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

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

Graduate Studies in Mathematics Volume 98

Editorial Board David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 53-01, 53-02; Secondary 51Axx, 51Bxx, 53Axx, 37Kxx, 39A12, 52C26.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-98

Library o f Congress C atalogin g-in-P u b lication D a ta Bobenko, Alexander I. Discrete differential geometry: integrable structure / Alexander I. 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. I. Suris, Yuri B., 1963- II. Title. QA672 .B63 2008 516.3'62— dc22

2008029305

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13 12 11 10 09 08

Contents

Preface

xi

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

xiii xiii xv xvii xxi xxii xxiii 1 2 2 3 4 5

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

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

vi

Contents

1.4.5. 1.5.

Ribaucour transformations of orthogonal nets

Principally parametrized sphere congruences

17 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 2.4.1. 2.4.2. 2.4.3.

66 66 70 72

2.5.

Exercises

73

2.6.

Bibliographical notes

82

asymptotic nets Notion and consistency of discrete asymptotic nets Discrete Lelieuvre representation Transformations of discrete A-nets

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

vii

3.2.

Q-nets in quadrics

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

Discrete asymptotic nets in Pliicker line geometry

118

3.9.

99

3.10. Exercises

120

3.11. Bibliographical notes

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.

Bibliographical notes

183

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

5.9.

Bibliographical notes

207

Chapter 6. Consistency as Integrability 6.1.

Continuous integrable systems

209 210

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 6.7.

From 3D consistency to zero curvature representations and Backlund transformations

220 222

6.8. 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

6.9.

3D consistent equations with noncommutative fields

235

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

239

6.11. Proof of the classification theorem 6.11.1. 3D consistent systems, biquadratics and tetrahedron property 6.11.2. Analysis: descending from multiaffine 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.

Pliicker 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

357 357 360 361 362 364 366 367

Appendix. Solutions of Selected Exercises A.I.

369

Solutions of exercises to Chapter 2

369

A.2. Solutions of exercises to Chapter 3

376

A.3.

Solutions of exercises to Chapter 4

377

A.4.

Solutions of exercises to Chapter 6

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 TU Berlin and TU 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 intelli­ gent 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. All the readers are encouraged to read or at least to skim the Introduc­ tion (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.

Introduction

W h a t is discrete differential geom etry. A new field of discrete dif­ ferential 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), dis­ crete 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, nu­ merics and computer graphics by Janzen-Polthier (2007). For a given smooth geometry one can suggest many different discretiza­ tions with the same continuous limit. Which is the best one? From the theo­ retical point of view, one would strive to preserve fundamental properties of the smooth theory. For applications the requirements of a good discretiza­ tion 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” ) intro­ duced by Pinkall-Polthier (1993) in their investigation of discrete minimal

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 the­ ory 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 clas­ sical 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 differ­ ential 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 Darboux (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 sur­ prising that remarkable discretizations yield remarkable discrete equations.

Introduction

xv

The main message of discrete differential geometry, addressed to the inte­ grable 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 develop­ ment of the integrability aspects of discrete differential geometry. The classi­ cal 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 exten­ sive volumes by Darboux (1910, 1914-27), Bianchi (1923), etc. One can say that the local differential geometry of special classes of surfaces and coordi­ nate systems has been completed during this period. Mathematicians of that era have found most (if not all) geometries of interest and knew nearly every­ thing about their properties. It was observed that special geometries such as minimal surfaces, surfaces with constant curvature, isothermic surfaces, or­ thogonal 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 real­ ized 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 equa­ tions. The theory of integrable systems (called also the theory of solitons) is a vast field in mathematical physics with a huge literature. It has applica­ tions 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 equa­ tion 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 con­ stant 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 sys­ tems 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 Backlund-Darboux transformations. These special trans­ formations are often used to generate new solutions from the known ones. It is a characteristic feature of soliton (integrable) partial differ­ ential 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 integrabil­ ity?” , Zakharov (1991), clearly demonstrates). Different scientists suggest different properties as the defining ones. Usually, one refers to some addi­ tional 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 struc­ tures. In the theory of solitons the problem is to discretize an integrable dif­ ferential 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 corre­ sponding linear problems. This idea appeared first in Ablowitz-Ladik (1975).

Introduction

xvii

Its various realizations based on the bilinear method, algebro-geometric inte­ gration, integral equations, R-matrices, and Lagrangian mechanics were de­ veloped 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 alge­ braic curves based on algebro-geometric methods of integration of difference equations was given in Krichever (2006). From discrete to sm ooth . 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 Wunder­ lich (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 wit­ nessed a rapid development reflected in numerous publications. In particu­ lar, 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 (i.e., maps from the regular cubic lattice Z m 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 direc­ tions. A characteristic feature of the theory is that all lattice directions are considered on an equal footing with respect to the defining geometric prop­ erties. Due to this symmetry, discrete surfaces and their transformations become indistinguishable. We associate such a situation with the multidi­ mensional consistency (of geometric properties, and of the equations which serve for their analytic description). In each case, multidimensional con­ sistency, and therefore the existence and construction of multidimensional

xviii

Introduction

discrete nets, is seen to rely on some incidence theorems of elementary ge­ ometry. 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 elemen­ tary geometric properties of the original multidimensional discrete nets. In particular, difficult classical theorems about the permutability of BacklundDarboux type transformations (Bianchi permutability) for various geome­ tries 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 differ­ ential geometry these elementary roots remain hidden. The limiting process taking the discrete master theory to the classical one is inevitably accompa­ nied 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-geomet­ ric 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 differen­ tial 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 differ­ ential 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 equa­ tions may serve as a constructive and almost algorithmic definition of their integrability. This idea was introduced in Bobenko-Suris (2002a) (and inde­ pendently in Nijhoff (2002)). This definition of integrability captures enough structure to guarantee such traditional attributes of integrable equations as zero curvature representations and Backlund-Darboux transformations (which, in turn, serve as the basis for applying analytic methods such as in­ verse 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 cor­ responding difference and differential equations. However one should not underestimate the difficulty of the convergence theorems required for a rig­ orous justification of this philosophy. Solutions to similar problems are substantial in various areas of differen­ tial 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)). Alexan­ drov’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

Discrete Differential Geometry

Integrability

integrable equations

zero-curvature representation BacklundDarboux transformations surfaces and their transformations

y V

/

hierarchies of commuting flows

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 geom­ etry) 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 curva­ ture 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 Z 2. 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 Zd. 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).

Structu re o f this b o o k . 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 consid­ ered 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 con­ stant negative Gaussian curvature. There are no proofs in this chapter. The analytic proofs are usually tedious and can be found in the original litera­ ture. 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 (Eu­ clidean, spherical, hyperbolic, Mobius, Pliicker, 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 dis­ crete analogs of curvature line parametrized surfaces and orthogonal nets, and give a description of discrete asymptotic nets within the framework of Pliicker 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 isothermic 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 equa­ tions 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 Backlund-Darboux transformations from the consistency principle. Moreover, we obtain a com­ plete 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 intro­ duction to projective geometry and the geometries of Lie, Mobius, Laguerre and Pliicker. We also include a number of classical incidence theorems rel­ evant to discrete differential geometry. H ow to read this b o o k . Different audiences (see the Preface) should read this book differently, as suggested in Figure 0.3. Namely, Chapter 1 on clas­ sical 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

xxiii

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 Miinchen, with no knowledge of differential geometry required. Those interested primarily in applications of discrete dif­ ferential 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

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

A cknow ledgem ents. Essential parts of this book are based on results ob­ tained jointly with Vsevolod Adler, Tim Hoffmann, Daniel Matthes, Chris­ tian 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 dis­ cussions 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 grate­ fully 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 M a t h e o n “Mathematics for key technologies” in Berlin.

Chapter 1

Classical Differential Geometry

In this chapter we discuss some classical results of the differential geome­ try 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; or­ thogonal nets, including curvature line parametrized surfaces), the situation is somewhat different. The corresponding permutability theorems (Theo­ rems 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 quadrilater­ als. In our opinion, this is not the proper setting in the three-dimensional

1

2

1. Classical Differential Geometry

context, and the nonuniqueness of the fourth net in these theorems reflects this. The natural setting for permutability is given in the second part, where the permut ability 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 / : Rm —> RN by u — G Km, and we set = d/dUi. All nets are supposed to be sufficiently smooth, so that all the required partial derivatives exist. We write = {u e R m :

Ui =

0 for i ^ n , ... , i s}

for s-dimensional coordinate planes (coordinate axes, if s = 1). 1.1. C o n ju g a te n ets 1.1.1. N otion o f con ju gate nets. We always suppose that the dimension of the ambient space N > 3. D efinition 1.1. (C on ju gate net) A map f : Rm —> RN is called an m-dimensional conjugate net in RN if at every u G Rm and for all pairs l < i ^ j < m we have didjf G span ( 2 support conjugate nets). From Definition 1.1 it follows that the conjugate nets are described by the (linear) differential equations (1.1)

didjf = Cjidif + Cijdjf,

i # j,

with some functions Cij : Rm —►R. Compatibility of these equations, i.e. the requirement di(djdkf) = dj(didkf), is expressed by the following system of (nonlinear) differential equations: (1.2)

diCjk = CijCjk + CjiCik — CjkCik,

i y^ j ^ k ^ i.

Note that the latter equations for the coefficients do not contain / any­ more. 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 / uniquely: (Q i) the values of / on the coordinate axes 25* for 1 < i < ra, i.e., m smooth curves f\^. with a common intersection point / ( 0); (Q2) the values of Cij, cji on the coordinate planes 23^ for all 1 < i < j < ra, i.e., m(m — 1) smooth real-valued functions Cij\^. of two variables. 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 RN of a conjugate net should be interpreted as an affine part of the projective space WPN — P(IRAr“hl), with R^ +1 being the space of homogeneous coordinates. Equations (1.1) hold then for the standard lift ( /, 1) G R^ +1 of the conjugate net / ~ [ / : 1] £ WPN, while an arbitrary lift f = A( /, 1) £ R n + 1 is characterized by a more general linear system (1.3)

didjf = Cjidif + Cijdjf + pijf,

i+ j

(with the corresponding compatibility conditions for the coefficients Qj, pij, which generalize equations (1.2)). We will not pursue this description fur­ ther. 1.1.2. A ltern ative analytic d escrip tion o f con ju gate nets. A classical description of conjugate nets makes use of the following construction. Given the functions , define functions gi : Rm —> M* as solutions of the system of differential equations (1.4)

d{Qj — Cijgj ,

i 7^ j.

Compatibility of this system is expressed as diCjk — djc^ and is a conse­ quence of equations (1.2) (whose right-hand sides are symmetric with re­ spect to the flip i j). Solutions gi can be specified by prescribing their values arbitrarily on the corresponding coordinate axes 33*. Define vectors Wi = g~ldif. It follows from (1.1) and (1.4) that these vectors satisfy the following differential equations: (1.5)

diWj — -CjiWi,

i 7^ j .

9j

Thus, defining the rotation coefficients as ( 1 .6)

7ji

= ~ cjii yj

we end up with the following system: (1.7) (1.8) (1*9)

dif

=

giWi,

d{Wj =

7 jiW i,

fygj

=

Qilij >

i 7^ j, i 7^ j-

4

1. Classical Differential Geometry

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

di'Jkj — Ifki^fij -i

^/ J / ^

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 conju­ gate nets: while the functions Cij describe the local geometry of a net, this is not the case for the rotation coefficients 7^. Indeed, to define the latter, one needs first to find gi as solutions of differential equations (1.4). 1.1.3. Transform ations o f con ju gate nets. The most general class of transformations of conjugate nets was introduced by Jonas and Eisenhart. D efinition 1.2. (Fundam ental transform ation) Two m-dimensional conjugate nets / , / + : Mm —> RN are said to be related by a fundamental transformation (F-transformation) if at every point u G Mm of the domain and for each 1 < i < m the three vectors d if, d if+ and Sf — f + — f are coplanar. The net f + is called an F-transform of the net f . This definition yields that F-transformations are described by the fol­ lowing (linear) differential equations: (1.11)

dif + = aidif + bi(f+ - f ) .

Of course, the functions a^, bi : Rm —>R must satisfy (nonlinear) differential equations, which express the compatibility of (1.11) with (1.1): (1*12)

&iCLj

— (Q>i

(1.13)

dibj

=

(1.14)

(ijC^j

Q>j)Cij ~j“ bi{cLj

1),

ctb j + c p i - b j b i ,

— OjCjj "i- 6^(flj

1).

The following data determine an F-transform / + of a given conjugate net f uniquely: (Fi) a point / + (0); (F2) the values of a^, bi on the coordinate axes for 1 < i < m, i.e., 2m smooth real-valued functions bi [3 . 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 (/(u ), f(u + ^e^), f(u + tiCi + c^ej), f (u + tjej)), while the condition of Definition 1.2 can be interpreted as planarity of in­ finitesimally narrow quadrilaterals ( f ( u ) , f ( u + e^ei),/+ (w + e^e*), f + (u)).

1.1. Conjugate nets

5

One can iterate F-transformations and obtain a sequence / , / + , ( / + )+ , 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. T h eorem 1.3. (P erm utability o f F -transform ations) 1) Let f be an m-dimensional conjugate net, and let and f ^ be two of its F-transforms. Then there exists a two-parameter family of conjugate nets / ( 12) that are F-transforms of both f ^ and f^2K Corresponding points of the four conjugate nets f , f ^ \ f ^ and f ^ are coplanar. 2) Let f be an m-dimensional conjugate net. Let f^ \ f ^ and f ^ be three of its F-transforms, and let three further conjugate nets f^12\ f ^ and f (13) be given such that f ^ is a simultaneous F-transform, of f ^ and f^ \ Then there exists generically a unique conjugate net / ( 123) that is an F-transform of f^l2\ f ^ and f ^ . The net / ( 123) is uniquely defined by the condition that for every permutation (ijk ) of (123) the corresponding points of f ( %\ f ^ \ f ^ and / ( 123) are coplanar. The situations described in this theorem can be interpreted as conjugate nets of dimension M = m + 2, resp. M = 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 the­ ory. D efinition 1.4. (C om bescu re transform ation) We will say that two Tri­ dimensional conjugate nets / , f + : Mm —> RN are related by a Combescure transformation if at every point u G Mm and for each 1 < i < m the vectors dif, d if+ are parallel. The net / + is called parallel to f , or a Combescure transform of the net f . 1.1.4. Classical form ulation o f F -transform ation. Our formulation of F-transformations is rather different from the classical one, due to Jonas and Eisenhart, based on the formula /,

,4-

,

4>

1. Classical Differential Geometry

6

whose data are: an additional solution 0 : Rm —> R of (1.1), a Combescure transform p : Rm —> RN of / , and the function ijj : Rm —►R, associated to 4> in the same way as p is related to / . 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 6'j \

bj

bj b^ + Cji

C%j CLj

. (Xi

Q'j

The symmetry of the right-hand sides of (1.16), (1.13) yields the existence of the functions 0 , : Rm —►R such that /

s

(1.17)

did) bi — = —, + ++ cpi+ c± c±dj<j>+ ,

for all 1 < i 7^ j < m. Thus, an F-transformation yields some additional scalar solutions (j) and of the equations describing the nets / and / + , respectively. Of these two, the solution 0 is directly specified by the original net / and the initial data (F2). Indeed, the data (F2) yield the values of (j) along the coordinate axes, through integrating the first equations in (1.17); these values determine the solution of (1.18) with the known coefficients c^ uniquely. Further, introduce the quantities

d-aoj

* = -£■

Then a direct computation based on (1.11), (1.12)-(1.14), and (1.17) shows that the following equations hold: (1.21)

dip

=

atidif,

(1.22)

diip

=

aidi(f),

where (1.23)

on = a i ~ 1 <j)+

1.2. Koenigs and Moutard nets

7

Thus, p is a Combescure transform of / , and ^ is a function associated to 0, in Eisenhart’s terminology. Another computation leads to the relation (1.24)

diCtj =: Cij{oti

otj).

