation N+I (u,x)

=

L

UiXi =

0,

i=l

where (UI, ... ,UN,UN+d E (JR.N+l)* \ {O}, and (-,.) denotes the pairing between the dual spaces JR.N+I and (JR.N+I)*. Actually, only the equivalence class [UI : ... : UN : UN+I] is relevant in this description, and a hyperplane can be identified with this equivalence class. One calls [UI : ... : UN : UN+I] the homogeneous coordinates of a hyperplane u. For instance, the hyperplane at infinity has homogeneous coordinates [0 : ... : 0 : 1]. Thus, the set (JR.lP'N)* of all projective hyperplanes is isomorphic to JR.lP'N, again. Interchanging the roles of points from JR.lP'N with hyperplanes from (JR.lP'N)* is the projective duality. For any 1 ~ d ~ N -1, a projective d-space in JR.lP'N is a projectivization of a vector (d + I)-space in JR.N+I. There are two dual ways to describe a projective d-space . • Let Xl, . .. ,Xd+l be d + 1 points of JR.lP'N in general position with representatives Xl, ... , Xd+l in the space of homogeneous coordinates. The general position condition means that the vector space L: = span(xI, ... ,Xd+l) has dimension d + 1. Then lP'(L:) is the d-dimensional space through Xl, ... , Xd+l. The points of lP'(L:) are given, in homogeneous coordinates, by all possible linear combinations X = cdl + ... + Cd+IXd+1 with (Cl l • • • , cd+d # (0, ... ,0).

333

9.1. Projective geometry

• Alternatively, let U1, ... , UN-d be N - d hyperplanes of (IRIP'N)* in general position, with representatives '111, ... , UN-d in the space of homogeneous coordinates. Again, the general position condition means that the vector space 1;~ = span( '11 1, ... , UN -d) has dimension N - d. Then the vector space

1; =

{x

E

IR N+ 1 : ('111, x) = ... = (UN-d,X) =

o}

has dimension d + 1, and 1P'(1;) is a projective d-space defined as the intersection of the hyperplanes U1,···, UN-d' If d 1 + d 2 2: N, then the intersection of a d 1-space with a d 2 -space in IRIP'N is a projective space of dimension 2: d1 + d2 - N, with an equality in the case of general position. For instance, intersection of a projective line with a projective hyperplane, in the general position, is a point. Here "general position" means that the line does not belong to the hyperplane.

The projective duality extends from the relation between points and hyperplanes to the relation between projective spaces of any two complementary dimensions d 1 and d 2 such that d 1 + d2 = N - 1. A famous and striking duality principle of projective geometry says that to any statement about incidence of projective spaces corresponds the dual statement where every object is replaced by its dual, with a simultaneous inversion of all incidence relations. For instance, in the plane projective geometry, the dual statements are obtained by replacing the notions according to the following (incomplete) vocabulary: a point a line a line a point a point on a line a line through a point the intersection point of two lines the line connecting two points three lines have a common point three points are collinear In the three-dimensional projective geometry, the incomplete vocabulary looks as follows: a plane a point a line a line a plane a point a line in a plane a line through a point the line connecting two points the intersection line of two planes the intersection point of a line the plane through a line with a plane and a point the intersection point of three planes the plane through three points four points are coplanar four planes have a common point

9. Foundations

334

Projective transformations or collineations of ~JlDN are induced by nondegenerate linear maps on the space of homogeneous coordinates: fj = rx,

r

E

GL(N + 1,~).

Theorem 9.1. (Fundamental theorem of projective geometry) a) Let, : ~JlDN --t ~JlDN be an injective map such that ,(~JlDN) does not lie in a hyperplane, and for any three collinear points Xl, X2, X3 their images ,(Xl), ,(X2), ,(X3) are also collinear. Then, is a projective transformation. b) For any two sets {Xl, ... , XN+2} C ~JlDN and {YI, ... ,YN+2} C ~JlDN such that in each set no N + 1 points lie in a hyperplane, there is a unique projective transformation, such that Yk = ,(Xk) for all k = 1, ... ,N + 2. A projective transformation of a line is characterized by the property of preserving the cross-ratios of four points. Finally, we briefly discuss the notion of quadric in a projective space. Let Q : ~N+l X ~N+I --t ~ be a nondegenerate symmetric bilinear form; we will denote the matrix of this form by Q. The set of points X E ~JlDN with homogeneous coordinates E ~N+I satisfying the quadratic equation

x

N+I

(9.1)

L

Q(x, x) =

QjkXjXk = 0

j,k=l is called a (nondegenerate) quadric Q C ~JlDN. Of course, only those nonempty quadrics are interesting which correspond to indefinite bilinear forms. In particular, a nondegenerate quadric in ~JlD2 is called a conic. Two points x, Y E ~N with homogeneous coordinates x, fj E ~N+l are called conjugate with respect to a quadric if N+I

L

Q(x, fj) =

QjkXjYk =

o.

j,k=l The points conjugate to a given point X E ~JlDN build the polar hyperplane of x. Thus, the polar hyperplane is defined as JlD(x~), where the orthogonal complement is taken according to the scalar product Q: x~ = {fj E ~N+I : Q(x, fj) =

O}.

Homogeneous coordinates of the polar hyperplane can be chosen as

U= Qx. Thus, the polar point

X

of a hyperplane u has homogeneous coordinates A

X

=

Q-I u. A

9.2. Lie geometry

335

Two hyperplanes u, v with homogeneous coordinates it, v E ~N+l are called conjugate with respect to the quadric Q if N+l

L

(Q-l )jkUjVk = O.

j,k=l

Each of them contains the polar point of the other. A tangent hyperplane to the quadric Q is self-conjugate, so its homogeneous coordinates satisfy the quadratic equation N+l

L

(9.2)

(Q-l )jkUjUk = O.

j,k=l

A quadric can be viewed either as the set of points satisfying (9.1) or as the envelope of its tangent hyperplanes satisfying (9.2). The polarity relation can be generalized from points and hyperplanes to projective spaces of arbitrary dimensions. For a projective d-space U = JP>(U) one defines a polar subspace as JP>(UJ..), where the orthogonal complement is understood with respect to the scalar product Q. Polarity can be regarded as a generalization of duality.

9.2. Lie geometry 9.2.1. Objects of Lie geometry. The following geometric objects in the Euclidean space ~N are elements of Lie geometry: • Oriented hyperspheres. A hypersphere in ~N with center c E ~N and radius r > 0 is described as S = {x E ~N : Ix - cl 2 = r2}. It divides ~N into two parts, inner and outer. Declaring one of the two parts of ~N to be positive, we come to the notion of an oriented hypersphere. Thus, there are two oriented hyperspheres S± for any S. One can take the orientation of a hypersphere into account by assigning a signed radius ±r to it. For instance, one can assign positive radii r > 0 to the hyperspheres with the inward field of unit normals and negative radii r < 0 to the hyperspheres with the outward field of unit normals . • Oriented hyperplanes. A hyperplane in ~N is given by the equation p = {x E ~N : (v, x) = d}, with a unit normal v E §N-l and d E R Clearly, the pairs (v, d) and (-v, -d) represent one and the same hyperplane. It divides ~N into two half-spaces. Declaring one of the two half-spaces to be positive, we arrive at the notion of an oriented hyperplane. Thus, there are two oriented hyperplanes p± for any P. One can take the orientation of a hyperplane into

336

9. Foundations

account by assigning the pair (v, d) to the hyperplane with the unit normal v pointing into the positive half-space.