The same argument as above shows that the data (F2) yield the values of + , and thus the values of a*, on the coordinate axes . This uniquely specifies the functions a* everywhere on Rm as solutions of the compatible linear system (1.24) with the known coefficients C{j. This, in turn, allows for a unique determination of the solutions p, ip of equations (1.21), (1.22) with the initial data p(0) = / + (0) — / ( 0) and ^(0) = 1 (here the data (Fi) enter into the construction). Thus, the classical formula (1.15) is recovered. 1.2. K o e n ig s a n d M o u t a r d n ets 1.2.1. N otion o f K oen igs and M ou tard nets. A geometrically impor­ tant subclass of two-dimensional conjugate nets, very popular in the classical differential geometry, can be most directly defined as follows. D efinition 1.5. (K oen igs net) A map f : R2 —> RN is called a Koenigs net if it satisfies a differential equation (1.25)

did2f = (d2 log 1/) d if + (di logi/) d2f

with some scalar function u : M2 —>R + . In other words, a Koenigs net is a two-dimensional conjugate net with the coefficients C21, cyi satisfying d\C2\ — ^2Ci2. 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 / uniquely: (Ki) the values of / on the coordinate axes S i, ®2, with a common intersection point / ( 0);

two smooth curves

(K2) a smooth function v : R2 -> R+. Leaving aside numerous geometric properties of Koenigs nets, discovered by the classics, we formulate here only the following characterization. T h eorem 1.6. (C hristoffel dual for a K oen igs net) A conjugate net f : R2 —> R^ is a Koenigs net if and only if there exists a scalar function v : R2 —►R+ such that the differential one-form df* defined by * _ d' f

*

1. Classical Differential Geometry

8

is closed. In this case the map /* : R2 —> , 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:

(i.27)

(^ + d2)r

ft/* I ft/, ft/* lift/, i (ft - %)/, (ft - d2)r i (ft + ft)/.

If one considers the ambient space RN of a Koenigs net as an affine part of RP^, then there is an important choice of representatives for / ( /, 1) in the space R N+1 of homogeneous coordinates, namely (1-28)

y = v - \ f , 1).

Indeed, a straightforward computation shows that the representatives (1.28) satisfy the following simple differential equation: (1.29)

d\&2 y = q\2V

with the scalar function Mj/V+1 is called a Moutard net if it satisfies the Moutard differential equation (1.29) with some q\ 2 : R2 -> 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 R ^ -1-1, then it is not difficult to figure out the condition for a scalar function v : R2 —►R, under which / = vy satisfies an equation of the type (1.1): v~ l has to be a solution of the same Moutard equation (1.29), and then ft f t / = (ft log v)d\ f + (ft log v)d 2f . For instance, one can choose v~l 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 R^. Of course, Moutard nets can be considered also in their own right, i.e., one does not have to regard the ambient space R^ -1-1 of a Moutard net as the space of homogeneous coordinates for MFN. Nevertheless, such an interpretation is useful in most cases. The following data determine a Moutard net y uniquely: (Mi) the values of y on the coordinate axes 231, *B2, i.e., two smooth curves y\M that has the meaning of the coefficient of the Moutard equation. 1.2.2. Transform ations o f K oen igs and M ou tard nets. Moutard in­ vented a remarkable analytic device for transforming Moutard nets. D efin ition 1.8. (M ou ta rd transform ation) Two Moutard nets : ]R2 —►RN are called Moutard transforms of one another if they satisfy (lin­ ear) differential equations (1.30)

diy+ + diy

=

pi(y+ - y ) ,

(1.31)

d2y+ - d 2y

=

p2 {y+ + y),

with some functions pi,p 2 : M2 —> K (or similar equations with all plus and minus signs interchanged, which is also equivalent to renaming the coordi­ nate axes 1 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): (1-32)

dip 2 = d2pi

(1-33)

=

- q i 2 + p i p 2,

=

- q u + Zpm -

The following data determine a Moutard transform y+ of a given Moutard net y: (M Ti) a point y+ (0) G RN; (M T2) the values of the functions pi on the coordinate axes *Bi for i = 1, 2, i.e., two smooth functions Pil^ of one variable. If the Moutard nets y, y+ in R7V+1 are considered as lifts of Koenigs nets / = [y], / + = [y+] in MN, then a geometric content of the Moutard transformation can be easily revealed. Introduce two surfaces jp (1) , Jp(2) : M2 —* R n with the homogeneous coordinates F (1) = [y+ + y\,

F (2) = [y+ - y).

Then for every u G M2 the points F ^ lie on the line ( / / + ), and equations (1.30), (1.31) show that this line is tangent to both surfaces F^\ F ^ . One says that these surfaces are focal surfaces of the line congruence ( / / + ) . Now an easy computation shows that on each such line the four points / , /+ , F (2) build a harmonic set, that is, (1-34)

g( / , F (1), / + , F (2)) = - 1,

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

1. Classical Differential Geometry

10

D efinition 1.9. (K oen igs transform ation) Two Koenigs nets / , / + : M2 —> R n are said to be related by a Koenigs transformation if the focal points F^l\ F o f the line congruence ( f f + ) separate the points f , f + harmonically. It can be shown that any Koenigs transformation is analytically repre­ sented as the Moutard transformation (1.30), (1.31) by a suitable choice of Moutard lifts y , y + . T h eorem 1.10. (P erm utability o f M ou tard transform ations) 1) Let y be a Moutard net, and let y ^ and y ^ 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 ^ and y^2\ 2) Let y be a Moutard net. Let y^l\ y ^ and y ^ be three of its Moutard transforms, and let three further Moutard nets y^l2\ y ^ and y (13) be given such that y W is a simultaneous Moutard transform of y ^ and y ^ . Then generically there exists a unique Moutard net ? /123) that is a Moutard trans­ form of y(12\ j / 23) and y(13\ 1.2.3. Classical form ulation o f the M ou tard transform ation. Due to the first equation in (1.32), for any Moutard transformation there exists a function 6 : R2 —►M, unique up to a constant factor, such that

(1 (1-35)

pi = —dl° —,

p2 = —d2° Y'

The last equation in (1.32) implies that 6 satisfies (1.29). This scalar solution of (1.29) can be specified by its values on the coordinate axes (i = 1, 2), which are readily obtained from the data (MT2) 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 6 of this equation, via (1.30), (1.31) with (1.35). Note that these equations can be equivalently rewritten as (1.36)

d1 (0y+ ) = - e 2di ( | ) ,

d2 (ey+ ) = e 2d2 ( J ) .

From these equations one can conclude that y+ solves the Moutard equation (1.29) with the transformed potential (1.37)

q+2 = qx2 - 2 did 2 \og6 =

6 + = J.

In our formulation, the origin of the function 0 becomes clear: it comes from Pi) P 2 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. A s y m p t o t ic n ets D efinition 1.11. (A -su rface) A map f : R2 —> R3 is called an A-surface (an asymptotic line parametrized surface) if at every point the vectors d f f , d\f lie in the tangent plane to the surface f spanned by d\f, d2f . Thus, the second fundamental form of an A-surface in R3 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 re­ formulated so as to contain projectively invariant notions only. Therefore, A-surfaces actually belong to the geometry of the three-dimensional projec­ tive space. In our presentation, however, we will use for convenienpe addi­ tional structures on R3 (the Euclidean structure and the cross-product). A convenient description of A-surfaces is provided by the Lelieuvre representa­ tion which states: there exists a unique (up to sign) normal field n : R2 —>R3 to the surface / such that (1.38)

d if = d\n x n,

d2f — n x d2n.

Cross-differentiation of (1.38) reveals that d\d2n x n = 0, that is, the Lelieu­ vre normal field satisfies the Moutard equation (1.39)

did2n = qi2n

with some q\ 2 : 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 / : R2 —>R3. T h eorem 1.12. (Lelieuvre norm als o f A -surfaces are M ou tard nets) A-surfaces / : R2 —> R3 are in a one-to-one correspondence (up to transla­ tions of f ) with Moutard nets n : R2 —>R3, via the Lelieuvre representation (1.38). An A-surface / 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 i^), which we denote in this context by (A i^)' (A i) the values of the Lelieuvre normal field on the coordinate axes ®i, 2$2, i.e., two smooth curves n |^. with a common intersection point n(0); (A2) a smooth function q\ 2 : R2 —►R that has the meaning of the coefficient of the Moutard equation for n. D efinition 1.13. (W eingarten transform ation) A pair of A-surfaces /, / + : R2 —> R3 is related by a Weingarten transformation if for every u G R2, the line (f( u) f+ (u )) is tangent to both surfaces f and / + at the corresponding points. The surface / + is called a Weingarten transform of the surface f . The lines ( / (u) f + (u)) are said to build a W-congruence.

1. Classical Differential Geometry

12

It can be demonstrated that the Lelieuvre normal fields of a Weingarten pair / , / + of A-surfaces satisfy (with the suitable choice of their signs) the following relation: (1.40)

/ + - / = n+ x n.

Differentiating the last equation and using the Lelieuvre formulas (1.38) for / and for / + , one easily sees that the normal fields of a Weingarten pair are related by (linear) differential equations: (1.41)

d\n+ + d\n

=

pi(n+ —n),

(1.42)

d2n r - d 2n

— p 2 (n+ + n),

with some functions pi,p 2 •R2 —>M. Thus: T h eorem 1.14. (W eingarten transform ation = M ou ta rd transfor­ m ation for Lelieuvre norm als) The Lelieuvre normal fields n , n+ of a Weingarten pair f , f + of A-surfaces are Moutard transforms of one another. A Weingarten transform / + of a given A-surface / is reconstructed from a Moutard transform n+ of the Lelieuvre normal field n. The data necessary for this are the data (M Ti^) for n: (W i) a point n+ (0) £ R3; (W 2) the values of the functions p\ on the coordinate axes i.e., two smooth functions Pil^ of one variable.

for i = 1, 2,

The following statement is a direct consequence of Theorem 1.10. T h eorem 1.15. (P erm utability o f W eingarten transform ations) 1) Let f be an A-surface, and let f ^ and f ^ 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 ^ and f ^ . 2) Let f be an A-surface. Let f ^ \ f ^ and f ^ be three of its Wein­ garten transforms, and let three further A-surfaces f^12\ f ^ and f ( 13^ be given such that f ^ is a simultaneous Weingarten transform of f ^ and f^ \ Then generically there exists a unique A-surface / ( 123) that is a Wein­ garten transform of f ( 12\ f ^ and f ( 13K 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. O r th o g o n a l n ets 1.4.1. N otion o f orth ogon a l nets. An important subclass of conjugate nets is fixed in the following definition.

1.4. Orthogonal nets

13

D efinition 1.16. (O rth ogon al net) A conjugate net f : Rm —> RN is called an m-dimensional orthogonal net in RN if at every u G Rm and for all pairs 1 < i ^ j < m we have dif _L d j f . Such a net is called an orthogonal coordinate system ifm = 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 / i—> / = //| /| 2. 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 c^ = c^ — 2(dif, /)/| /| 2, 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 RN (or, better, of the conformal 7V-sphere §^ , which is a compactification of RN) are represented by elements of the projectivized light cone P(LAr+1,1) in the projectivized Minkowski space P(R7V+1,1). The light cone h N+1'x = {£ G R * +M : ( £ ,0 = 0} is of central importance in Mobius geometry (the absolute quadric). Let { e i , ... ,ejv+2} denote the standard basis of the Minkowski space fljjVH-1,1 denote aiSo eo = \(e ^+2 — e N+1) and e ^ = ^(eM-\-2 + ew+i). The Euclidean space RN is identified, via (1.43)

7r0 : R N 3 f h-y / = / + e0 + |/|2eoo G

,

with the section of the cone L^ +1,1 by the affine hyperplane {£o = 1}, where £o is the eo-component of £ G R^ +1,1 in the basis { e i , . . . , eyv, eo, eoo}. An elegant characterization of orthogonal nets is due to Darboux: T h eorem 1.17. (M ob iu s-geom etric characterization o f orthogonal nets) A conjugate net f : Rm —> RN is orthogonal if and only if the scalar function |/|2 satisfies the same equation (1.1) as f does, or, equivalently, if the lift f — 7To ° f : Rm —►Q q is a conjugate net in In other words, the image of an orthogonal net in the projectivized light cone P(LAr"h1,1) is a conjugate net in P(RAr+1,1). In particular, any lift / = A / of / in L ^ +1,1, not necessarily normalized as in (1.43), satisfies linear differential equations (1.3). This criterion makes the invariance of orthogo­ nal 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 Differential 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 P(Lj/v+1,1). Choosing various quadrics, we come to the classical geometries of Klein’s Erlangen program including the hyperbolic, spherical, Lie, Pliicker, Laguerre, etc. geometry. 1.4.2. A n alytic description o f orth ogon a l nets. For an analytic de­ scription of an orthogonal net / : Mm —> RN, introduce metric coefficients hi = \dif \ and (pairwise orthogonal) unit vectors — h~ld i f . Then the following equations hold: (1.44)

d if

=

hm ,

(1.45)

diVj

=

(3jiVi,

i 7^ j,

(1.46)

dihj

=

hiPij,

i^j,

(1-47)

dif3kj

=

(3ki/3ij,

i /j/f c /i ,

which are analogous to (1.7)—(1.10). Indeed, equation (1.45) holds since / 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)

difoj -f- djflji = —(diVi, djVj) ,

i ^ j.

To derive (1.48), one considers the identity didj(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 vari­ ables z, j only, and therefore it makes sense to require that it be fulfilled on the coordinate plane 3 ^ . One can easily check that if a solution to the system (1.44)-(1.47) satisfies (1.48) on all coordinate planes for 1 < i < j < tn, then it is fulfilled everywhere on Mm. The meaning of the orthogonality condition (1.48) is that the coordinate surfaces / \K for all 1 < i < j < m. 1.4.3. Spinor fram es o f orth ogon al nets. The Mobius-geometric de­ scription 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 (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 ‘K qq, the isotropy subgroup of eoo in Spin'f (A^r + 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 = \dif\ satisfy also hi = |9j/|, where / = / + eo + |/|2eoo- Hence, the vectors Vi = h~ldif = Vi + 2(f,Vi)e 00 have the (Lorentz) length 1. Since ( /, / ) = 0, one readily finds that ( /, v%) = 0 and hi = — /). T h eorem 1.18. (Spin or fram e o f an orth ogon al net) For an orthogonal net f : Rm —> RN, i.e., for the corresponding conjugate net f : Mm —> (Q)^, there exists a function x/j : M771 —>!Koo (called a frame of f ) , such that (1.49)

/

=

(1.50)

Vi

— 'ip~leiip,

1 < i < m,

and satisfying the system of differential equations: (1.51)

dity — —ei'ipSi,

Si = \dii)i,

1 < i <m .

Note that for an orthogonal coordinate system (rn = N ) the frame ^ 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 = (diVj,Vi) = —(diVi, Vj). With the help of the frame ^ we extend the set of vectors {bi : 1 < i < m} to an orthonormal basis {% : 1 < k < N } of Tj Q q : (1.52)

ilk = '0_ 1e/c'0,

1 < k < N.

Respectively, we extend the set of rotation coefficients according to the for­ mula Pki = (diVk,Vi) = —(diVi, Vk) = -(diVi.t/j l e kip),

I < i <m ,

1 < k < N .

1. Classical Differential Geometry

16

Recall that we also have: hi = ~{diVi,f) = -{diVi,ifi~1 eo'ip},

1 < i < m.

Thus, introducing vectors Si = ipSiip^1, we have the following expansion with respect to the vectors e^: (1.53)

Si = i)Si^~l = lipidiVi) ^ - 1 = - \ '^TPkiek + M ook^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 < z, j < m and 1 < k < N, and that the orthogonality constraint (1.48) can be now put as (1.54)

difaj + djfiji = - ^ 2 PkiPkj ■

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

1.4.4. Curvatures o f surfaces and curvature line param etrized sur­ faces. Two-dimensional (ra = 2) orthogonal nets in M3 are nothing but sur­ faces 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.

Figure 1.1. Principal directions through touching spheres.

Curvature lines are subject of Lie geometry, i.e., are invariant with re­ spect to Mobius transformations and normal shifts. To see this, consider an infinitesimal neighborhood U of a point / of an oriented smooth surface in R3, and the pencil of spheres S(n) with the curvatures k , touching the surface at / ; see Figure 1.1. The curvature k, as well as the signed radius r = 1/k;, is assumed positive if S( k ) lies on the same side of the tangent plane as the normal n, and negative otherwise; the tangent plane itself is 5(0). For big kq > 0 the spheres S( k,q) and S(—kq) intersect U in f only.

1.4. Orthogonal nets

17

The set of the touching spheres with this property (intersecting U in f only) has two connected components: M+ containing S(ko) and M _ containing S( —kq) for big k0 > 0. The boundary values = inf {k : S(k) G M + },

k 2 — sup

: S(k) G M _ }

are the principal curvatures of the surface in / . The directions in which S( k i) and S( k2) touch U are the principal directions. Curvature lines are integral curves of the principal directions fields. The symmetric functions v K = « i « 2,

TT K\ K2 H = — -—

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) be­ long 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 / with the Gauss map n. For sufficiently small t the formula ft = f + tn defines smooth surfaces parallel to / . The infinitesimal area of the parallel surface ft turns out to be a quadratic polynomial of t and is described by the classical Steiner formula (1.55)

dA{ft) = (1 - 2Ht + K t 2 )dA(f),

Here dA is the infinitesimal area of the corresponding surface and H and K are the mean and the Gaussian curvatures of the surface / , respectively. 1.4.5. R ib a u cou r transform ations o f orth ogon al nets. An important class of transformations between orthogonal nets is specified in the following definition. D efinition 1.19. (R ib a u cou r transform ation) A pair of m-dimensional orthogonal nets / , / + : Mm —> WN 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 G Mm and for every 1 < i < m the

18

1. Classical Differential Geometry

straight lines spanned by the vectors d i f , d if+ at the respective points f , / + are interchanged by the reflection in the orthogonal bisecting hyperplane of the segment [/, / +]. The net / + is called a Ribaucour transform of f . The nets / , f + serve as two envelopes of a Ribaucour sphere congruence Sm :R m -> {m-spheres in RN}. In other words, f(u), f + (u) G Sm(u), and the tangent m-spaces to Sm(u) at f{u), resp. f + (u), are spanned by dif(u), resp. by dif+ (u), i — 1 , .. ., ra. To describe a Ribaucour transformation analytically, we write: (1.56)

d j+ = n (d j - 2

>

with some functions r* : Rm —> R*. It is convenient to define the metric coefficients of the transformed net as h f — rihi — sign(r*)|R defined as 9t = (hf — hi)jt = (rt — l)hi/i. Equations (1.57) imply equations for the metric coefficients: (1.58)

h f = hi + Bit,

dil = -{vi ,y) (h + + hi).

Compatibility of the system (1.57) yields that 6t have to satisfy certain differential equations: (1.59)

0+ = f a - 2(Vi, y)9j,

8 ^ = ± ft(0+ + /% ).

The following data determine a Ribaucour transform / + of a given orthog­ onal net / uniquely: (Ri) the point / + (0); (R2) the values of 9i on the coordinate axes 2 * for 1 < i < m , i.e., ra smooth functions 0* of one variable. According to the general philosophy, iterating Ribaucour transforma­ tions 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. T h eorem 1.20. (P erm utability o f R ib a u cou r transform ations) 1) Let f be an m-dimensional orthogonal net, and let f ^ and f ^ be two of its Ribaucour transforms. Then there exists a one-parameter family of orthogonal nets / ( 12) that are Ribaucour transforms of both f ^ and f ( 2\ The corresponding points of the four orthogonal nets f , f ^ \ f ^ and / ( 12) are concircular.

1.5. Principally parametrized sphere congruences

19

2) Let f be an m-dimensional orthogonal net. Let f ^ and f ^ be three of its Ribaucour transforms, and let three further orthogonal nets f^12K f ^ and f ( 13^ be given such that f ^ is a simultaneous Ribaucour transform of /W and f^ \ Then generically there exists a unique orthogonal net / ( 123) that is a Ribaucour transform of f^l2\ f (23) and f ^ . The net / ( 123) is uniquely defined by the condition that for every permutation (ijk) of (123) the corresponding points of f ( % \ f ^ \ f ^ and / ( 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 hyper­ sphere congruences. The classical case is, of course, that of two-parametric families of spheres in M3. Nonoriented spheres in K3 can be represented as elements of P (R ^ t), where (1.60)

Rout = {$ € M4’1 : (s, s) > 0}

is the space-like part of E4,1. D efinition 1.21. (P rin cip ally param etrized sphere congruence) A map (1.61)

S : R 2 —> {nonoriented spheres in R3}

is called a principally parametrized sphere congruence if the corresponding map s : R 2 —>P(M ^t) is a conjugate net, i.e., if for any lift of it to M40yt, (1.62)

8182 s e span(s, d\s, &2s).

A principal parametrization exists and is unique for a generic congru­ ence. The classical description of this is as follows. In an arbitrary paramet­ rization of a congruence, consider two neighbors S(u + du\), 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\(u) and C^u). Thus, in the projective model of Mobius geometry of R3, based on the Minkowski space M4,1 of pentaspherical coordinates, these circles are de­ scribed as L4,1 fl (span(s, dis))1 , resp. L4,1 D (span(s, c^s))1 ; i.e. their points are represented by elements x E L4,1 satisfying (1.63)

C\ :

(s,x) = 0,

(d\s, x) = 0,

C2 :

{s,x) = 0,

{d2S , x ) = 0 .

resp. (1.64)

20

1. Classical Differential Geometry

These two circles intersect in two points. Such pairs of points comprise the two enveloping surfaces of the congruence, described in the Mobiusgeometric formalism as L4,1 fl (span(s, d\s, d2s)) ± . In other words, the en­ velopes are represented by the elements x £ L4,1 satisfying (1.65)

(«,£) = 0,

(d \ s , x ) = 0 ,

(d2s,x) = 0.

Now, the principal parametrization is characterized by the following con­ dition: when an infinitesimal displacement is made along one of the coor­ dinate 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\(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, d2x) = 0,

(d\ 5, d2x) = 0,

which, compared with (1.63), demonstrates the claim. A convenient choice of representatives s of hyperspheres S is the Eu­ clidean one, in terms of the centers c and radii r: (1.66)

s : R2 -> Rout n {Co = 1},

s = c + e0 + (|c|2 - r ^ e ^ .

The condition for this to be a conjugate net in R4,1 leads to the following classical statement. T h eorem 1.22. (P rin cip ally param etrized sphere congruences; cen­ ters and radii) A map (1.61) is a principally parametrized sphere congru­ ence 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 \c\2 — r 2 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. D efinition 1.23. (K -su rface) 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 £ 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

T h eorem 1.24. (K -su rfaces = A -surfaces w ith C h ebyshev p rop ­ erty) An asymptotic line parametrized surface f : R2 —>R3 is a K-surface if and only if the functions fa = \dif\ (i — 1, 2) depend on Ui only: fa — fa(ui). One of the approaches to the analytical study of K-surfaces is based on the investigation of the angle 4>(u\,u2) between asymptotic lines which is governed by the equation d\d2(j) = —Kfa(ui)f 32 (u2 ) sin. After a reparametrization of asymptotic lines one arrives at the famous sine-Gordon equation (1.67)

d\d2(j) — sin (j).

Another description is based on the Gauss maps. T h eorem 1.25. (G auss m ap o f a K -surface is a M ou tard net) The Lelieuvre normal field n : R2 —►R3 of a K-surface with K — —1 takes values in the sphere § 2 C M3, 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, |c^n| = fa (i = 1,2), with the same functions Pi = fa(v,i) 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 (ui,v, 2 ))- It is important to observe that the coefficient qi 2 of the Moutard equation (1.39) satisfied by a Lorentz-harmonic map n is completely determined by its first order derivatives: (1.68)

qu — ( b N+1^ of a Darboux pair of isothermic surfaces / , f + : R2 —> R^ are related by a Moutard transfor­ mation, i.e., there exist two functions pi,p 2 •R2 —►R such that (1.79)

d\

+ d\s = pi(s+ - 5),

^25^ - d2s = P2 (s+ + s).

Conversely, for a Moutard net s in the light cone L ^ +1,1, any Moutard transform with values in h N+1,1 is a lift of a Darboux transform f + of the isothermic surface f . Note that the quantity (s, s+) is constant (does not depend on u E and is related to the parameter c of the Darboux transformation: (s, s+ ) = —c/2. The formulas

(1om

„

§ 2. T h eorem 1.37. (Parallel constant m ean curvature surfaces) Let f : R2 —> R3 be a surface with constant mean curvature H q ^ 0 and without umbilic points, and let n : R2 —►§ 2 be its Gauss map. Then (i) every parallel surface ft = f + t n is linear Weingarten, i.e., its mean and Gaussian curvature functions Ht,Kt satisfy a linear relation aHt + /3Kt = 1 with constant coefficients a, (3; (ii) the parallel surface / jH0 - = / + j120 r n is Christoffel dual to f and has constant mean curvature

H

q;

(iii) the mid-surface f i k : = , + W 0n has constant positive Gaussian curvature Ko = 4i/o • We summarize considerations of these chapter in the following table: Koenigs net f in R^ A-surface / in R3 Orthogonal net / in R^ Principally parametrized sphere congruence S in R3 K-surface / in R3 Isothermic surface / in R^ Minimal surface / in R^

Moutard net y in R ^+1 Moutard net n in R3 conjugate net / in ~ P(LiV+1,1) . „ . ^41 ' conjugate net s m Moutard net n in § 2 Moutard net s in L ^ +1,1 Isothermic net n in § 2

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 con­ stant mean curvature surfaces. In the context of Lie and Pliicker 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. Classical Differential Geometry

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 clas­ sical 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 theo­ rem 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 transfor­ mation 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 trans­ formations in terms of Lelieuvre normals see the classical books by Dar­ boux (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 differ­ ential geometry based on asymptotic line parametrization is in Ferapontov (2000a). Section 1.4: Orthogonal nets and their transformations. A funda­ mental monograph on orthogonal coordinate systems is Darboux (1910). A

1.9. Bibliographical notes

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 inte­ grable systems were investigated in Zakharov (1998). Algebro-geometric or­ thogonal 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 de­ scription 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., Darboux (1914­ 27), 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 Dar­ boux (1914-27) and Bianchi (1923). In particular, the Darboux transfor­ mations 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 Darboux transformations was established by Bianchi. The constant cross-ratio property in the permutability theo­ rem 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 the­ ory of isothermic surfaces to the theory of integrable systems was found in Ciesliriski-Goldstein-Sym (1995). A spinor zero-curvature representation for isothermic surfaces was found in Bobenko-Pinkall (1996b). A relation of

30

1. Classical Differential Geometry

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). Isother­ mic 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 HertrichJeromin-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 es­ sentially no difference between nets and their transformations, which can be regarded just as various parts of multidimensional discrete nets char­ acterized by certain elementary geometric properties. The very possibility to impose these properties on a multidimensional net, which usually re­ lies on certain incidence theorems of elementary geometry, is the ultimate reason for the existence of transformations and their remarkable permuta­ tion 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:

(nf)(u)

f{u + ei),

(6if)(u) = f(u + e^ -

f(u),

where e* is the i-th coordinate vector of Zm. Often we write /*, fij for t*/, TiTjf, etc. The notation for (discrete) 5-dimensional coordinate planes is the same as in the continuous case: = {u e Z m : Ui = 0 for %^ i i , ... , i s}. 31

32

2. Discretization Principles. Multidimensional Nets

We will denote by Czi...zs the elementary 5-dimensional cube with the 2s vertices u + e ^ e ^ + .. - + ^ se*s, €i £ {0 ,1 }.

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 R N of our nets to be > 3. Definition 2.1. (Q-net) A map f : Zm —>

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 £ Zm and for every pair 1 < i ^ j < m the four points f , Tif, T jf, and TiTjf are coplanar:

(2.1)

SiSjf = c i f + CijSjf,

i

j,

or, equivalently,

(2.2)

TiTjf - f =

(1 + Cji)Sif

+

(1 + Cij)6jf,

i i - j.

Here it is convenient to think that the real numbers c*j, cji, as well as equation (2.1) itself, are assigned to elementary squares Q{j of Zm parallel to the coordinate plane *Bij.

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 am­ bient space R^ is interpreted as an affine part of RP^ = P(RN_hl), then

2.1. Discrete conjugate nets (Q-nets)

33

an arbitrary lift / = p (/, 1) G RN+l of a Q-net / to the space of homo­ geneous coordinates R ^ 1 is characterized by the following condition: for every u G Z m and for every pair i =£ j, the four elements / , T;/, Tjf, and TiTjf are linearly dependent (span a three-dimensional vector subspace): (2.3)

TiTjf = QLjiTif + OtijTjf + fiijf.