• Points. One considers points x E radius.

]RN

as hyperspheres of a vanishing

• Infinity. One compactifies the space ]RN by adding the point at infinity 00, with the understanding that a basis of open neighborhoods of 00 is given, e.g., by the outer parts of the hyperspheres Ixl 2 = r2. Topologically the compactification so defined is equivalent to a sphere §N. • Contact elements. A contact element is a pair consisting of a point x E ]RN and an (oriented) hyperplane P through x; alternatively, one can use a normal vector v to P at x. A contact element represents the (equivalence class of) hypersurfaces through the point x with the tangent hyperplane P at x. In the framework of Lie geometry, a contact element can be identified with a set (a pencil) of all hyperspheres S through x which are in oriented contact with P (and with one another), thus sharing the normal vector v at x; see Figure g.1.

p

Figure 9.1. Contact element.

9.2.2. Projective model of Lie geometry. All the above elements are modelled in Lie geometry as points, resp. lines, in the (N + 2)-dimensional projective space lP'(]RN+l,2) with the space of homogeneous coordinates ]RN+l,2. The latter is the space spanned by the N + 3 linearly independent vectors el, ... , eN +3 and equipped with the pseudo-Euclidean scalar

9.2. Lie geometry

337

product

1, (ei, ej) = { -1, 0, It is convenient to introduce two (9.3)

eo

= ~(eN+2 -

i=jE{1, ... ,N+1}, i=jE{N+2,N+3}, i =1= j.

isotropic vectors e OCl

eN+l),

= ~(eN+2 + eN+d,

for which (eo, eo) = (e OCl , e OCl ) = 0, (eo, e oo ) = -~. The models of the above elements in the space ~N+l,2 of homogeneous coordinates are as follows: • Oriented hypersphere with center c E ~N and signed radius r E R' (9.4)

S=

c+ eo + (lcl 2 -

• Oriented hyperplane (v, x)

p=

(9.5)

v

r2)e oo

= d with v

+ reN+3. E §N-l and d E ~:

+ 0 . eo + 2de oo + eN+3.

• Point x E ~N :

x = x + eo + Ixl 2e oo + O· eN+3'

(9.6) • Infinity

00:

(9.7)

00 = e oo . • Contact element (x, P):

(9.8)

span(x,p).

In the projective space lP'(~N+l,2) the first four types of elements are represented by the points which are equivalence classes of (9.4)-(9.7) with respect to the relation ~ '" 'rJ {:} ~ = A'rJ with A E ~* for ~, 'rJ E ~N+l,2. A contact element is represented by the line in lP'(~N+l,2) through the points with representatives x and p. We mention several fundamentally important features of this model: (i) All the above elements belong to the Lie quadric lP'(lI.P+l,2), where (9.9)

JL N +1,2 = {~E ~N+l,2: (~,~) = O}. Moreover, the points of lP'(JL N+ 1,2) are in a one-to-one correspondence with the oriented hyperspheres in ~N, including degenerate cases: proper hyperspheres in ~N correspond to points oflP'(JL N+ 1,2) with both eo- and eN+3-components nonvanishing, hyperplanes in ~N correspond to points of lP'(JLN+l,2) whose eo-component vanishes, points in ~N correspond to points of lP'(JLN+l,2) with vanishing eN+3-component, and infinity corresponds to the only point of lP'(JLN+l,2) with both eo- and eN+3-components vanishing.

9. Foundations

338

(ii) Two oriented hyperspheres 5 1 ,52 are in oriented contact (i.e., are tangent to each other with the unit normals at tangency pointing in the same direction) if and only if

leI -

(9.10)

c21 2 = (rJ - r2) 2 ,

and this is equivalent to (S],S2)

=

O.

(iii) An oriented hypersphere 5 = {x E IRN : Ix-cl 2 = r2} is in oriented contact with an oriented hyperplane P = {x E IRN : (v, :r) = d} if and only if

(c,v) - r - d = O.

(9.11)

Indeed, equation of the hyperplane P tangent to 5 at Xo E 5 reads: (xo - c, x - c) = r2. Denoting by v = (c - xo) / r the unit normal vector of P (recall that the positive radii are assigned to spheres with inward unit normals), we can write the above equation as (v, x) = d with d = (c, (c - XO)/T) - T = (c, v) - T, which proves (9.11). Now, the latter equation is equivalent to (s,p) = O. (iv) A point x can be considered as a hypersphere ofradius r = 0 (in this case the two oriented hyperspheres coincide). An incidence relation x E 5 with a hypersphere 5 (resp. x E P with a hyperplane P) can be interpreted as a particular case of oriented contact of a sphere of radius T = 0 with 5 (resp. with P), and it takes place if and only if (x,t,) = 0 (resp. (x,p) = 0). (v) For any hyperplane P, (oo,p) = O. One can interpret hyperplanes as hyperspheres (of an infinite radius) through 00. More precisely, a hyperplane (v, x) = d can be interpreted as a limit, as r -+ 00, of the hyperspheres of radii T with centers located at c = TV + 1t, with (v, u) = d. Indeed, the representatives (9.4) of such spheres are

s

+ u) + eo + (2dT + (u, u) )e oo + TeN+3 (v + O(l/r)) + (l/T)eo + (2d + O(l/T))e oo + eN+3 p + O(l/T). (rv

Moreover, for similar reasons, the infinity 00 can be considered as a limiting position of any sequence of points x with Ixl -+ 00. (vi) Any two hyperspheres 5 1 ,52 in oriented contact determine a contact element (their point of contact and their common tangent hyperplane). For their representatives SI, S2 in IR N + l ,2, the line in IP'(IR N + I ,2) through the corresponding points in IP'(JLN+1.2) is isotropic, i.e., lies entirely on the Lie quadric IP'(JLN+1,2). This follows from

9.2. Lie geometry

339

Such a line contains exactly one point whose representative x has vanishing eN+3-component (and corresponds to x, the common point of contact of all the hyperspheres), and, if x of- 00, exactly one point whose representative p has vanishing eo-component (and corresponds to P, the common tangent hyperplane of all the hyperspheres). In case when an isotropic line contains 00, all its points represent parallel hyperplanes, which constitute a contact element through 00. Thus, if one considers hyperplanes as hyperspheres of infinite radii, and points as hyperspheres of vanishing radii, then one can conclude that: ~

Oriented hyperspheres are in a one-to-one correspondence with points of the Lie quadric JP>(IL,N+l,2) in the projective space JP>(~N+l,2).

~

Oriented contact of two oriented hyperspheres corresponds to orthogonality of (any) representatives of the corresponding points in JP>(~N+l,2).

~

Contact elements of hypersurfaces are in a one-to-one correspondence with isotropic lines in JP>(~N+l,2). We will denote the set of all such lines by J:~+1,2.