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 / : Z 2 —>RN. Suppose its two coordinate lines, f\ ^ and f \^2 , are given. To extend the surface into the quadrant Z+, say, one proceeds by induction whose step consists in choosing f u in the plane through / , f\ and / 2, provided the latter three points are known (and are in general position). The planarity condition is expressed as (2.4)

S\S2f = C2lS\f + C\2S2 f.

So, on each such step one has two free real parameters C21, c 12, attached to the elementary square G\2 (u) of the lattice Z 2. Thus, one can define a Q-surface / by prescribing its two coordinate lines / f ® 1? / f ^ > and two real-valued functions c\2 ,c 2\ defined on all elementary squares of Z 2. The planes ( /, / 1, / 12, f 2) of a Q-surface can be assigned to elementary quadrilaterals of Z 2, 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 iV = 3, this view of Q-surfaces makes them an example of the following notion. Definition 2.2. (Q*-net) A map (2.5)

P : Z 2 —> {planes in M3}

is called a Q*-net if at each u G Z 2 the four planes P, Pi,P 2 ,Pi 2 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 MP3 rather than in R3. Each plane P in RP3 is described as an element p G (RP3)*, with some representatives p G R4 in the space of homogeneous coordinates. The condition for P, Pi, P2, P12 to have a common point is equivalent to the condition for p, pi, p2, P\ 2 to span a three-dimensional vector subspace. In other words, a Q*-net is nothing but a Q-net in the dual space p : Z 2 —►(RP3)*, while the intersection points of the planes of a Q*-net constitute a Q-net in RP3.

34

2. Discretization Principles. Multidimensional Nets

It will be important to remark that the combinatorics of Q-surfaces may well be more complicated than that of I? . Definition 2.1 can be literally extended to maps / : V(T>) —> RN, 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 / are given, that is, f\ provided / , fi and fij are known. We now show that the planarity condition determines the point /123 uniquely. Remark. (General position assumption) In the spirit of local differ­ ential 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 / , fi span a three-dimensional space, and that no three points fu fijJ ik are collinear. Theorem 2.3. (Elementary hexahedron of a Q-net) Given seven points f , fi and fij (1 < i < j < 3) in RN, such that each of the three quadrilater­ als ( /, f i , f i j , fj) is planar, define three planes TiUjk as those passing through the point triples (fi, fij, fik), respectively. Then these three planes intersect at one point: (2.6)

/123 = T1II23 H T2II13 fl T3II12 .

Proof. Planarity of the quadrilaterals ( f , f i , f i j , f j ) assures that all seven initial points / , fi and fij belong to the three-dimensional affine space II123 through the four points / , fi. Hence, the planes TiHjk lie in this three­ dimensional space, and therefore generically they intersect at exactly one point. □ 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.

2.1. Discrete conjugate nets (Q-nets)

35

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 Z 3 depicted in Figure 2.2. These equations yield that Si(SjSkf) can be expressed as linear combinations of S^f (1 < £ < 3). Equating coefficients of 5ef in expressions for 6i(5j5kf) for the three cyclic permutations (z, j, k) of the indices (1, 2,3), we see that the equations for Cjk split off from the equations for / : (2.7)

$iCjk — (TkCij)Cjk “b (TkCji^Cik

ij'iCjkjCik')

M J/ ^

^

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 5ef (1 < £ < 3) are linearly independent. Consider the numbers {cjk} on the three faces adjacent to / as known, and the numbers {riCjk} on the three faces adjacent to /123 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 /123 in the general position case), this system generically admits a unique solution. The resulting map (2.8)

{cjk} ' ^ {TiCjk}

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., for all 1 < i < j < 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 provided / , fi and fij are known. However, this inductive process works only if one does not encounter contradictions. To see the

36

2. Discretization Principles. Multidimensional Nets

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 / , fi and fij (1 < i < j < 4) one determines all uniquely. After that, one has, in principle, four different ways to determine /1234 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 /1234 automatically coincide. We call this property the 4D consistency. D efinition 2.4. (4D con sisten cy) A 3D system is called J^D consistent if it can be imposed on all three-dimensional faces of an elementary hypercube of Z 4.

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 {cjk} > —►{TiCj/J, the 4D

2.1. Discrete conjugate nets (Q-nets)

37

consistency means that the two values t^t^c^) and TjijiCkt) coincide for any permutation (z, j, /c, £) of the indices (1, 2 ,3, 4). Theorem 2.5. (4D consistency of Q-nets) The 3D system governing Q-nets is J^D consistent. Proof. In the construction above, the four values in question are

/1234 = T1T2II34 n T1T3II24 n T1T4II23, and the three others are obtained by cyclic shifts of indices. Thus, we have to prove that the six planes TiTj 11^ intersect in one point. First, assume that the ambient space RN has dimension N > 4. Then, in general position, the affine space II1234 through the five points / , fi (1 < i < 4) is four-dimensional. It is easy to understand that the plane TiTjUki ls the intersection of two three-dimensional subspaces TiUj^t and TjUike- Indeed, the subspace TiUjke through the four points f i , f ij, /^ , fn contains also , fijt, and / ^ . Therefore, both r^IIj^ and TjUikt contain the three points f i j , fijic and / ^ , which determine the plane TiTjUki- 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 II1234. This intersection consists in the general case of exactly one point. This argument yields also the 4D consistency of the map {cjk} »-> { ^ c ^ }. This map, however, does not depend on the dimension N of the ambient space of / . The 4D consistency of the map {cjk} *-> { t\Cjk} yields, in turn, the 4D consistency of the construction of Q-nets also in the case N — 3. □ The m-dimensional consistency of a 3D system for m > 4 is defined analogously to the case m = 4. Remarkably and quite generally, the 4­ dimensional consistency already implies m-dimensional consistency for all m > 4. Theorem 2.6. (4D consistency yields consistency in all higher di­ mensions) 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 + l)-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. Initial data for a 3D system on the 5D cube C12345 with the fields on vertices consist of the fields / , fi and fij for all 1 < i < j < 5. From these

38

2. Discretization Principles. Multidimensional Nets

data one first gets ten fields for 1 < i < j < k < 5, and then five fields fijhi for 1 < i < j < k < I < 5 (the fact that the latter are well defined is nothing but the assumed 4D consistency for the 4D cubes 6^ / ) . Now, one has ten possibly different values for /12345, coming from ten 3D cubes TiTjQkim• To prove that these ten values coincide, consider five 4D cubes Tfijktm- F°r instance, for the 4D cube T1C2345 the assumed consistency assures that the four values for /12345 coming from four 3D cubes ^1^26345,

^1^36245,

^1^46235,

^1^56234

are all the same. Similarly, for the 4D cube T2C1345 the 4D consistency leads to the conclusion that the four values for /12345 coming from TlT2e 345,

T2T3ei45,

^ 746135,

^2X56134

coincide. Note that the 3D cube T1T2C345, the intersection of T1C2345 and 72C1345, is present in both lists, so that we now have seven coinciding values for / i 2345- Adding similar conclusions for other 4D cubes t*Cj^m, we arrive at the desired result. □ Theorems 2.5, 2.6 yield that Q-nets are ra-dimensionally consistent for any ra. 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 ( Q f ) the values of / on the coordinate axes 23; for 1 < i < m; (Q^) the values of c;j, Cji on all elementary squares of the coordinate planes T$ij , for 1 < i < j < m. 2.1.2. Transformations of Q-nets. A natural generalization of Defini­ tion 1.2 would be the following. Definition 2.7. (Discrete fundamental transformation) Two m-dimensional Q-nets / , / + : Zm —►RN are said to be related by a fundamental transformation (F-transformation) if at every u G Z 771 and for every 1 < i < m the four points f , Tif, / + and Tif+ are co-planar. The net / + is called an F-transform of the net f . However, this relation can be rephrased as follows: set F (^,0) = f(u) and F(u, 1) = / + (u); then F : Z m x {0 ,1 } —> RN constitutes (two layers of) an M-dimensional Q-net, where M = m + 1. Thus, in the discrete case

39

2.1. Discrete conjugate nets (Q-nets)

there is no difference between conjugate nets and their F-transformations. The situation of Definition 2.7 is governed by the equation (2.9)

Sif+ = a,i6i f + bi(f + - / ) ,

where the coefficients a*, bi are nothing but a* = 1 + cmi , h = CiM• These coefficients are naturally attached to the elementary squares of Zm x {0 ,1 } parallel to the coordinate plane It is also convenient to think of them as attached to the edges of Z m parallel to (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)

Sidj

=

(Tjbi)(a,j - 1) + (Tjdi - Tiaj)cij,

(2.11)

Sibj

=

cfjbj + c p i - (Tibj)bi,

(2.12)

djc^

=

(Tjdi)cij + (Tjbi)(a,j - 1).

The following data are needed to specify an F-transform / + of a given m-dimensional Q-net / : ( F f ) the value of /+ (0 ); (F^) the values of a*, bi on all edges of the respective coordinate axis 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 and f ^ be two of its discrete F-transforms. Then there exists a two-parameter family of Q-nets / ( 12) that are discrete F-transforms of both f ^ and f ( 2K The corresponding points of the four Q-nets f , f ( l\ f ^ and f ^ are coplanar. 2) Let f be an m-dimensional Q-net Let f ( l\ f ^ and f ® be three of its discrete F-transforms, and let three further Q-nets f^l2\ f ^ and f ^ be given such that f ^ is a simultaneous discrete F-transform of /M and fd \ Then generically there exists a unique Q-net / ( 123) that is a discrete F-transform of f ( 12\ f ^ and f^13\ The net / ( 123) is uniquely determined by the condition that for every permutation (ij k ) of (123) the corresponding points of f ( l\ f ^ \ f ^ and / ( 123) are coplanar. Proof. In the first part of the theorem, we define an (m + 2)-dimensional Q-net F : Z m x {0, l } 2 —►RN such that

F(u, 0,0)

= /(«), F(u,l,0) = /^(u),

F(u,0,l) = fW(u).

40

2. Discretization Principles. Multidimensional Nets

The only additional initial datum which is required for the construction of F is the value of F (u , 1,1) = f ( 12 \ u ) at one point u € Z m, say at u = 0 . The point /^ 12^(0 ) G R N can be chosen arbitrarily in the plane through the points / ( 0 ), / ^ ( 0 ), / ( 2)( 0 ), which leads to the two-parameter family of Q-nets f ( u \ The left part of Figure 2.5 illustrates the extension of / ( 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 (ra + 3)-dimensional Q-net F : Z m x {0, l } 3 —>R N such that F (u , 0 ,0 , 0 ) = f(u ), and the shift of F in the (ra + i)- th coordinate direction is denoted by the superscript i of / , so that, e.g., F (u , 1,1,1) = f^ 123 \ u ). For each u e Z m the point / ( 123) is uniquely determined by the other seven points / , /W , and which is illustrated in the right part of Figure 2.5. □

Ti

f ( 2)

Ti / ( i2>

i/

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

2.1.3. A lternative analytic descrip tion o f Q -nets. In a complete anal­ ogy 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 Wi par­ allel to the edges Sif of a Q-net / such that (2.13)

SiWj — /"yjiWi

with some real coefficients 7^, called discrete rotation coefficients of a Q-net. Clearly, in general SiWj is a linear combination of Wi and w j , and the re­ quirement 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 (fifij) with the line through the tip of the vector wj parallel to the vector wf, see Figure 2.6.

2.1. Discrete conjugate nets (Q-nets)

41

Figure 2.6. Geometric construction of the vectors TiWj.

This construction is well defined also in the multidimensional situation. To show this, one has to prove that Ti(TjWk) = Tjfawk)- But the tips of both vectors can be defined as the intersection of the line (fijfijk) with the plane passing through the tip of the vector w^ parallel to the plane 11^ of the face ( /, fr, f a , / ,) ; see Figure 2.7.

fk

Figure 2.7. Vectors Wk are well defined in an elementary hexahedron of a Q-net.

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

Wi = g ^ S i f .

2. Discretization Principles. Multidimensional Nets

42

Substituting (2.14) into ( 2 .1), we see that requirement (2.13) is equivalent to (2.15)

ngj = (1 + C{j)gj

&%9j — Cij9j ?

^ 7^ JS

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

= —

ngj

{n c k j} 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. C ontinuous lim it. 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 —» 0 . Analogously, equations (2.9)-(2.12) are similar to (1.11)—(1.14). Accord­ ingly, 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 differ­ ential 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 £jN be the space of lines in RP^; it can be identified with the Grassmannian Gr(iV + 1, 2) of two-dimensional vector subspaces of R ^ -1"1. D efinition 2.9. (D iscrete line con gru en ce) A map £ : Zm HN is called an m-dimensional discrete line congruence in WPN (N > 3) if every two neighboring lines £, £{ (for each u G Z m and fo r each 1 < i < m ) intersect (are coplanar).

For instance, the lines £ ^ =

connecting neighboring points / = : Z m —►R ¥ N constitute a discrete line congruence. Likewise, the lines £ = ( / / + ) connecting the corresponding points / = f(u ) and f + = f^~{u) of two Q-nets / , f + : Zm —► RP^ related by an F-transformation build a discrete line congruence. (ffi)

f(u ) and fi — f ( u + e*) of an arbitrary Q-net /

A two-dimensional discrete line congruence is called generic if any two diagonally neighboring lines (that is, l\ and £2, as well as £ and £12 ) do not intersect. Each of these two pairs of lines span a three-dimensional space V 12 , 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 of maximal possible dimension m + 1. This space contains all lines assigned to the vertices of the Tri­ dimensional 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 £, £\, £2 of a congruence, one

44

2. Discretization Principles. Multidimensional Nets

Figure 2 .8 . Four lines of a generic discrete line congruence.

has a two-parameter family of lines admissible as the fourth line £\2: connect by a line any point of £\ with any point of £2. Thus, given any two sequences of lines £: Z x {0 } —>L N and £: {0 } x Z - ^ &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 £ : Z 2 —» &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: T h eorem 2.10. (D iscrete line congruences are d escribed b y a 3D system ) Given seven lines £, £i, £ij G L N (1 < i < j < 3) satisfying the conditions fo r a generic line congruence, there is a generically unique line

^123 that intersects all three £ij.

P roof. All seven lines, and therefore also the three-dimensional spaces TiVjk = span(£ij,£ik), lie in Vi23. 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)

^123 = T1V23 n T2V13 n T3V12.

It is now not difficult to realize that this line does, indeed, intersect all three of £ij. For instance, t\V23 H T2V13 = span(^i2, ^13) D span(^i2, ^23) is a plane containing £\2\ therefore its intersection with T3V12 (the line ^123) intersects £\2. □

2.2. Discrete line congruences

45

A similar argument shows (see Exercise 2.10): T h eorem 2 . 11 . (D iscrete 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 transforma­ tions of discrete line congruences with remarkable permutability properties. D efin ition 2 . 12 . (F -tran sform ation o f line congruences) T w om -d im ensional line congruences £, t f : Z m —> L N are said to be related by an F transformation if fo r every u G Z m the corresponding lines £ = £{u) and £+ = £+ (u) intersect, i.e., if the map L : Z m x {0 ,1 } —> C N defined by L(u, 0) = £{u) and L(u, 1) = £+ (u ) is a two-layer (m + I)-dimensional line congruence.

It follows from Theorem 2.10 that, given a line congruence its Ftransform £+ is uniquely defined as soon as its lines along the coordinate axes are suitably prescribed. According to Definition 2.9, any two neighboring lines £ = £{u) and of a line congruence intersect at exactly one point F = £n £i G MP^, which is thus combinatorially associated with the edge (u, u + e*) of the lattice Z m: F = F ( u ,u + ei). It is, however, sometimes more convenient to use the notation F (u , u + e*) = F^l\ u ) for this point, thus associating it to the vertex u of the lattice (and, of course, to the coordinate direction i). £i = £(u + ei)

D efinition 2.13. (F ocal net) For a discrete line congruence £ : Z m —> &N , the map F ^ : Z m —>WPN defined by F ^ ( u ) = £{u) fl £{u + e*) is called its i-th focal net.

This is illustrated in Figure 2.8. T h eorem 2.14. (F ocal nets are Q -nets) For a nondegenerate discrete line congruence £ : Z m —> &N , all its focal nets F ^ : Zm —►RP^, 1 < k < m , are Q-nets.

P ro o f. The proof consists of two steps. 1. First, one shows that for the k-th focal net F^k\ all elementary quadrilaterals ( F ^ , F^k\ F ^ \ F ^ ) are planar. This is true for any line

congruence. Indeed, both points

and F ^ lie on the line £ ^ while both

points F j k^ and lie on the line Therefore, all four points lie in the plane spanned by these two lines £f~ and £ ^ 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\ F ^ k\ F ^ k\ F j k^), with both i ^ j different from fc, are

46

2. Discretization Principles. Multidimensional Nets

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

spaces Vij = span(^, li, i j j i j )

and

rkVij = span(4 , £ik, £jk, £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. □

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

C orollary 2.15. (Focal net o f F -transform ation o f a line con gru ­ ence) Given two generic line congruences £,£+ : Z m —» HN in the rela­ tion of F-transformation, the intersection points F = £ n £+ form a Q-net F : Z m -> R F N .

A different aspect of the close relations between Q-nets and line congru­ ences is given in the following theorem. T h eorem 2.16. (E xtending a Q -net to a line con gru en ce) Given a Q-net f : Z m —> R N , there exist discrete line congruences £ : Z m —> &N such that f ( u ) G £(u) fo r every u G Z m. Such a congruence is uniquely determined by prescribing the lines through the points /f® . along the coordinate axes.

P ro o f. 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 £ i , £ ‘2 and a point f u in the three-dimensional space Vi2 spanned by these lines, there exists a unique

2.3. Discrete Koenigs and Moutard nets

47

line £12 through f \2 intersecting £\ and £2 (this line is the intersection of the planes through / 12, £\ and through f u , £2 )• Having constructed the three lines £12, £1 3 , £23 , Theorem 2.10 provides us with the unique line £123 given by (2.23). The point /123 is the intersection of three planes (2.6). The incidence /123 G ^123 follows from TiUjk C r^V^. □ Q-nets with line congruences introduced in Theorem 2.16 play an im­ portant 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. D efinition 2.17. (D iscrete C om bescu re transform ation) Two m -dim ­ ensional Q-nets / , / + : Z m —> R N are said to be related by a Combescure transformation if at every u G Zm and fo r every 1 < i < m the edges 8 { f and S if+ are parallel. The net / + is called parallel to f or a Combescure transform o f f .

The following result is straightforward. T h eorem 2.18. (C om bescu re transform ations and line congruences) Given a Q-net f : Zm —> R N with a discrete line congruence £ : Zm —►&N , i.e., f ( u ) G £(u) fo r every u G Zm, there exists a one-parameter family of parallel Q-nets f t : Z m —» such that ft(u ) G £(u). Such a net is uniquely determined by its one vertex ft(u 0) G ^(^o)* Conversely, a Combescure pair o f Q-nets / , / + determines a line congruence by £{u) = ( f ( u ) f + (u )) for every u G Zm. 2.3. D is c r e t e K o e n ig s a n d M o u t a r d n ets 2.3.1. N otion o f dual quadrilaterals. D efinition 2.19. (D ual 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* ) | | (AB),

(£*C*) || ( B C ) ,

(C*D*) || { C D ) ,

(D M *) || (DA) ,

and the noncorresponding diagonals are parallel:

(2.25)

(A*C*) || ( B D ) ,

(S*D*) || ( AC) .

Lem m a 2 . 2 0 . (E xistence and uniqueness o f dual quadrilateral) For any planar quadrilateral (A , B , C , D ) , a dual one exists and is unique up to scaling and translation.

2. Discretization Principles. Multidimensional Nets

48

C

Figure 2.10. Dual quadrilaterals.

P roof. Uniqueness of the form of the dual quadrilateral can be argued as follows. Denote the intersection point of the diagonals of (A, £?, C, D ) by M — ( A C) fl ( B D ). Take an arbitrary point M* in the plane as the des­ ignated intersection point of the diagonals of the dual quadrilateral. Draw two lines £\ and £2 through M* parallel to ( A C) and ( B D ) , 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 £\ will be B*; draw the line through B * parallel to ( B C ) ; its intersec­ tion point with £2 will be C*; draw the line through C* parallel to ( C D ) ; its intersection point with £\ will be D *. It remains to see that this construc­ tion closes, namely that the line through D* parallel to ( D A ) intersects £2 at A*. Clearly, this property does not depend on the initial choice of ^4* 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 e\ and e 2 some vectors along the diagonals, and introduce the coefficients a , . . . , 6 by (2.26)

M A — oce1,

M B = fie 2,

M D — 5e 2,

M C — 7 ei,

so that (2.27)

— (3 e 2 — aei, C D = 5e 2 —7 ei, AB

BC

=

— (3e2,

D A — ae\ — 8 e 2.

Construct a quadrilateral (A*, B * , C * , D * ) by setting (2.28) M * A * = —^2- , a

71/T* 7~>* = ATS*

p

,

71/#■*sy* = —^2- , M*C* 7

II/T* TTi* M*D* =

o

.

2.3. Discrete Koenigs and Moutard nets

49

Its diagonals are parallel to the noncorresponding diagonals of the original quadrilateral, by construction. The corresponding sides are parallel as well: ------ ►

1

1

2

1 —►

A*B*

=

- - ei + - e p a

B*C*

=

-------> C*D* =

- - e 2 + ^ei = ^ B C , 7 0 01 1 1 1 — > — ei + - e 2 = — CD,

WA*

- ~ e 2 + \ e l = ^-DA.

=

7

o

a

0

= — AB, ap

70

da

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

□

Note that the quantities a , . . . , S 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 e\, e2. Well defined are their ratios, which can be viewed also as ratios of the directed lengths of the corresponding segments of diagonals, say 7 : a = Z(M, C) : /(M , A) and S : (3 = /(M , D) : Z(M, B ). It is natural to associate these ratios with directed diagonals: D efinition 2 . 2 1 . (R a tio o f diagonal segm ents) Given a quadrilateral (.A , B , C, D ), with the intersection point o f diagonals M = ( AC ) Pi ( B D ) , we set

(2.29)

,(XC) -

,

q (B D ) -

.

Changing the direction of a diagonal corresponds to inverting the associated quantity q.

Note that (2.30)

(A ,B ,C ,D )

convex

0 and q ( B D ) < 0; and a nonembedded one with q( A C) > 0 and q ( B D ) > 0 . 2.3.2. N otion o f discrete K oen igs nets. D efinition 2 . 2 2 . (D iscrete K oen igs net) A Q-net f : Z m —> R N is called a discrete Koenigs net if there exists a Q-net /* : Z m —> R N , called Christoffel dual to f , such that any elementary quadrilateral o f the net /* is dual to the corresponding quadrilateral o f the net f :

2. Discretization Principles. Multidimensional Nets

50

M

D

A

D B

B

A

Figure 2 .1 1 . 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 Z m 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 = (u \,. . . , um) with an even value of \u\ = u\ H------- \-um black and those with an odd value of |u |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 Z m and the set of edges consisting of black diagonals of all elementary quadrilaterals of Z m, and the analogous white graph Z ^ d. The geometry of the elementary quadrilaterals of a Q-net / : Z m —> R N induces, according to Definition 2 .21, the quantities q (ratios of directed lengths of diagonal segments) on all directed diagonals, white and black. D efinition 2.23. (D iscrete on e-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 i f p ( - e ) = —p{e) for every directed edge e G E . It is called exact if fo r 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 G E . It is called exact if fo r every cycle of directed edges the product of the values of q along this cycle is equal to one.

52

2. Discretization Principles. Multidimensional Nets

Figure 2 .1 2 . Four quadrilaterals around a vertex of a two-dimensional net.

Similarly, we find:

A2 _ ol2^2 A3 &3/ V

A3 _ 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 . 2 1 .* a) Let f : Z 2 —>MFN be a discrete Koenigs net. Show that its Laplace invariants (defined as in Exercise 2.15) satisfy the relation hh-\ — k k - 2.

b) Let M : Z 2 —>RP^ 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 hh 2 = kk\. 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 . 2 2 * Consider a planar quadrilateral ( / / 1/ 12/ 2), and denote L\ — ( f f i ) fl ( / 2/ 12) and L 2 = ( f f 2) D ( / 1/ 12). Consider four further points L 3 G ( / / 1), La e ( / 2/ 12), £5 G ( / / 2), L 6 G ( / 1/ 12). Show that the six points L i , ... ,L 6 belong to a conic (see Figure 2.26) if and only if the following relation is satisfied: Q(f »

f i , L s ) q { f 2, L 4 , f \2, L{) = q ( f , L 2, f 2, L$)q(fi, Lq, / 12, L 2).

2. Discretization Principles. Multidimensional Nets

80

h-2

k h

/l_ l

k

*

ki

/

h k 2 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 : Z 2 —> WPN built by the intersection points of the diagonals of a discrete Koenigs net possesses the following geometric property: for any u G Z 2, the six points L ^ M ( u ) , L R N of y is constructed as the solution of the compatible system (2.86)

9 n y + - y

=

9

—

(y+ - T i y ) ,

r 2y + +

y =

^

{y+ +

r 2y ) ,

or, equivalently, (2.87)

S^eyf) =

82{6y+ )

=

6( t 20)S2 { ^ ) .

It satisfies the Moutard equation (2.88)

ri r2y + + y + = a+ ( n y + + r2y + )

with the transformed potential

(2 RQ) (2.89)

n+ = n i2 012

0 ( Ti t 2 6 )

'

The function 9+ = 1 /9 is an additional scalar solution of the Moutard equation (2.88) with the potential a^2. Hint: An additional scalar solution 9 : Z 2 —> R of the discrete Moutard equation (2.56) can be defined by the system (2.90)

A

TiU

=

bi,

i =

1,2.

2.28. The following three theorems constitute the beginning of an infinite sequence (Cox’s chain of theorems). C o x ’s f i r s t t h e o r e m . Let IIi, II2, II3, II4 be four planes in general position through a point / . Let fij be an arbitrary point 011 the line n^DlIj.

82

2. Discretization Principles. Multidimensional Nets

Let Tlijk denote the plane through f y , , fjk . Then the four planes II123, II124, II134, II234 all pass through one point / i 234C o x ’s SECON D T H E O R E M . Let II i, . . . , II5 be five planes in general po­ sition through / . Then the five points / 1234, / 1235, / 1245, / 1345, /2345 all lie in one plane II12345. C o x ’s t h i r d T H E O R E M . Let II i, . . . , Efe be six planes in general position through / . Then the six planes II12345? IIi 2346>11x2356? II12456? 1113456, II23456 all pass through one point /i23456Prove this chain of theorems, by establishing their relation with the content of Section 2.4.

2.29.* Prove that a projection of an A-net to any plane from a point not lying in this plane is a (planar) Koenigs net. 2.6. Bibliographical notes Section 2.1: Discrete conjugate nets. Two-dimensional nets with pla­ nar quadrilaterals as discrete analogs of conjugate nets on smooth surfaces were introduced by R. Sauer in the 1930s in a series of papers starting with Sauer-Graf (1931). They were mainly devoted to infinitesimal deformations of smooth and discrete surfaces. The results are summarized in the books Sauer (1937) and Sauer (1970). Multidimensional Q-nets were introduced in Doliwa-Santini (1997). The fundamental Theorem 2.5 is also due to them. F-transformations of Q-nets and their special cases (named after Laplace, Levy, Combescure, etc.) were investigated in Doliwa-Santini-Manas (2000) on the base of the multidimensional consistency. The geometry of Laplace transformations was previously clarified by Doliwa (1997). Q*-nets were discussed in Doliwa-Santini (2000). Equation (2.3) interpreted as a discrete version of the two-dimensional Schrodinger equation was considered in Krichever (1985), where its algebrogeometric solutions were constructed. The discrete Darboux system (2.20) was first derived by Bogdanov-Konopelchenko (1995) without any relation to geometry, while a geometric interpretation in terms of Q-nets was found in Doliwa-Santini (1997). For further analytic studies of Q-nets and their transformations see, for example, Doliwa-Manakov-Santini (1998), DoliwaManas-Martinez Alonso (1999). Some fragments of the general picture of Q-nets were known within the theory of transformations of smooth surfaces. In particular, Eisenhart (1923) observed that the corresponding points of the permutability diagram of two F-transformations are coplanar and three F-transformations close up in a tree-dimensional hexahedron with planar faces. A modern presentation of this topic is in Rogers-Schief ( 2002).