9.2.3. Lie sphere transformations. According to F. Klein's Erlangen Program, Lie geometry is the study of properties of transformations which map oriented hyperspheres (including points and hyperplanes) to oriented hyperspheres and, moreover, preserve the oriented contact of hypersphere pairs. In the projective model described above, Lie geometry is the study of projective transformations of JP>(~N+l,2) which leave JP>(IL,N+l,2) invariant, and, moreover, preserve orthogonality of points of JP>(lI...N+ 1,2) (which is understood as orthogonality of their lifts to 1I...N+ 1,2 C ~N+l,2; clearly, this relation does not depend on the choice of lifts). Such transformations are called Lie sphere transformations. Theorem 9.2. (Fundamental theorem of Lie geometry) a) The group of Lie sphere transformations is isomorphic to the factor O(N + 1, 2)/{±I}. b) Every line preserving diffeomorphism of JP>(lI....N+ 1,2) is the restriction to JP>(lI.,N+l,2) of a Lie sphere transformation.

Since vanishing of the eo- or eN+3-component of a point in JP>(lI.,N+l,2) is not invariant under a general Lie sphere transformation, there is no distinction between oriented hyperspheres, oriented hyperplanes and points in Lie geometry.

9. Foundations

340

9.2.4. Planar families of spheres; Dupin cyclides. Considerations of this subsection hold for the geometrically most significant case N = 3. Definition 9.3. (Planar family of spheres) A planar family of (oriented) spheres in]R3 is a set of spheres whose representatives s E JP'(JL4,2) are contained in a projective plane JP'(~), where ~ is a three-dimensional vector subspace of ]R4,2 such that the restriction of (-, .) to ~ is nondegenerate. Thus, a planar family of spheres is a conic section JP'(~ there are two possibilities:

n JL 4,2).

Clearly,

(a) The signature of (', ')IE is (2,1), and so the signature of (-, also (2,1).

')IE~

is

(b) The signature of (-, .) IE is (1, 2), and so the signature of (-, .) IE~ is (3,0). It is easy to see that a planar family is a one-parameter family, parametrized by a circle §1. Indeed, if e1, e2, e3 is an orthogonal basis of ~ such that (e1, e1) = (e2, e2) = -(e3, e3) = 1 (say), then the spheres of the planar family come from the linear combinations s = a1 e1 + a2e2 + e3 with (a1e1

+ a2e2 + e3, a1e1 + a2e2 + e3) = 0

¢}

ai + a~ = 1.

In the second case mentioned above, the space ~~ has only a trivial intersection with JL 4,2, so the spheres of the planar family JP'(JL4,2 n ~) have no common touching spheres. Definition 9.4. (Cyclidic family of spheres) A planar family of spheres is called cyclidic if the signature of (-, ')IE is (2,1), so that the signature of (-, ')IE~ is also (2,1). For any cyclidic family JP'(JL 4 ,2 n ~) there is a dual cyclidic family JID(JL 4,2 n ~~) such that any sphere of the first family is in oriented contact with any sphere of the second. The family JP'(JL 4,2 n~), as anyone-parameter family of spheres, envelopes a canal surface in ]R3, and this surface is an envelope of the dual family JP'(JL 4,2 n ~~), as well. Such surfaces are called Dupin cyclides. Examples: a) Points of a circle build a planar cyclidic family of spheres (of radius zero). The dual family consists of all (oriented) spheres through this circle, with centers lying on the line through the center of the circle orthogonal to its plane; see Figure 9.2, left. The corresponding Dupin cyclide is the circle itself. It can be shown that any Dupin cyclide is an image of this case under a Lie sphere transformation. b) Planes tangent to a cone of revolution build a planar cyclidic family of spheres, as well. The dual family consists of all (oriented) spheres tangent

9.3. Mobius geometry

341

to the cone, with centers on the axis of the cone; see Figure 9.2, right. The corresponding Dupin cyclide is the cone itself.

Figure 9.2. Left: A cyclidic family of spheres through a circle. Right: A cyclidic family of spheres tangent to a cone.

9.3. Mobius geometry 9.3.1. Objects ofM6bius geometry. Mobius geometry is a subgeometry of Lie geometry, with points distinguishable among all hyperspheres as those of radius zero. Thus, Mobius geometry studies properties of hyperspheres in ]RN invariant under the subgroup of Lie sphere transformations preserving the set of points. The following geometric objects are elements of Mobius geometry of ]RN: • Points x E ]RN. • Infinity

00

which compactifies

]RN

into

§N.

• (Nonoriented) hyperspheres S = {x E ]RN : centers c E ]RN and radii r > o. • (Nonoriented) hyperplanes P normals v E §N -1 and d E lR..

= {x

E ]RN :

Ix - cI2

= r2} with

(v, x) = d}, with unit

The Mobius group Mob(N) of ]RN consists of point transformations generated by reflections in hyperplanes P = {x E ]RN : (v, x) = d}:

(9.12)

x

f--r

x- 2

(v, x) - d ( ) v, v,v

and by inversions in hyperspheres S = {x

(9.13)

E]RN :

Ix - cl 2 = r2}:

r2

Xf--rC+

Ix-c 12 (x-c).

Clearly, Mob(N) contains as a subgroup the group E(N) of Euclidean motions of ]RN, which is generated by reflections in hyperplanes. It contains

9. Foundations

342

also dilations, since they can be represented as compositions of inversions in two concentric hyperspheres. For N 2: 3, the Liouville theorem says that Mob(N) coincides with the group of conformal diffeomorphisms. Yoo

]RN

Figure 9.3. Stereographic projection.

One compactifies ]RN by adding the point 00, thus arriving at the Nsphere §N. It is convenient to model §N as embedded in ]RN +1 : §N

= {Y E ]RN+l : (y,y) = 1}

(we use one and the same notation for the scalar products in ]RN and in ]RN+l; its meaning in each case should be clear from the context). The (inverse) stereographic projection 0" : ]RN - t §N \ {Yoo} from the north pole Yoo = eN +1 is defined by (9.14)

see Figure 9.3. The formula

2

O"(x)

= eN+l + 1x 12 + 1 (x

- eN+d

= eN+l + 1x -

2 eN+l

12

(x - eN+d

shows that one can view the stereographic projection 0" also as the restriction to ]RN of the inversion of ]RN+l in the hypersphere with center eN+l and radius.)2. Setting 0"(00) = Yoo makes 0" to a diffeomorphism 0" : ]RN U {(X)} - t §N. Hyperplanes and hyperspheres in ]RN are mapped by the stereographic projection 0" to hyperspheres in §N, the images of hyperplanes being hyperspheres through Yoo' Thus, hyperplanes in ]RN can be interpreted as hyperspheres through 00.


343

Elements of Mobius geometry of §N are: • Points y E

§N .

• (Nonoriented) hyperspheres S C

§N.

Any hypersphere S C §N, except for great ones, may be described as the intersection of §N with an affine hyperplane {y E IRN+l : (s, y) = 1}. The point s E IR N +1 , which is the pole of this hyperplane with respect to §N, lies outside of §N, and S C §N is the contact set of §N with the tangent cone to §N with apex s. Also, S c §N is the intersection of §N and the orthogonal N-sphere S C IRN +1 with center s and radius p such that p2 = (s, s) - 1, see Figure 9.4. (For a great hypersphere S C §N, which is the intersection of §N with a hyperplane {y E IR N +1 : (s, y) = O}, the latter hyperplane also plays the role of the orthogonal N-sphere S, and the tangent cone becomes a cylinder.)

(8, y) = 1

S

Figure 9.4. Hypersphere S C §N and the corresponding point s E with an orthogonal N-sphere S through S.