2.6. Bibliographical notes

83

Section 2.2: Discrete line congruences. Theory of discrete line con­ gruences has been developed in Doliwa-Santini-Manas (2000). Section 2.3: Discrete Koenigs nets. In his research of deformations of quadrilateral nets / : 1 ? —> R 3 Sauer considered a special class of flexible nets possessing infinitesimal deformations with rigid vertex stars ( “eckenstarr wackelige Netze” ). He showed that their characteristic property is the existence of a net with parallel corresponding edges and noncorresponding diagonals ( “antiparalleles Netz” ). Sauer also observed that this class of nets is preserved under projective transformations. See Sauer (1933, 1934, 1935). Recently an equivalent geometric characterization of this class in terms of mixed areas was given in Pottmann-Liu-Wallner-Bobenko-Wang (2007). This class was generalized for multidimensional nets / : Zm —> R^ in Bobenko-Suris (2007b, 2007c) and Doliwa (2007a). In particular, Definition 2.22, the algebraic description of Theorem 2.30, as well as the projective geometric characterization of Koenigs nets for m = 2 (Theorem 2.26), is due to Bobenko-Suris (2007c). The geometric characterization for m > 3 (Theorem 2.29) is due to Doliwa (2007a). Note that a different discretization of Koenigs nets was suggested in Doliwa (2003). In fact a net which comprises intersection points of diagonals of a discrete Koenigs net in the sense of Definition 2.22, is a Koenigs lattice in the sense of Doliwa (2003). As a matter of fact, the first Q-nets appeared in the modern literature on integrable discrete differential geometry were special (circular) Koenigs nets introduced in Bobenko-Pinkall (1996b) as discrete isothermic surfaces, see Chapter 3. In the context of integrable systems, the discrete Moutard equation (2.44) was introduced in Miwa (1982), Date-Jimbo-Miwa (1983). This equa­ tion expresses the permutability properties of the Moutard transformation for the differential Moutard equation; see Moutard (1878), Bianchi (1923). A three-dimensional permutability diagram of Moutard transformations was discovered in Ganzha-Tsarev (1996) and Nimmo-Schief (1997). The latter paper interprets this diagram as a discrete 3D equation on a multidimen­ sional lattice. A nice geometric interpretation of the discrete Moutard equa­ tion and of the star-triangle map in terms of reciprocal figures of graphical statics was given in Konopelchenko-Schief ( 2002). In discrete differential geometry T-nets and M-nets in R 3 appeared as Lelieuvre normal fields of discrete asymptotic nets (see Section 2.4) in Doliwa (2001a) and Doliwa-Nieszporski-Santini (2001). The special case of M-nets in the sphere S 2 appeared earlier in Bobenko-Pinkall (1996a) as the Gauss

84

2. Discretization Principles. Multidimensional Nets

map of discrete K-surfaces. Another special case related to discrete isother­ mic surfaces is due to Schief (2001). In all generality, multidimensional T-nets are treated in Bobenko-Suris (2007b).

Section 2.4: Discrete asymptotic nets. Two-dimensional discrete as­ ymptotic nets were introduced by Sauer (1937, 1970) as a discrete analog of surfaces parametrized along asymptotic lines. The discrete Lelieuvre rep­ resentation is due to Konopelchenko-Pinkall (2000). Weingarten transfor­ mations of discrete asymptotic nets were introduced in Doliwa ( 2001a) and Doliwa-Nieszporski-Santini (2001). Mobius pairs of tetrahedra were invented by Mobius (1828); they are well studied in classical projective geometry; see Blaschke (1954). Section 2.5: Exercises. Ex. 2 .2 : See Adler (2006). Ex. 2.3: This is a discrete version of a result by Eisenhart (1923), p. 55. Ex. 2.4: Eisenhart’s formulation of the F-transformation is given in Eisenhart (1923), Chapter II. Ex. 2.7: See Doliwa-Santini (2000). Ex. 2.8: See Schief (2003b). Ex. 2.13-2.15: Laplace transformations and Laplace invariants of Qnets were introduced in Doliwa (1997); see a detailed exposition in DoliwaSantini-Manas (2000). A far-going generalization of discrete Laplace trans­ formations has been developed in Dynnikov-Novikov (1997). Ex. 2.19: See Doliwa-Grinevich-Nieszporski-Santini (2007). Ex. 2.20: These “dual Koenigs nets” appeared in Sauer (1937, 1970) as Q-nets admitting infinitesimal deformations with rigid faces ( “flachenstarr wackelige Netze” ). He proved that such nets are projectively dual to Qnets admitting infinitesimal deformations with rigid vertex stars ( “eckenstarr wackelige Netze” ). Blaschke (1920) gave a characterization of flexible octahedra ( “wackelige Achtflache” ) which can be treated as a projectively dual version of Theorem 2.29 (see also Kokotsakis (1932)); indeed, an octa­ hedron can be regarded as an elementary cell of a three-dimensional Q*-net. For a relation of flexible Q-nets to integrable systems see Schief-BobenkoHoffmann (2008). An interesting class of special dual Koenigs nets was studied in Doliwa-Nieszporski-Santini (2004). Ex. 2 .22: The property of the Q-net M established in this exercise was used in Doliwa (2003) as a definition of Koenigs lattices. Ex. 2.23: The property of discrete line congruence (M M + ) established in this exercise is characteristic for discrete Koenigs transformations in the sense of Doliwa (2003).

2.6. Bibliographical notes

85

Ex. 2.26: See Doliwa (2007a). Ex. 2.27: See Nimmo-Schief (1997), Doliwa (2001a), Doliwa-NieszporskiSantini (2001). Such a formulation is traditional in the classical surface theory. In the discrete setup this formulation seems artificial and awkward since the symmetry of the multidimensional T-nets is lost. Ex. 2.28: See Cox (1891) and Coxeter (1969). Classical chains of in­ cidence theorems (see Coolidge (1916) and Berger (1987)) admit a natural interpretation in terms of the multidimensional consistency. Ex. 2.29: This is a discrete version of a classical result of Koenigs (1892a, b).

Chapter 3

Discretization Principles. Nets in Quadrics

Numerous examples of multidimensional nets discussed in the previous chap­ ter provide a firm basis for the following fundamental discretization princi­ ple. 1. Multidimensional consistency principle: Discretizations of surfaces,

coordinate systems, and other smooth parametrized objects should be extendable to multidimensional consistent nets. Now, we would like to extend it to the second principle. 2 . Transformation group principle: Smooth geometric objects and

their discretizations should belong to the same geometry, i.e. be invariant with respect to the same transformation group. Let us explain why such different imperatives as the consistency principle and the transformation group principle can be simultaneously imposed for discretization of classical geometries. The transformation groups of various geometries, including those of Lie, Mobius and Laguerre, as well as Pliicker line geometry, hyperbolic geometry and so on, can be modelled as subgroups of the projective transformation group of the suitable higher-dimensional space. Classically, such a subgroup is described as consisting of projective transformations which preserve some distinguished quadric called absolute. Remarkably, multidimensional Q-nets and discrete line congruences, being manifestly projectively invariant objects, can be restricted to an arbitrary quadric. This is the reason why the above two fundamental discretization principles work together for the classical geometries. 87

3. Discretization Principles. Nets in Quadrics

88

In this chapter, we start with a detailed discussion of circular nets, which are the main instance of conjugate nets in quadrics. We introduce circular nets with elementary geometric means, and then arrive at their Mobius geometric characterization. Motivated by this example, we formulate the general results concerning the reduction to quadrics, to continue with fur­ ther classes of multidimensional nets illustrating the second discretization principle.

3.1. Circular nets 3.1.1. Notion and consistency of circular nets. Definition 3.1. (Circular net) A map f : Zm —> R N is called an Tri­ dimensional circular net in cular, i.e., if at every u £

, if all its elementary quadrilaterals are cir­

Zm and fo r every pair o f indices 1 < i ^ j < m

the four points f , Tif, Tjf and TiTjf are concircular.

A piece of a circular net is shown in Figure 3.1.

Figure 3.1. A circular net.

To better understand the constructive aspects of circular nets, we con­ sider again various values of ra. m — 2 : discrete curvature line parametrized surfaces. Two-dimen­ sional circular nets (ra — 2 ) can be viewed as discrete analogs of curvature lines parametrized surfaces. It is natural to regard the lines passing through the centers of the circles orthogonally to their respective planes as the nor­ mals to the discrete circular surface. This normals behave in a way charac­ teristic for the normals to a smooth surface along curvature lines; namely, for any two neighboring quadrilaterals of a circular net, the discrete normals intersect. Indeed, both normals lie in the bisecting orthogonal plane of the

3.1. Circular nets

89

common edge. The intersection point is the center of a sphere containing both circles; see Figure 3.2.

Figure 3.2. Normals of two neighboring quadrilaterals of a circular net intersect.

Suppose two coordinate lines f \ ^ x and f \*%2 on a circular surface / are given. An elementary induction step for extending the circular surface to the quadrant Z \ consists in choosing f \2 on the circle through / , fi and /2 . In doing so, one has the freedom of choosing one real parameter at each such step, for instance, the cross-ratio of four points q\2 q(f, / 1 , /1 2 , f 2), which is naturally assigned to the elementary square G(u). Thus, to define a circular surface / , one needs to prescribe the coordinate lines f \ ^ and f\ R N is an M-dimensional circular net, where M = ra-hi. So, in the discrete case there is no difference between orthogonal nets and their Ribaucour transformations. To specify a Ribaucour transform / + of a given ra-dimensional circular net / , one clearly needs the following data: ( R f ) the value of / + (0); (R^) the values of qiM = q(f, f , /+ , /+ ) on “vertical” elementary squares attached to all edges of the coordinate axes 23; for 1 < i < ra. Multidimensional consistency of circular nets yields the following theo­ rem.

Theorem 3.6. (Permutability of discrete Ribaucour transforma­ tions) 1) Let f be an m-dimensional circular net, and let f ^ of its Ribaucour transforms. circular nets f ^

and f ^

be two

Then there exists a one-parameter family of

that are Ribaucour transforms o f both f ^

corresponding points o f the four circular nets f , f ( l\ f ^

and f ( 2\

The

and f ^

are

concircular.

2) Let f be an m-dimensional circular net. Let f ^ \ f ^ and f ^ be three of its discrete Ribaucour transforms, and let three further circular nets f^l2\

/ ( 23) and / ( 13) be given such that f W is a simultaneous discrete Ribaucour transform of /W and f ^ \ Then generically there exists a unique circular net / ( 123) that is a Ribaucour transform of f ( 12\ f (23) and f^ 13\ The net f(i23) ^ uniqueiy determined by the condition that the corresponding points ° f f^ K f ^ \

(123).

f ^

and / ( 123) are concircular fo r any permutation ( i j k ) of

93

3.1. Circular nets

Proof, This proof is absolutely similar to the proof of Theorem 2.8. In the first part, we define an (m + 2)-dimensional circular net F : Z m x {0, l } 2 —> R n such that F(u,0,0) = f(u),

F(u, 1 ,0 ) = f m (u),

F{u,0,1) = f {2)(u).

The only additional initial datum required for the construction of F is the value of F(u, 1,1) = /^ 12^(u) at u — 0 , which can be chosen arbitrarily on the circle through the points / (0 ) , / ^ ( 0), f ( 2 \ 0 ) . This leads to the one-parameter family of circular nets f^ l2\ □

3.1.3. Analytic description of circular nets. Theorem 3.7. (Circularity criterion) Let / i , / 2 , / 3,/4 G E4 be four coplanar points:

(3.1)

a i f i + (*2/2 + £*3/3 + < 24/4 = 0

with

a\ + a 2 + equivalently, the point / ( 0) and the functions gi on the edges of the coordinate axes £ ;) for i = 1 , . . . , m, and o n e rotation coefficient 7 ^ (say) for each plaquette of the coordinate planes the second rotation coefficient

7 ^ on

‘B i j

for

1
)vi = Ay.(f) = n f ,

so that (3.24) holds everywhere on Z m.

□

3.2. Q-nets in quadrics Upon a stereographic projection, circular nets can be characterized as Qnets in the sphere C R ^ -1"1 , while in the Mobius-geometric formulation of Theorem 3.9, they are nothing but Q-nets in the quadric C R^"1"1,1 (or in C R ^ 4"1,1). We will see later in this chapter that many other geometrically relevant nets turn out to be reductions of Q-nets to some quadrics or to intersections of quadrics. This is due to the following general fact: Q-nets can be consistently restricted to an arbitrary quadric in R ^ . A deep reason for this is, in turn, the following fundamental result, well known in the classical projective geometry: Given seven generic points in CP3, there exists a unique eighth point which belongs to any quadric through the original seven points . Heuristically, this can be understood as follows. The equation Q = 0 of a quadric in Q C CP3 has ten coefficients (we identify a quadric with its equation, whose left-hand side is a homogeneous quadratic polynomial in 4 variables). Therefore, nine points in general position define a unique quadric Q through them. Similarly, eight points in general position define a pencil (one-dimensional linear family) of quadrics AiQi + A2Q2 through them, and, likewise, seven points in general position define a two-dimensional linear family of quadrics A1Q1 + A2Q2 + A3Q3 through them. Generically, the solution of the system of three quadratic equations Qi = 0,

Q2 — 0,

Q3 = 0

for the intersection of three quadrics in CP3 consists of eight points. These eight points lie on every quadric of the two-dimensional family through the original seven points. We will not try to make this argument rigorous, prov­ ing instead the following, somewhat weaker, statement which is sufficient for our purposes. T h eorem 3.12. (A ssocia ted p oints) Given eight distinct points which are the set of intersections of three quadrics in

RP3, all quadrics through

any subset o f seven o f the points must pass through the eighth point. Such sets of points are called associated.

P ro o f. Let A\, A2, •. • >A$ be the set of intersections of three quadrics Qi,

Q2, Q3. Note that no three of the eight points Ak can be collinear, since otherwise the set of intersection of the three quadrics would contain a whole line and not just eight points (see Exercise 3.7). For similar reasons no five of

3. Discretization Principles. Nets in Quadrics

100

the eight points

can be coplanar. Indeed, five coplanar points no three of

which are collinear determine a unique conic. The intersection of the three quadrics Qi, Q2, Q3 would contain this conic and not just eight points. Choose any subset of seven points A \ , A 2 , . . . , A7. We show that any quadric Q through these seven points must belong to the family Q = A1Q1 + A2Q2 + A3Q3 ; as a consequence, the eighth intersection point Ag will automatically lie on Q. Suppose that, on the contrary, Q is linearly independent of Qi,Q2,Q3Consider the family of quadrics (3 .26)

Q' = A1Q1 + A2Q2 “I- A3Q3 + fxQ.

Due to the assumed linear independence, one could find a quadric in this family through any prescribed triple of points in RP3. We show that this would lead to a contradiction. First assume that no four points among A i , A 2, . . . , A r are coplanar. Choose three points B , C , D in the plane of A \, A2, A3 so that the six points

A\, A2, A3, B, C, D do not lie on a conic. Find a quadric Q' in the fam­ ily (3 .26) through B , C, D.

This quadric must be reducible, one compo­

nent being the plane of A\, A2, Az (indeed, otherwise Q! would cut this plane in a conic through A\, A 2, A3, B, C, D, which contradicts the choice of

B, C, D). The other component of Q' must be a plane containing four points A4, A$, A q, A y, a contradiction. The remaining case, when there are four coplanar points among A\, A2 , . •., Aj, is dealt with analogously. Let A\, A 2, A3 and A4 be coplanar. Denote the plane through these four points by II. Take two points B, C in the plane II so that the six points A\, A 2, A3, A ± , B , C do not lie on a conic, and take a point D not coplanar with A§, A§, Aj (which is always possible, because the latter three points are not collinear). Then there exists a quadric Q' in the family ( 3 .26 ) through B , C , D .

Again, this quadric

must be reducible, consisting of two planes, one of them being the plane II. The other component of Q! must be a plane containing A§, A§, Ai , D, a contradiction again (this time to the choice of D).

□

Theorem 3.13. (Elementary hexahedron of a Q-net in a quadric) If seven points f , f i , and fij (1 < i < j < 3) of an elementary hexahedron of a Q-net belong to a quadric Q C WPN, then so does the eighth point /i23Proof. The eight points / , fi, fij , /123 lie in a three-dimensional space, and are known to form the intersection of three (degenerate) quadrics — the pairs of planes Ilj^ U n lljk for (jk) = ( 12), (23 ), (31 ). Therefore, they are associated points. According to Theorem 3 . 12, any quadric Q through the seven points f , fi, fij automatically goes through the eighth point.