]RN+l,

The Mobius group Mob(N) of spheres S C §N, given by

§N

is generated by inversions in hyper-

(9.15) Transformation (9.15) coincides with the restriction to §N of the inversion of IR N + 1 in the N-sphere S, which is orthogonal to §N and intersects §N along the hypersphere S. A hypersphere S in IRN (or in §N) can also be interpreted as the set of points XES. This allows us to introduce lower-dimensional spheres:

9. Foundations

344

• Spheres. A k-sphere is a (generic) intersection of N -k hyperspheres Si (i = 1, ... ,N - k).

They are further objects of Mobius geometry (in contrast to Lie geometry). This means that the class of k-spheres is preserved by Mobius transformations.

9.3.2. Projective model of Mobius geometry. In modelling elements of Mobius geometry (of either of the spaces jRN U{ oo} or §N), one can use the Lie-geometric description and just omit the eN+3-component. The resulting objects are points of the (N + I)-dimensional projective space IP(jRN+l,l) with the space of homogeneous coordinates jRN+l,l. The latter is the space spanned by N + 2 linearly independent vectors el, ... , eN+2 and equipped with the Minkowski scalar product

1, ~=~E{I, ... ,N+I}, { (ei,ej)= -1, z=]=N+2, 0,

i

=1=

j.

We continue to use notation (9.3) in the context of Mobius geometry. Elements of Mobius geometry of jRN are modelled in the space jRN+l,l of homogeneous coordinates as follows: • Point x E

jRN: X~

(9.16) • Infinity

~ = XEue = x

+ eo + Ix 12 eCX).

00:

00 = eCX).

(9.17)

• Hypersphere with center c E jRN and radius r

S=

(9.18)

SEue

= C + eo + (lcl 2 - r2)eCX).

• Hyperplane (v,x) = d with v E

P = PEue =

(9.19)

> 0:

V

§N-l

and d E R

+ 0 . eo + 2deCX).

In the projective space IP(jRN+l,l) these elements are represented by points which are equivalence classes of (9.16)-(9.19) with respect to the usual relation ~ rv TJ ¢} ~ = ATJ with A E jR* for ~,TJ E jRN+1,l. Fundamental features of these identifications are the following: (i) The infinity 00 can be considered as a limit of any sequence of x for x E jRN with Ixl - t 00. The points x E jRN U {oo} are in a one-toone correspondence with the points of the projectivized light cone IP(lLF+l,l), that is, with the straight line generators of

(9.20)

rrP+l,l

=

{~E jRN+l,l : (~,~)

= o}.

345


The points x E ]RN correspond to the points of lP'(IL,N+l,l) with a nonvanishing eo-component, while 00 corresponds to the only point of lP'(lL N +1,l) with the vanishing eo-component. Euclidean representatives (9.16) have an important property:

(9.21)

(Xl, X2) =

(ii) Hyperspheres

-!IX1 - x21 2,

s and hyperplanes p belong to lP'(]R:%t 1,1), where

]R:%t 1,1 = {~ E

(9.22)

VX1, X2 E ]RN.

]RN+1,1 :

(~,~)

> o}

is the set of space-like vectors of the Minkowski space ]RN+1,1. Hyperplanes can be interpreted as hyperspheres (of an infinite radius) through 00. (iii) Two hyperspheres 8 1 ,82 with centers Cl, C2 and radii r1, r2 intersect orthogonally if and only if

(9.23) which is equivalent to (Sl' S2) = O. Similarly, a hypersphere 8 intersects orthogonally with a hyperplane P if and only if its center lies in P:

(9.24)

(c, v) - d = 0, which is equivalent to (s,p)

=

O.

(iv) A point x can be considered as a limiting case of a hypersphere with radius r = o. An incidence relation x E 8 with a hypersphere 8 (resp. x E P with a hyperplane P) can be interpreted as a particular case of an orthogonal intersection of a sphere of radius r = 0 with 8 (resp. with P). We have: x E8

~

x E P

~

(x, s) = 0, (x, p) = o.

Switching from the Euclidean space ]RN to the sphere §N corresponds to a different choice of representatives for the points of lP'(]RN+1,1): • Point y E

§N:

f) = f)Sph = Y + eN+2·

(9.25) • Hypersphere 8

(9.26)

=

{y E §N: (s,y)

s = SSph = s + eN+2. • Great hypersphere 8

(9.27)

= 1}:

S=

=

{y E §N : (8, y)

SSph = 8

= O}:

+ O· eN+2.

Features of this choice of representatives:

9. Foundations

346

(i) In formulas (9.25), (9.26), Y and 8 are points ofIR N+ 1 with (y, y) = 1 and (8,8) > 1, which is equivalent to f) E ll..N+ u and s E IR~l~l,l, respectively. Also elements (9.27) (still defined up to a real factor) l TlllN+l,l . L)e I ongt 0 8 E !No. out A

(ii) Incidence relation: yES

(f), ,~)

{::;>

= O.

Indeed, the relation (f), s) = 0 for f) from (9.25) and for s from (9.26) is equivalent to (8, y) = 1. Similarly, the relation (f), s) = o for elements ,9 with vanishing eN+2-component, as in (9.27), is equivalent to (8, y) = 0, which characterizes great hyperspheres. To sum up: in the Minkowski space IRN+l,l of homogeneous coordinates, points and hyperspheres (different from hyperplanes) of the Euclidean space IRN find their place in the affine hyperplane (~, e oo ) = in particular,

-!;

IRN

(9 .28) 7ro

TlllN '" :!No. ~

~ «J!~

x

f---t

=

{~ E lLN+l,l : ~N+2

X =X Euc A

A

=.T

= x

-

~N+l

=

I},

+ eo + 1x 12 e=

1 2 + 2(lxl -

1) eN+l

1 2 + 2(lxl + 1) eN+2

N

E «J!o

(Euclidean metric d~r + ... + d~Jv being induced from the ambient IRN+1,1). The model «J!~ of the Euclidean space IRN can be viewed as a paraboloid in an (N + I)-dimensional affine subspace through eo spanned by el,"" eN, e oo · Similarly, points and hyperspheres of §N (different from great hyperspheres) find their place in the affine hyperplane (~,eN+2) = -1 of the Minkowski space IR N +l,I; in particular, §N ~

(9.29)

7rl :

§N

«J!'i = {~E lL N+ 1,1 3 y

~N+2 = I},

Y = YSph = Y + eN+2 E A

f---t

:

A

rn,N

'1,£1 .

The model «J!'i of the N-sphere §N can be viewed as a copy of §N in the (N + I)-dimensional affine subspace through eN+2 spanned by el,"" eN+l. Note that the correspondence between «J!~ and «J!'i along the straight line generators of lLN+1,1 induces the stereographic projection (j (compare (9.28) with (9.29) and with (9.14)). In particular, the generators of lLN+l,l through the points eo and e oo correspond to the zero and the point at infinity in IR N , and to the south pole Yo = -eN+l and the north pole Yoo = eN+l on §N, respectively. Turning to projective models of lower-dimensional spheres, recall that a hypersphere Sin IRN (or in §N) can also be interpreted as the set of points xES, and therefore it admits, along with the representation 5, the dual

347


Figure 9.5. Projective model of Mobius geometry.

representation as a transversal intersection of IP(JL N + 1,1) with the projective N-space IP(s.l), polar to the point S with respect to IP(JL N+l,1); here, of course, s.l = {x E ]R N +1, 1 : (s, X) = O}. This can be generalized to model lower-dimensional spheres .

• Spheres. A k-sphere is a (generic) intersection of N - k hyperspheres Si represented by Si E ]R~t1,1 (i = 1, ... , N - k). Such an intersection is generic if the (N - k )-dimensionallinear subspace of ]RN+1,1 spanned by Si is space-like: "

LI

' ' ) = span (S1,··., SN-k C

lIll N

+1,1 .

ll'\,.out

As a set of points, this k-sphere is represented as IP(JLN+l,l where

n

n I;.l),

N-k

I;.l=

Sf={XE]RN+I'l: (SI,X)="'=(SN-k,X)=O}

i=l

is a (k + 2)-dimensional linear subspace of ]RN+I,1 of signature (k+1,1). Through any k + 2 points Xl, ... , Xk+2 E ]RN in general position one can draw a unique k-sphere. It corresponds to the (k + 2)dimensional linear subspace I;.l = span(x1, ... ,Xk+2), of signature (k + 1, 1), with k + 2 linearly independent isotropic vectors Xl, ... , xk+ 2 E JL N+ I, 1. In the polar formulation, this ksphere corresponds to the (N - k)-dimensional space-like linear

9. Foundations

348

subspace k+2

2: =

nxf

=

{s E JRN+l,l:

(S,Xl) = ... = (S,Xk+2) =

o}.

i=1

To conclude, we mention that for hyperspheres s yet another choice of representatives in JRN+l,1 is sometimes used: one fixes the Lorentz norm of S. For any K, > 0, introduce the quadric (9.30)

lL~+I,1 = {~E JRN+l,l: (~,~) = K,2},

and choose the representative of a hypersphere in lL~+I,I: (9.31)

s = sMob =

~(s + eN+2) p

=

~ (c + eo + (lcl2 - r2)e oo ) E lL~+l,1. r

Actually, equation (9.31) contains two representatives of any hypersphere, corresponding to opposite values of p, resp. r, and therefore it represents oriented hyperspheres, each choice of the sign corresponding to one of the two possible orientations of a given hypersphere. Strictly speaking, this choice leads us outside of the projective model of Mobius geometry, and is a remainder of the Lie-geometric approach. We call p E JR (resp. r E JR) the oriented spherical (resp. Euclidean) radius of the hypersphere. For any two (oriented) hyperspheres 8 1 , 8 2 , the scalar product of their representatives SMob is a Mobius invariant: if K, = 1, then (SI,S2) = _1_((SI,S2) -1) = - 12 (ri PIP2 rl r 2

+ r~ -ICI -

c212)

is the cosine of the intersection angle of 8 1 , 8 2 , if they intersect, and the inversive distance between 8 1 , 8 2 , otherwise.

9.3.3. Mobius transformations. Mobius geometry is the study of properties of (nonoriented) hyperspheres invariant with respect to projective transformations of JPl(JR N+1,I) which map points to points, i.e., which leave JPl(lL N+ l ,1) invariant. Such transformations are called Mobius transformations.

Theorem 9.5. (Fundamental theorem of Mobius geometry) a) The group of Mobius transformations is isomorphic to O(N + 1,1)/ ~ O+(N + 1, 1), the group of Lorentz transformations of JR N +l ,1 preserving the time-like direction.

{±I}

b) Every conformal diffeomorphism of §N ~ JRN U {oo} is induced by the restriction to JPl(lLN+l,l) of a Mobius transformation.

The group O+(N + 1, 1) is generated by reflections, (9.32)

As : JRN+l,l -. JR N +l ,l,

As(x)

=

x _ 2((!, ~)) s. s,s


349

These reflections preserve the light cone lI.P+1,l and map straight line generators to straight line generators. Therefore, they induce some transformations on lP'(JLN+1,l) c::::: Ql'i', resp. on Ql~. The induced transformations on Ql~ c::::: ffi.N are obtained from (9.32) by direct computations with representatives (9.16) for points and representatives (9.18) for hyperspheres, and are given by (9.13) (inversion in the hypersphere S = {x E ffi.N : Ix-cl 2 = r2}); similarly, if s = P is the hyperplane (9.19), then the transformation induced on ffi.N by Ap is easily computed to be as in (9.12) (reflection in the hyperplane P = {x E ffi.N : (v, x) = d}). Similarly, the induced transformations on Ql'i' c::::: §N are obtained by a straightforward computation with representatives (9.25) for points and (9.26) for hyperspheres:

~

=

As(Y)

(

Y-8+

IY - 812 ) p2

and so the induced transformation on

§N

8

+ IY -p2 81

2

eN+2,

is given by (9.15).

Since (non)vanishing of the eo-component of a point in lP'(ffi.N+1,l) is not invariant under a general Mobius transformation, there is no distinction in Mobius geometry between hyperspheres and hyperplanes. The elements of the isotropy subgroup Oto(N + 1,1) of Lorentz transformations which fix e oo are generated by reflections in the hyperspheres (9.19), which induce reflections in the hyperplanes of ffi.N. Therefore, Oto(N + 1,1) is identified with E(N), the group of Euclidean motions of ffi.N. It is convenient to work with spinor representations of these groups. Recall that the Clifford algebra e£( N +1, 1) is an algebra over ffi. with generators el, ... ,eN+2 E ffi.N+1,l subject to the relation

e

e

This implies that 2 = -(e, e); therefore any vector E ffi.N+l,l \ JLN+l,l l = -e/(e,e). The multiplicative group generated by has an inverse the invertible vectors is called the Clifford group. We need its subgroup generated by the unit space-like vectors:

e-

9 = Pin+(N + 1,1) = {7/J = 6·· ·en: el = -I}, and its subgroup generated by the vectors orthogonal to e oo :

900 = Pin~(N + 1,1) = {7/J = 6 ... en :

e; = -1,

(ei, e oo ) = O}.

These groups act on ffi.N+1,l by twisted conjugations: A1/I('1]) = (_l)n7/J-l'1]7/J. In particular, for a vector with = -lone has:

e

Ae('1]) =

e

-e- l '1]e = e'1]e = '1] -

2(e, '1])e,

e.

which is the reflection in the hyperplane orthogonal to Thus, 9 is generated by reflections, while 900 is generated by reflections which fix eo, and

9. Foundations

350

therefore leave Ql{j invariant. Actually, 9 is a double cover ofO+(N +1,1) :::::: Mob(N), while 900 is a double cover of Ot,(N + 1,1) :::::: E(N). Orientation preserving transformations from 9, 900 form the subgroups J(

= Spin+(N + 1,1),

J(oo

=

Spin~(N + 1,1),

which are singled out by the condition that the number n of vectors ~i in the multiplicative representation of their elements 7jJ = 6 ... ~n is even. The Lie algebras of the Lie groups J( and J(oo consist of bivectors:

~ ~oo

spin(N+I,I)

= span{eiej: i,jE{O,I, ... ,N,oo}, i#j},

spinoo(N+I,I) = span{eiej: i,jE{I, ... ,N,oo},

i#j}.