□

3.3. Discrete line congruences in quadrics

1 01

T h eorem 3.14. (R ed u ction o f Q -nets to a quadric) If the coordinate

surfaces f\'&ij of a Q-net f : Z m —> MFN belong to a quadric Q? then so does the entire f . P ro o f. For m = 3 this follows from Theorem 3 . 13. The claim for m > 4 follows from the m-dimensional consistency of Q-nets.

□

Another version of Theorem 3.13 can be formulated as follows. It is based on an obvious fact that for a nonisotropic line £ with a nonempty in­ tersection with Q this intersection consists generically of two points (because it is governed by a quadratic equation).

T h eorem 3.15. (E lem entary R ib a u cou r transform ation in a quadric)

Let ( /, / i , /12, f 2) be a planar quadrilateral in a quadric Q. Let £, £\,£2, £\2 be nonisotropic lines in R F N containing the corresponding points f , f i , f 2, fi2 and such that every two neighboring lines intersect. Denote the second in­ tersection points of the lines with Q by / + , / + , f ^ f ^ , respectively. Then the quadrilateral ( / + , f^ , f ^ f 2 ) is also planar. P r o o f. The eight points / , / i , / 2, / i 2, lie in a three-dimen­ sional space, and are known to form the intersection of three quadrics, one of them being Q, and two others being the pairs of planes s p a n (^ i) U span(£2,^i2) and span(^, £2) U span(^i, £\2). Therefore, they are associated points. The pair of planes ( /, /1 , /12, / 2) and ( / + , / jf, f £ ) contains seven of them; therefore the eighth point, f ^ , must lie in the plane ( / + , f i , f 2 )- □ As a global corollary of this local statement, we immediately obtain the following:

T h eorem 3.16. (R ib a u cou r transform ation o f a Q -net in a quadric)

Consider a quadric Q C RP^ and a Q-net f : Z m —> Q. Let a discrete congruence of nonisotropic lines £ : Z m —> £j N be given such that f{u) £ £(u) for all u E Z m. Denote by f + (u) the second intersection point of £{u) with Q, so that £(u) fl Q = {/('u), f+(u)}. Then / + : Z m —►Q is also a Q-net. 3.3. Discrete line congruences in quadrics Consider a quadric Q C RP^ which is generated by a symmetric non­ degenerate bilinear form (•, •) in the space R ^ +1 of homogeneous coordinates. Suppose that the signature of the corresponding quadratic form contains at least two positive and two negative entries. In this case the quadric Q carries isotropic lines £ C Q; actually, one can draw at least two such lines through any point of Q. We denote the set of isotropic lines on Q by £ q . A good insight in the construction possibilities of discrete line congru­ ences in Q is provided by the following statement.

102

3. Discretization Principles. Nets in Quadrics

Lemma 3.17. Let £

G £ q be an isotropic line, and let f \ G Q be a point not lying on £. If the plane through £ and f \ is nonisotropic, then there exists a unique isotropic line £\ through f \ intersecting £.

Proof. Let f , g be the homogeneous coordinates of two arbitrary points / , j G f, so that the line £ is given by the linear combinations a f + (3g. Relation ( a f + (3g, f \ ) = 0 yields a = _ (ffi/i) P

(fJi)

(By assumption, at least one of the scalar products ( / , / i ) and ( g, f \ ) does not vanish.) Thus, there exists a unique point G £ such that (f^l \ f i ) = 0. Now t \ is the line through f \ and □ Now one can see that, given three lines t, l \ , £2 of a congruence of isotropic lines, one generically has a one-parameter family of lines admissible as the fourth line £\2: through any point of £\ passes a unique isotropic line which intersects t 2. Thus, given any two sequences of isotropic lines < : Z x {0} —> £q and £ : {0} x Z - > with a common line ^(0,0) such that any two neighboring lines intersect, one can extend them to a two­ dimensional congruence of isotropic lines i : 1? —* £ q in an infinite number of ways: on each step of the induction procedure one has a freedom of choosing a line from a one-parameter family (as opposed to a two-parameter family in the case of general, i.e., nonisotropic line congruences). At the same time, the 3D system which describes line congruences (see Theorem 2.10) can be consistently reduced to the isotropic case without any additional restrictions.

Theorem 3.18. (3D discrete line congruences in a quadric) Given seven isotropic lines £, l{, iij G Cq (1 < i < j < 3) such that £ intersects each of the space V123 spanned by £, t \ , £2l £3 has dimension four, and each £i intersects both £{j and £ik, generically there is a unique isotropic line £ \ 23 G £ q that intersects all three £{j.

Proof. The unique line ^123 G &N intersecting all three £ij (see Theorem 2.10) has three points in Q, namely ^123 C\£ij. Therefore, it must be isotropic, ^123 G £ q (see Exercise 3.7). □ Another important construction of isotropic line congruences is given in the following theorem.

Theorem 3.19. (E xtending a Q-net in a quadric to an isotropic congruence) Given a Q-net f : Zm —> Q, there exist discrete congruences of isotropic lines £ : Zm —►£ q such that, for every u G Zm, we have

3.4. Conical nets

103

f ( u) £ £(u). Such a congruence is uniquely determined by prescribing an isotropic line ^(0 ) through the point / ( 0 ).

Proof. The construction hinges on Lemma 3.17. According to it, one can construct the isotropic lines of the congruence along the coordinate axes £ \^ r Furthermore, according to Theorem 2.16 there exists a unique line congruence through the points / with these initial data. It remains to show that all the lines of this congruence are isotropic. This follows from a consideration of one 2D face. If the lines £\, £2 are isotropic, then the line £12 contains at least three points on Q, namely /12 and the two further points = £\ fl £12 and / 2(1) = £2 fl £12 • Therefore £12 is isotropic. □

3.4. C onical n ets Our next example of Q-nets in quadrics will be the so-called conical nets. They can be considered as a subclass of Q-nets in R 3 satisfying a certain condition on four quadrilaterals adjacent to one vertex. However, it turns out to be more convenient to take a dual point of view from the outset and to pay the main attention to the planes of the quadrilaterals rather than to their vertices.

D efinition 3.20. (Conical net) A map P : Zm —►{oriented planes in R3} is called a conical net if for every u G Zm and for each pair of indices 1 < i i=- j < m the four planes P, Pi , Pi j , Pj intersect at a common point and, moreovery are in oriented contact with a cone of revolution (with the tip in the common point of all four planes); see Figure 3.7.

F igure 3.7. Four planes of a conical net.

Recall that an oriented plane P in R 3 can be described by a pair ( /\ d) 6 § 2 x R, where

P = {x € R 3 : (v, x ) = d},

104

3. Discretization Principles. Nets in Quadrics

so that v G § 2 is the unit normal vector to P, and d is the distance of P to the origin. Two orientations of one and the same plane correspond to two pairs ( v, d) and (—v, —d). Note that if one neglects the orientation of the planes, the conical nets form a subclass of Q*-nets. Slightly abusing terminology, we will ascribe conical nets to Q*-nets. One can think of two-dimensional conical nets as another (besides circu­ lar nets) discretization of curvature line parametrized surfaces. The axes of the cones of revolution mentioned in Definition 3.20 are thought of as dis­ crete normals to the surface, defined at the points of the dual lattice (Z2)*. For any two neighboring cones, there is a unique sphere touching both of them. The center of this sphere is the intersection point of the axes of the cones; see Figure 3.8. Indeed, two neighboring quadruples of planes share two planes, and the plane bisecting the dihedral angle between these two contains the axes of both cones, which therefore intersect (the orientations of the planes fix one of the two dihedral angles).

V

Figure 3.8. Axes of two neighboring cones of a conical net intersect.

A simple geometric criterion for a Q*-net P to be conical can be given in terms of the Gauss m ap of P, (3.27)

§2,

which is comprised by the (directed) unit normal vectors v to the planes P .

Theorem 3.21. (Conical nets have circular Gauss maps) A Q*-net P : Zm —» {oriented planes in M3} is conical if and only if the elem entary quadrilaterals of its Gauss m ap v : Zm —>8 2 are planar, that is, if the Gauss m ap is a circular net in S2. Proof. The angles between all four unit vectors v, Vj, Vij and the axis of the cone are equal; therefore their tips are at an equal (spherical) distance

3.4. Conical nets

105

from the point of § 2 representing the cone axis direction. Thus, the quadri­ lateral (y,Vi,Vij,Vj) in § 2 is circular, with the spherical center of the circle given by the direction of the axis of the tangent cone. □ In order to determine a conical net, it is enough to prescribe a circular Gauss map v : Zm —►§ 2 and, additionally, to give the numbers d (i.e., to specify the planes P = { x G M3 : (v, x) — d} with the given normals v) along the coordinate axes 2 ; of Zm. These data allow one to reconstruct the conical net uniquely. This is done via a recursive procedure, whose elementary step consists in finding the fourth plane Pij provided three planes P, Pi , Pj and the normal direction Vij of the fourth plane are known. But this is easy: Pij is the plane normal to through the unique intersection point of the three planes P, P i , Pj. It is important to observe that Definition 3.20 actually belongs to Laguerre geometry. This means that the property of touching a common cone of revolution for given planes is invariant under Laguerre transformations, in particular, under shifting all the planes by the same distance in their cor­ responding normal directions (normal shift). Recall (or see Section 9.4) that a plane P — {x 6 M3 : (v, x) = d} is represented in the projec­ tive model of Laguerre geometry (Blaschke cylinder model) by the point in P(L3,1,1) C P(R3,1,1) with the representative p = v + 2de(yo + e 6 in the space of homogeneous coordinates.

Theorem 3,22. (Conical net, Laguerre-geom etric characterization) A net P : Zm —> {oriented planes in M3} is conical if and only if the corresponding points p : Zm —* P(L3,1,1) form a Q-net in P(R3,1,1).

Proof. Representatives p 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 —> M 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 , vi j , 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 , Pj lies on Pij 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 ev­ ery quadruple of planes (P, Pi, P^, Pj) 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. □ Thus, conical nets constitute a further example of multidimensional Qnets restricted to a quadric (the absolute quadric in the projective model of Laguerre geometry).

106

3. Discretization Principles. Nets in Quadrics

3.5. P rin cip a l c o n ta c t e lem en t n e ts 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 R3. On the other hand, a surface can be viewed as the envelope of the sys­ tem 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 ge­ ometry should be understood as a map P : 1? —> {oriented planes in M3}.

A substantial part of such a description of a surface is its Gauss map v : I? -► §2,

consisting of unit normals v to the planes P = {x <ER:! : (v. x) = d}. There are also geometries where contact elements are distinguished. A contact element can be interpreted as a pair (/, P ) consisting of a point / of the surface along with an (oriented) tangent plane P through / (or, equivalently, a normal vector v to P at / ) . 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 / which are in oriented contact with P (and with one another), thus sharing the normal vector v at /; see Figure 9.1 in Section 9.2. Another instance is Pliicker line geometry, where a contact element is understood in terms of lines in RP3 and their intersections, as a one-parameter family (pencil) of lines in P through / . 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 (/, P ) : 1? —> {contact elements in M3}.

3.5. Principal contact element nets

107

The stage where the projective model of Lie geometry takes place is the Lie quadric P(L4,2) C P(R4,2). In particular, contact elements are modelled as isotropic lines (lines in P(R4,2) which lie in P(L4,2)). We will denote the set of isotropic lines of P(R4,2) by £ q2. 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 m ap £ : Z2 —> £q 2 is called a principal contact elem ent net if it is a discrete congruence of isotropic lines in P(R4,2), that is, if every two neighboring lines intersect:

(3.28)

£(u) H £{u + a ) = s w (u) G P ( L 4’2),

Vu G Z 2,

Vi -

1, 2.

Naturally associated to a congruence of isotropic lines £ are two focal nets (3.29)

s (i) : I ? -*■ P(L4,2),

cf. Definition 2.13. The points in R3.

i = 1,2;

€ P(L4,2) represent oriented spheres

A direct translation of Definition 3.23 into the language of the geometry of spheres in R 3 looks as follows.

Definition 3.24. (Principal contact element net, Euclidean model of sphere geometry) A map (/, P ) : Z 2 —> {contact elements in R3} is called a principal contact elem ent net if every two neighboring contact elem ents (/, P ), (r*/, r*P) have a sphere in common, that is, a sphere in oriented contact with both planes P , T{P at the corresponding points f , n f.

Thus, the normals to the neighboring planes P, Pi at the corresponding points / , fi intersect at a point (the center of the sphere S ^ ), and the distances from to / and to fi are equal; see Figure 3.9. The spheres are naturally assigned to the edges of Z 2 parallel to the z-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 P(L4,2)) results in the following definition.

108

3. Discretization Principles. Nets in Quadrics

F igure 3.9. Principal curvature sphere.

D efinition 3.25. (Ribaucour transform ation, projective m odel of sphere geom etry) Two principal contact element nets t, \ Z 2 —> £q ’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)

i{u) n i + {u) = s(u) € P(L4’2),

Vu e Z2.

Again, a direct translation of this into the conventional geometric terms reads:

Definition 3.26. (Ribaucour transform ation, Euclidean m odel of sphere geom etry) Two principal contact element nets (/, P ), ( / +, P + ) : Z2 —> {contact elements in M3} are called Ribaucour transforms of one another if every two corresponding contact elements (/, P ) and ( / +, 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 5 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, 0) and (u , 1) of the lattice Z2 x {0,1}. In the projective model, their representatives

3.5. Principal contact element nets

109

build the focal net of the three-dimensional line congruence for the third coordinate direction. According to Theorem 2.14, both focal nets of a principal contact element net are Q-nets in P(L4,2) C P(R4,2), and the same holds for the spheres s of a generic Ribaucour transformation. This motivates the follow­ ing definition.

D efinition 3.27. (D iscrete 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 : Z m —►P(L4,2),

s = c + e 0 + (|c |2 - r 2 ) e oo + re6,

is a Q-net in P(R4,2).

Thus, we can formulate:

Corollary 3.28. (R-congruences of principal curvature spheres and o f a R ibaucour transform ation) 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 of the i-th coor­ dinate direction is degenerate in the sense that the plane of every its ele­ mentary quadrilateral (s ^ , s\l\ s ^ ) contains two isotropic lines i uUj ,

3. Discretization Principles. Nets in Quadrics

110

so that the three-dimensional vector space

E= s

p

a

n

C R 4,2

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

3.6. Q -con gru en ces o f sp h eres 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 —> {oriented spheres in R3}

is a discrete R-congruence if and only if the centers c : Zm —> R 3 form a Q-net in R3, and the two real-valued functions, |c |2 - r 2 : Zm —> R

and

r : Zm —►R,

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 geome­ try rather than of Lie geometry. Recall (or see Section 9.3) that spheres in R3 can be represented in the Mobius-geometric formalism as elements of P(R 40i ) , where

(3.33)

R4^ = {s € R4’1 : (S, s) > 0}

is the space-like part of R4,1. 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 —> {nonoriented spheres in R3}

is called a Q-congruence of spheres if the corresponding m ap

(3.35)

s : Zm —>P(Rout)>

s = c + e 0 + (|c |2 - r 2)eoo,

is a Q-net in P(R4,1).