9.4. Laguerre geometry Laguerre geometry is a sub geometry of Lie geometry, with hyperplanes distinguished among all hyperspheres, as the hyperspheres through 00. Thus, Laguerre geometry studies properties of hyperspheres invariant under the subgroup of Lie sphere transformations which preserve the set of hyperplanes. The following objects in JRN are elements of Laguerre geometry. • (Oriented) hyperspheres S = {x E JRN : Ix - cl 2 = r2} with centers c E JRN and signed radii r E JR, can be put into correspondence with (N + I)-tuples (c, r). • Points x E JRN are considered as hyperspheres of radius zero, and are put into correspondence with (N + I)-tuples (x,O). • (Oriented) hyperplanes P = {x E JRN : (v, x) = d}, with unit normals v E §N-l and d E JR, can be put into correspondence with (N + I)-tuples (v, d).

In the projective model of Lie geometry, hyperplanes are distinguished as elements of J!D(lL,N+l,2) with vanishing eo-component. Thus, Laguerre geometry studies the subgroup of Lie sphere transformations preserving the subset of J!D(lLN +1,2) with vanishing eo-component. There seems to exist no model of Laguerre geometry where hyperspheres and hyperplanes would be modelled as points of one and the same space. Depending on which of the two types of elements is modelled by points, one comes to the Blaschke cylinder model or to the cyclographic model of Laguerre geometry. We will use the first model, which has the advantage of a simpler description of the distinguished objects of Laguerre geometry, which are hyperplanes. The main advantage of the second model is a simpler description of the group of Laguerre transformations.

9.4. Laguerre geometry

351

The scene of both models consists of two (N + I)-dimensional projective spaces with dual spaces of homogeneous coordinates, ]RN,l,l and (]RN,l,l)*, which arise from ]RN+1.2 by "forgetting" the eo-, resp. eoo-components. Thus, ]RN,l,l is spanned by N + 2 linearly independent vectors el" .. , eN, eN+3, e oo , and is equipped with a degenerate bilinear form of signature (N, 1, 1) in which the above vectors are pairwise orthogonal, the first N being space-like: (ei' ei) = 1 for 1 :s: i :s: N, while the last two being time-like and isotropic, respectively: (eN+3,eN+3) = -1 and (eoo,e oo ) = 0, Similarly, (]RN,l,l)* is assumed to have an orthogonal basis consisting of el",· ,eN, eN+3, eo, again with an isotropic last vector: (eo, eo) = O. Note that one and the same symbol (', ,) is used to denote two degenerate bilinear forms in our two spaces, We will overload this symbol even more and use it also for the (nondegenerate) pairing between these two spaces, which is established by setting (eo, eexJ = - ~, in addition to the above relations. (Note that a degenerate bilinear form cannot be used to identify a vector space with its dual.) In both models mentioned above, • Hyperplane P = (11, d) is modelled as a point in the projective space JPl(]RN,l,l) with a representative

P= v + 2de oo + eN+3'

(9.33)

• Hypersphere S = (c, r) is modelled as a point in the projective space JPl( (]RN,l,1 )*) with a representative

(9.34)

Each of the models appears if we consider one of the spaces as a preferred (fundamental) space, and interpret the points of the second space as hyperplanes in the preferred space, In the Blaschke cylinder model, the preferred space is the space JPl(]RN,l,l) whose points model hyperplanes P C ]RN. A hypersphere S C ]RN is then modelled as a hyperplane {~E JPl(]RN,I,l) : (s,~) = O} in the space JPl(]RN,I,l). Basic features of this model are the following: (i) Oriented hyperplanes P C ]RN are in a one-to-one correspondence with the points p of the quadric JPl(lLN,I,I), where (9.35)

lLN,I,l = {~ E ]RN,I,I : (~,~) = O}.

(ii) Two oriented hyperplanes PI, P2 C ]RN are in oriented contact (parallel) if and only if their representatives PI, 'P2 differ by a vector parallel to e OC)) that is, if (PI, P2) = O. (iii) An oriented hypersphere S C ]RN is in oriented contact with an oriented hyperplane P C ]RN if and only if if (p, s) = O. Thus,

352

9. Foundations

a hypersphere 8 is interpreted as the set of all its tangent hyperplanes. The quadric lP'(IL N,l,l) is diffeomorphic to the Blaschke cylinder (9.36)

Two points of this cylinder represent parallel hyperplanes if they lie on one straight line generator of Z parallel to its axis. In the ambient space ]RN+l of the Blaschke cylinder, oriented hyperspheres 8 C ]RN are in a one-to-one correspondence with the hyperplanes nonparallel to the axis of Z:

8

(9.37)

rv

{(v,d) E

]RN+l:

(c,v) - d - r =

a}.

An intersection of such a hyperplane with Z consists of points in Z which represent tangent hyperplanes to 8 C ]RN, as follows from (9.11). In the cyclographic model, the preferred space is the space of hyper-

spheres (]RN,l,l)*, so hyperspheres 8 C ]RN are modelled as points s E lP'( (]RN,l,l )*), while hyperplanes P C ]RN are modelled as hyperplanes {e : (p, e) = o} c lP'( (]RN,l,l )*). Thus, a hyperplane P is interpreted as the set of hyperspheres 8 which are in oriented contact with P. Basic features of this model are the following: (i) The set of oriented hyperspheres 8 C ]RN is in a one-to-one correspondence with the points

a

(9.38)

=

(c, r)

of the Minkowski space ]RN,l spanned by the vectors eI, ... ,eN, eN+3. This space has interpretation as an affine part oflP'((]RN,l,l )*).

(ii) Oriented hyperplanes P

C

]RN can be modelled as hyperplanes in

]RN,l:

(9.39)

7r

= {(c, r)

E ]RN,l :

((v, 1), (c, r)) = (v, c) - r = d}.

Thus, oriented hyperplanes P E ]RN are in a one-to-one correspondence with the hyperplanes 7r C ]RN,l which make angle 7r / 4 with the subspace]RN = {(x,D)} C ]RN,l.

(iii) An oriented hypersphere 8

C ]RN is in oriented contact with an oriented hyperplane P C ]RN if and only if a E 7r.

(iv) Two oriented hyperspheres 8 1 ,82 C ]RN are in oriented contact if and only if their representatives in the Minkowski space aI, a2 E ]RN,l differ by an isotropic vector: ia1 - a2i = 0. In the cyclographic model, the group of Laguerre transformations admits

a beautiful description:

9.5. Plucker line geometry

353

Theorem 9.6. (Fundamental theorem of Laguerre geometry) The group of Laguerre transformations is isomorphic to the group of affine transformations of ffi,.N,l: y f---4 AAy + b with A E O(N, 1), A > 0, and b E jRN.l.

9.5. Pliicker line geometry In this section we denote the homogeneous coordinates of a point x E ffi,.1P'3 by x = (xO,x 1 ,x2 ,x3 ) E ffi,.4. For the sake of notational convenience, we abbreviate V = JR.4. In the standard way, projective subspaces of ffi,.1P'3 are projectivizations of vector subspaces of V. In particular, let x, y E JR.1P'3 be any two different points, and let x, fj E V be their arbitrary representatives in the space of homogeneous coordinates. Then the line g = (xy) C ffi,.1P'3 is the projectivization of the two-dimensional vector subspace span(x, fj) C V. After H. Grassmann and J. Plucker, the latter subspace can be identified with (a projectivization of) the decomposable bivector (9.40) We choose a basis of A2V to consist of ei 1\ ej with 0 ::; i < j < 3. A coordinate representation of the bivector (9.40) in this basis is (9.41)

9=

2::gij

ei 1\ ej,

(ij)

The numbers (gOl, g02, g03, g12 , g13, g23) are called PlUcker coordinates of the line g. They are defined projectively (up to a common factor). Indeed, changing the choice of the two points defining 9 from x, y to x, y with the homogeneous coordinates xj = ax j + byj, yj = cx j + dyj, ad - bc =1= 0, would lead to a simultaneous multiplication of all gij by a common factor: gij

=

(ad - bc)gij.

Not every bivector represents a line in ffi,.1P'3, since not every bivector is decomposable, as in (9.40). An obvious necessary condition for a non-zero 9 E A2 V to be decomposable is (9.42)

9 1\ 9 = O.

It can be shown that this condition is also sufficient. In Plucker coordinates, this condition can be written as

(9.43) Summarizing, we have the following description of £}, the set of lines in JR.1P'3, within Plucker line geometry. The six-dimensional vector space A2V

354

9. Foundations

with the basis e.i I\ek is supplied with a nondegenerate scalar product defined by the following list of nonvanishing scalar products of the basis vectors: (eo 1\ e1, e2 1\ e3)

=

-(eo 1\ e2, el 1\ e3)

=

(eo 1\ e3, e1 1\ e2)

=

l.

It is not difficult to verify that the signature of this scalar product is (3,3), so that we can write A 2 V ':::' lR 3 ,3. Denote

(9.44) The points of the PlUcker' quadric lP'(IL3,3) are in a one-to-one correspondence with elements of I.., 3 . A fundamental feature of this model is the following: • Two lines g, h in lRlP'3 intersect if and only if their representatives in A2V are polar to one another: (9.45)

(9, h) =

gOlh 23 - g02h 13

+ g03h l2 + g 23 h Ol

_ g 13 h 02

+ g 12 ho:3 = O.

In this case the line P C lP'(A2V) through [9] and [h] is isotropic:

P c lP'(IL 3,3). To prove this, note that if the lines g, h intersect at the point z, then 9 = x 1\ Z and 11 = Y 1\ z, and then 91\12 = O. Conversely, if the lines g, h do not intersect, then their lifts to V span the whole of ~/, and so 91\11, =1= o. It remains to observe that 91\ h = (9, h.) eo 1\ e1 1\ e2 1\ e3. Next, we turn to important linear subsets of the Plucker quadric. • Any isotropic line P C lP'(IL3.3) corresponds to a one-parameter family of lines in lRlP'3 through a common point, which lie in one plane. Such a family of lines is naturally interpreted as a contact element (a point and a plane through this point) within the line geometry. • Other than in Lie geometry, in the present case of signature (3,3) there exist also isotropic planes, which are projectivizatiolls of 3dimensional vector subspaces of A2V that belong to IL 3.:3. There are two sorts of isotropic planes in the Pliicker quadric lP'(IL·3,:3). An isotropic plane can represent: 0:) a two-parameter family of all lines in lRlP'3 through some common point; such a family is naturally identified with that common point; (3) a two-parameter family of all lines in some plane in lRlP'3; such a family is naturally identified with that common plane. To see why the latter statement holds, consider three noncollinear points in the isotropic plane. Their pairwise connecting lines are all isotropic. Therefore these three points represent three pairwise intersecting lines in lRlP'3. If all three are concurrent, then we are in the situation 0:). Otherwise they lie in a plane in lRlP'3, and we are in the situation ;3).

355

9.5. PlUcker line geometry

Projective transformations of W'(lR3,3) which leave the Plucker quadric W'(1L 3 ,3) invariant can be distinguished depending on their action on the two types of isotropic planes.

Theorem 9.7. (Fundamental theorem of Plucker line geometry)

a) The group of projective transformations of lRW'3 is isomorphic to the subgroup of 0(3, 3)/(±I) consisting of transformations which preserve the types 0;) and (3) of the three-dimensional vector subspaces in 1L3 ,3. b) The group of correlative transformations of lRW'3 is isomorphic to the subgroup of 0(3, 3)/(±I) consisting of transformations which interchange the types 0;) and (3) of the three-dimensional vector subspaces in 1L3 ,3. Next, we discuss planar families of lines. Such a family of lines is represented by a conic section W'CEnIL3,3), where L; stands for a three-dimensional vector subspace of lR3 ,3 such that the restriction (-, .) I~ is nondegenerate. It is not difficult to realize that four pairwise nonintersecting (skew) lines in lRW'3 belong to a planar family (have linearly dependent representatives in 1L3 ,3) if and only if they belong to a regulus (one family of generators of a ruled quadric in lRW'3, i.e., of a one-sheet hyperboloid or of a hyperbolic paraboloid). The complementary regulus is represented by the dual planar family of lines lJD(L;..l n 1L3,3). Finally, we briefly mention the duality in Plucker line geometry. One can describe any projective subspace W'(L;) c JRW'3 through its dual subspace W'(L;..l) c (lRW'3) *, where L;..l C V* is the annihilator of the vector subspace L; c V. As a set of points, W'(L;) is the intersection of planes represented by lJD(L;..l). Thus, a plane U C JRW'3 can be described through an element of lJD(L;..l) C (JRlJD3)* with homogeneous coordinates fl = (UO,UI,U2,U3) = L:?=o ulei E V*. As a set of points, this plane consists of x E JRlJD3 with homogeneous coordinates x = (x O, xl, x 2, x 3) E V satisfying 3

(9.46)

LU1X1 =0. [=0

This description of U is dual to the description as the projectiivization of the three-dimensional vector subspace span(x, fj, z) C V, where x, fj, z are homogeneous coordinates of any three noncollinear points x, Y, z E u. In the spirit of the Grassmann-Plucker approach, the latter vector subspace can be represented by (a projectivization of) the decomposable three-vector it = x 1\ fj 1\ z E A3V. In the basis of A3V consisting of ei 1\ ej 1\ ek, o ::; i < j < k ::; 3, one has:

it =

L (ijk)

) ei u t"k

1\ ej 1\ ek ,

u ijk =

Xi

x")

Xk

Yi

Yj

Yk

Zi

-)

'Y"

Zk

9. Foundations

356

It is easy to see that the homogeneous coordinates that Uo

=

u 123 ,

U1

=

_u0 23 ,

U2

= u0 13 ,

Ul U3

can be normalized so

=

_u0 12

Similarly, in the dual description, any line 9 C lR.1fD3 can be viewed as an intersection of two planes u, v C lR.1fD3, and thus can be described through span( il, v) E V*, which, in turn, can be represented by (a projectivization of) the bivector (9.47) In coordinates: (9.48)

9=

il /\

v=

L gij e; /\ ej, (ij)

The sextuple of numbers (g01, g02, g03, g12, g13, g23) is called dual PlUcker coordinates of the line g. Remarkably, this new set of coordinates is related to the previously introduced Pliicker coordinates in a fairly simple way: if u, v are any two planes in lR.1fD3 intersecting along the line g, then their homogeneous coordinates Ui, Vi can be so normalized that the dual Plucker coordinates (9.48) of the line 9 = un v coincide, after a suitable reordering, with its coordinates (9.41): (9.49)

gOl g23

= g23, = gOl,

g02 g13

= =

_g13,

g03

_g02,

g12

= g12; = g03.

To see this, take 9 = (xy) = un v and choose points p E u, q E v so that p (j. v and q (j. u. We can normalize homogeneous coordinates of the planes u, v so that

(~~ ~i ~: ~:) (~: ~:) qO

q1

q2

q3

11,3

=

v3

(H) . 1 0

Now (9.49) follows from a well-known generalization of the Cramer rule, which says that the 2 x 2 determinants gij = xiyj - xjyi are proportional to the 4 x 4 determinants obtained from the matrix of the latter linear system by replacing the i-th and j-th columns by the columns on the right-hand side of the system. To conclude, we mention a couple of useful relations for the usual and dual Plucker coordinates of lines. They follow directly from definitions. • Homogeneous coordinates of the plane point p (j. 9 are given by

11,

through a line 9 and a

3

(9.50)

Uj

= L9jkpk k=O

(j = 0,1,2,3);

9.6. Incidence theorems

357

homogeneous coordinates of the intersection point x of a line 9 with a plane v not containing 9 are given by 3

x j = L9jkvk

(9.51)

(j=O,1,2,3).

k=O

• A point x belongs to a line 9 if and only if 3

(9.52)

L9jk Xk

=

k=O

°

(j = 0, 1, 2, 3);

a line 9 lies in a plane u if and only if 3 "~gJ"k Uk =

(9.53)

k=O

°

(j=O,1,2,3).

By the way, the last statement allows us to give a simple argument for the claim that any 9 E lL3,3 corresponds to a line in IRlP3 . Indeed, (9,9) = and 9 i= is equivalent to the fact that the rank of system (9.53) is equal to 2, and so the solution of this system delivers two different planes. They intersect along the line we are looking for.

°

°

9.6. Incidence theorems This section contains a collection of classical incidence theorems which lie in the basis of discrete differential geometry. We will often use the cross-ratio of four collinear points a, b, c, d, defined as (9.54)

l (a, b) l (c, d) q(a, b, c, d) = l(b, c) . l(d, a)'

and the fact that the cross-ratio is invariant under projective transformations. 9.6.1. Menelaus' and Ceva's theorems. Theorem 9.8. (Menelaus' theorem) Consider a triangle L::.(AIA2A3) in the plane. Let P12, P23 , P31 be some points on the side lines (AIA2), (A 2A 3), (A3Ad, respectively, different from the vertices Ai of the triangle. These three points are collinear if and only if

(9.55) Theorem 9.9. (Ceva's theorem) Consider a triangle L::.(AIA2A3) in the plane. Let P12 , P23 , P31 be some points on the side lines (A 1A 2), (A 2A3), (A3Ad, respectively, different from the vertices Ai of the triangle. The three

9. Foundations

358

Figure 9.6. Menelaus'theorem.

lines (A 1P23), (A 2P3t) , (A 3P 12) have a common intersection point if and only if (9.56)

Figure 9.7. Ceva's theorem.

Both Menelaus' and Ceva's theorems have a similar flavor: their hypotheses are of a seemingly affine-geometric nature (the left-hand sides of equations (9.55), (9.56) are expressed in terms of quotients of directed lengths), while their conclusions are projectively invariant. Actually, one can show that the numeric value of the cyclic product of the quotients of directed lengths on the left-hand sides of (9.55), (9.56) is itself a projectively invariant quantity. This is a consequence of the following theorem. Theorem 9.10. (Projective invariance of a cyclic product of directed lengths ratios for a triangle) Consider a triangle 6(AIA2A3) in

9.6. Incidence theorems 359

the piane, and let P 12 , P", P" be Some points on the lines (A,A,), (A,A ), 3 (A"A,J, respectively, different fmm the vertices A; of the bia"qle, a) Drno9.8. te byThen Q12 the intersection point of the line (A, A,) with (P P ); 8('e Figurr: 23 31

() 9,57

"~_q (A

I(A I , P12) I(A 2, P2.3 ) I(A:l , P3J) I(P12 , A 2) I(P2:l, A 3) 1(P..11, Ad

A Q )

P 1,

[2,

"

.

~

12,

b) Set 0 (A,P",) n (A,p,,) and denote by Ii" the intersection point of the line (A,A 2) with (A:30); 8ee Figure 9.9. Then

() 9,58

I(A" P,,) i(A" P2J ) I(A" P31) I(P", A,) - I(P", A;} - I(P", AJ)

~ q(A

PAR ) 1,

12,

"

Figure 9.8. A projectively invariant meaning of the cyclic product from J\.Iendaus' theorem.

Figuretheorem. 9.9. A projectiwly invariant meaning of the cyclic product frorn Ceva's

-12-

360

9. Foundations

Proof. Clearly, this theorem yields Menelaus' and Ceva's: the cross-ratios on the right-hand sides of equations (9.57), (9.58) are equal to 1 if and only if Ql2 = P 12 , resp. Rl2 = P 12 . For a proof, note that, since both sides of (9.57), (9.58) are invariant under affine transformations, it is enough to consider Al = (0,0), A2 = (1,0), A3 = (0,1), and then P l2 = (XI'O)' P 23 = (1- X2,X2), P31 = (0,1- X3) with some XI,X2,X3 E IR \ {O, I}. Then a straightforward computation confirms both claims of the theorem. 0 9.6.2. Generalized Menelaus' theorem. Upon using the results of Theorem 9.10 to "cut off vertices", one can prove the following result. Theorem 9.11. (Projective invariance of cyclic products of directed lengths ratios) Let AI, A 2, ... , An be n 2': 3 points in IRm such that no three consecutive points in cyclic order are collinear. Let P 12 , P 23 , ... , P nl be points on the lines (AIA2)' (A2A3),"" (AnAl), respectively. Then the product of ratios of directed lengths

rr n

l(Ai, Pi,i+l)

i=l

l(Pi,HI, AHI )

is invariant under projective transformations.

The geometric meaning of the situation where the cyclic product in Theorem 9.11 takes a special value (_l)n is given by the following result. Theorem 9.12. (Generalized Menelaus' theorem) Let AI"'" An be n points in general position in IRn - l , so the affine space through the points Ai is (n - I)-dimensional. Let Pi,i+l be some points on the lines (AiAi+d different from A, AHI (indices are taken modulo n). The n points Pi,i+l lie in an (n - 2)-dimensional affine subspace if and only if the following relation for the quotients of the directed lengths holds:

IT

l(Ai,Pi,i+l) .z=l l(Pi ' HI, AHd

= (-It.

Proof. The points Pi ,Hl lie in an (n - 2)-dimensional affine subspace if there is a nontrivial linear dependence n

L

n

J1i P i,i+1 = 0

with

i=l

L J1i = O. i=l

Substituting Pi,i+l = (1 - ~i)A + ~iAHI' and taking into account the general position condition, which can be read as linear independence of the --+ vectors AlAi, we come to a homogeneous system of n linear equations for n coefficients J1i: ~iJ1i

+ (1 -

~i+I)J1HI = 0,

i = 1, ... , n

361

9.6. Incidence theorems

(where indices are understood modulo n). Clearly it admits a nontrivial solution if and only if

n~ rr 1 c. i=l

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