Thus, a map (3.34) is a Q-congruence if and only if the centers c : Zm —►R 3 of the spheres S form a Q-net in R3, and the function |c |2 —r 2 satisfies the same equation (2.1) as the centers c. Clearly, as r —> 0 the latter characterization turns into the criterion of Theorem 3.7. A geometric description of Q-congruences is given in the following theorem.

3.6. Q-congruences of spheres

111

T h eorem 3.30. (T h ree ty p es o f Q -congruences) Four (nonoriented) spheres (Si, $ 2 , £3 , S 4 ) in M3 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 0 -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).

C on cep tu al proof. For a Q-congruence, the vector subspace S = span(Si, 52 , S3 , 54) C M4’1 is three-dimensional, so its orthogonal complement S -1 is two-dimensional. If S 1 lies in M4^ , i.e., if the restriction of the Minkowski scalar product to S -1 is positive definite (of signature (2 , 0 )), then S '1 represents a 1-sphere (a circle) orthogonal to our four spheres, and we have case (i). If, on the contrary, the restriction of the scalar product to S -1 has signature ( 1 , 1 ), so that £ lies in M4^ , then E represents a 0 -sphere which is the intersection of our four spheres, and we have case (ii). Finally, if the restriction of the scalar product to S '1 is degenerate, then £ H T,1 is an isotropic one-dimensional vector subspace, which represents the common point of our four spheres, and we have case (iii). □ C o m p u ta tio n a l proof. For a Q-congruence, the quadrilateral in M3 with vertices at the sphere centers ci, 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 C g I I such that |ci - C \2 - r \ = |c2 - C \2 - r 2 = |c3 - C \2 - r 2 = |c4 - C \2 - r \ .

(3.36)

Indeed, let the centers c; and the quantities |q |2 —r 2 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 lyi and ^23 in II, where lij = { x e II : (2x - Ci - Cj, Ci - Cj) = rj - rf}.

Now, if the common value of all four expressions in (3.36) is positive, say equal to i?2, then the four spheres under consideration are orthogonal to the circle in the plane II with center C and radius i?, so that we have case (i); see Figure 3.11. If the common value of (3.36) is negative, say equal to —i?2, then the pair of points on the line through C orthogonal to II, at the

112

3. Discretization Principles. Nets in Quadrics

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). □

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 s, Si, Sij in P(Ro’*t), the Q-property (planarity condition) uniquely defines the eighth point 5123 in P(R4,1), which, however, might get outside of P(M ^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. (Elem entary hexahedron o f a Q -congruence o f cir­ cular type) C onsider a three-dimensional Q-congruence of spheres with all faces of type (i), i.e., with orthogonal circles. For an elem entary hexahe­ dron 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 C12 (say) means that £ |2 = span(s, si, §2 )^ is a two-dimensional vector subspace lying in ^out- Therefore, ^ 2 3 = sp a n (5 ,si,s 2 ,«3)'L is a one-dimensional vector subspace lying in This subspace is con­ tained in all X ^, £ 13, £ 23, as well as in (r ( t 2 ^is)'L, (ti^ s)" 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. □ A more elementary geometric proof of this statement is sketched in Ex­ ercise 3.17.

3.7. R ibaucour congruences o f spheres We now return to the study of R-congruences of (oriented) spheres, i.e., of Q-nets in the Lie quadric P(L4,2) C P(R4,2). Thus, in the present section the notation s again refers to the Lie-geometric formalism! We restrict ourselves to nondegenerate R-congruences, for which the subspaces spanned by its elementary quadrilaterals £ = span(si, £2 , 53, 84) do not contain two-dimensional isotropic subspaces. Thus, we leave aside the principal sphere congruences. The case when the signature of (*> *) Is-1is (3,0), so that the spheres si, 52, 83 , £4 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 (-, -)|s and (•, -)|s _l are (2 , 1 ), so that to any elementary quadrilateral (Si, 52, S3 , 54) of a discrete R-congruence there corresponds a Dupin cyclide (see Definition 9.4 in Subsection 9.2.4).

T heorem 3.32. (G eom etric characterization of R -congruences of cyclidic type) Consider four (oriented) spheres (Si, S 2 , S 3 , S 4) i n R 3, such that there exists a nonpoint sphere So in oriented contact with three of them, say with Si, The four spheres (Si, £ 2 , S 3 , S 4) 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 So is in oriented contact with the fourth sphere, S 4 , as well.

Proof. Let a sphere So in oriented contact with the three spheres Si, S 2 , S 3 have center co and (finite) oriented radius ro 7^ 0. This means that the following conditions are satisfied: (3.37)

(ci,co) - ^(|cj |2 - r2) - ^(|c0 |2 - r%) - n r 0 = 0,

i = 1,2,3

114

3. Discretization Principles. Nets in Quadrics

(tangency of Si with So; cf. (9.10)). Now, using the fact that c; and |c;|2 —t*2 satisfy one and the same linear dependence of the type (3.1), we conclude that (3.37) is fulfilled for (04, 7*4 ) if and only if r* satisfy the same linear dependence (3.1) as c; and |c;|2 — r 2 do. This proves the theorem in the case when the common tangent sphere So for the three spheres S i,S 2 ,S 3 has a finite radius. The case when So has an infinite radius, i.e., is actually a plane, is treated similarly, with the help of equation (3.38)

(ci, v 0) - r* - d0 = 0,

which comes to replace (3.37).

□

Remark. We have seen that, generically, if three oriented spheres Si, S 2 , S 3 have a common sphere in oriented contact, then they have a one-parameter (cyclidic) family of common touching spheres, represented by a three-dimen­ sional linear subspace E1- of M4,2. It is easy to see that if the projection of E -1 onto e ^ is nonvanishing, then the family of spheres represented by E -1 con­ tains exactly two planes. (For a conical cyclidic family E all elements have vanishing eo-component and represent planes, while the family E -1 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 S i, S 2 , S 3 of an elementary quadrilateral, one has a twoparameter family for the fourth sphere S 4 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 is an open set in M4,1, while L4,2 is a hypersurface in R 4,2 (not containing isotropic planes).

Theorem 3.33. (Com m on tangent spheres o f two neighboring quad­ rilaterals 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 gen­ erated by the subspaces Ei and E 2 of signature (2,1). These quadrilaterals share two spheres, which span a linear space of signature (1,1). Generi­ cally, each of the planar families Ei and E 2 adds one space-like vector, so that the linear space Ei U E 2 spanned by all six spheres of the congruence is four-dimensional and has signature (3,1), so that its orthogonal comple­ ment (Ei U Es )-1 is two-dimensional and has signature (1,1). Intersection of

3.8. Discrete curvature line parametrization in various geometries

115

L 4,2 with a two-dimensional linear subspace of signature ( 1 , 1 ) gives, upon projectivization, exactly two spheres: indeed, if e i,e 2 form an orthogonal basis of (S i U l b ) 1 with (e i,e i) — —(e2 ,e 2) = 1, then the spheres in this space correspond to ot\e\ + a 2 e 2 with ( a \ e i + a 2 e 2 , a i e i + a 2 e2) — 0



a2 =

a \ \ a 2 —±l.

□

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. (Com m on tangent spheres o f an elem entary hexa­ hedron o f 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 as­ signed 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. □ 3 .8 . D is c r e te cu rvatu re lin e p a r a m e tr iz a tio n in Lie, M ob iu s an d L aguerre g e o m e tries 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 £, £\, £2, i \ 2 (in the projective model), or of four contact elements (/, P ), ( /i,P i) , ( f 2 , P 2), ( / 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 I 3. In this situation, one cannot draw any further conclusions about the four points / , / 1, f 2, f i 2 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 £, £\ , £2, £ \ 2 is three-dimensional. Then the following statement, illustrated in Figure 3.12, holds.

116

3. Discretization Principles. Nets in Quadrics

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 elem ent nets) For a principal contact element net £ : Z2 —>£ q 2’ or (/, P ) : Z 2 —►{contact elements in R3}, its points f : Z 2 —> R 3 form a circular net, while its planes P : Z 2 —> {oriented planes in R3} form a conical net.

Proof. In the nonumbilic situation, the four points / , / i , / 2 , / i 2 € P(L4,2) obtained as the intersection of the four isotropic lines £,£\-l £ci->£vi with the projective hyperplane P(e 0~) in P(R4,2) lie in a two-plane. They correspond to the points / , / i , / 2, f n £ R3. Moreover, omitting the inessential (vanish­ ing) eg-component, we arrive at a planar quadrilateral in the Mobius sphere P(L4,1) C P(R4,1), so that the corresponding quadrilateral in R 3 is circular, according to Theorem 3.9. Analogously, the four points P, Pi , P 2 , Pi 2 £ P(L4,2) obtained as the in­ tersection of the four isotropic lines £, £\ , £ 2 ,£ \2 with the projective hyper­ plane P (e^ ) in P(R4,2) lie in a two-plane. They correspond to the planes P , P \ , P 2 ->Pi2 C R3. Omitting the inessential (vanishing) eo-component, we arrive at the Laguerre-geometric description of planes as points in P(L3,1,1) C P(R3,1,1). According to Theorem 3.22, a planar quadrilateral there corre­ 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

117

LAGUERRE

LIE

MOBIUS

F igure 3.13. An elementary quadrilateral of a principal contact ele­ ment net with vertices / and tangent planes P in the projective model. The vertices / 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 t (Lie geometry). Principal cur­ vature spheres pass through pairs of neighboring points / , fi and are tangent to the corresponding pairs of planes P, P i .

In view of Theorem 3.35, it is natural to ask whether, given a circular net / : Z2 —» R3, or a conical net P : I? —►{oriented planes in R3}, there exists a principal contact element net (/, P ) : Z2 —> {contact elements in R3}, with prescribed half of the data ( / or P ).

Theorem 3.36. (E xtending circular and conical nets to principal contact elem ent nets) i) Given a circular net f : Z 2 —>R3, there exists a two-parameter family of conical nets P : Z 2 —> {planes in R3} such that f G P for all u G Z2, and (/, P ) : Z2 —►{contact elements in R3} is a principal contact element net. Such a conical net is uniquely determined by prescribing a plane P ( 0,0) through the point /(0 ,0 ).

ii) Given a conical net P : Z2 —> {oriented planes in R3}, there exists a two-parameter family of circular nets f : Z2 —> R 3 such that f G P for all u G Z2, and (/, P ) : Z 2 —►{contact elements in R3}

118

3. Discretization Principles. Nets in Quadrics

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

Proof, i) Here we have to solve at each construction step the following problem: Given a contact element (/, P ) and a point f i , find a plane Pi through fi such that there exists a sphere tangent to both planes P, Pi at the points / , f i , respectively. Solution: Pi is obtained from P by reflection in the bisecting orthogonal plane of the edge [/, fi]. The center of the sphere is found as the intersection of the normal to P at / with the bisecting orthogonal plane of the edge [/,/i]. Closing of this construction around a quadrilateral (/, / i , / 12, f 2) 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 (/, P) and a plane P i , find a point fi in P* such that there exists a sphere SW tangent to both planes P, Pi at the points / , f i , respectively. Solution: The point fi is obtained from / by reflection in the bisecting plane of the dihedral angle formed by P, Pi. The center of the sphere is found as the intersection of the normal to P at / with this bisecting plane. Again, the construction closes around a quadrilateral, due to the conical condition. □ Theorem 3.36 admits a far going generalization (recall that in the frame­ work 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 elem ent 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

(/, P) : Zm —> {contact elements in R3} such that, for every u G Zm; the sphere S belongs to the contact element (/, P ), i.e., P is the tangent plane to S at the point f G S. Such a principal contact element net is uniquely determined by prescribing a contact element ( / , P )( 0 ) containing the sphere 5(0).

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

□

3.9. D isc r e te a sy m p to tic n e ts in P liick er lin e g e o m e tr y A contact element in Pliicker line geometry is understood as a family of lines through a point / G R 3 lying in the plane ! P c R 3. A contact element can

3.9. Discrete asym ptotic nets in Pliicker line geometry

119

be identified with a pair (/, 7) such that / £ 7. In the present section, we will only consider contact elements in the sense of Pliicker line geometry. It is not difficult to realize that Definition 2.39 of a discrete asymptotic net allows for the following reformulation:

D efinition 3.38. (D iscrete A -net, Euclidean m odel of line geom e­ try) A map ( f , 7 ) : Z m —> {contact elements in R3} is called an A-net if each pair of neighboring contact elements (/, 7), (f i , 7i) has a line in common, that is, if the line ( f fi) belongs to both planes 7 , 7

This can be immediately translated into the language of the projective model of Pliicker line geometry, where contact elements are represented by the set £q ’3 of isotropic lines in the Pliicker quadric P(L3,3) C P(R3,3).

D efinition 3.39. (D iscrete A -net, projective m odel o f line geometry) A map £ : Z m —►£ 30’3 is called an A-net if it is a discrete congruence of isotropic lines in P(R3,3), that is, if every two neighboring lines intersect: (3.39) i {u) n t { u +

= l(i)(u) E P(L3,3),

Vu € Z m, Vi e { 1 , 2 , . . . , m} .

The elements of the focal nets /W : Zm —> P(L3,3) represent the lines ( f fi) of the A-net in M3. 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 R3. We proceed with a re­ formulation of Definition 2.44:

D efinition 3.40. (D iscrete W -congruence, Euclidean m odel of line geom etry) Two discrete A-nets (/, IP), ( / + , 7 + ) : Z m —> {contact elements in R3} are called Weingarten transforms of each other if for every pair of corre­ sponding contact elements ( f , 7 ) , ( f + , 7 + ) the line I = ( f f + ) belongs to both tangent planes 7 , 7 ^ . The connecting lines I : Z m —> {lines in R3} of a Weingarten pair are said to constitute a discrete W-congruence.

In the language of the projective model this reads:

D efinition 3.41. (D iscrete W -congruence, projective m odel of line geom etry) Two discrete A-nets £, £+ : Zm —> £q ’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)

£{u) n £+ {u) = i(u) € P(L3’3),

Vu € Zm.

3. Discretization Principles. Nets in Quadrics

120

The intersection points I : Z771 —> P(L3,3) represent the lines of a discrete W-congruence.

In the situation of Definition 3.40, both A-nets (/, CP) and ( / + ,IP+ ) are said to be focal nets of the W-congruence I — ( / / + ). More generally, a discrete A-net (/, IP) is called focal for a discrete W-congruence I if each line I belongs to the corresponding contact element (/, IP), that is, / G I and Z C IP. 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 th e Pliicker quadric) A generic W-congruence of lines is represented by a Q-net in the Pliicker quadric P(L3,3).

In particular, four vectors (/, k, Uj, Ij) in R3,3 representing the four lines of an elementary quadrilateral of a generic W-congruence are linearly depen­ dent. This means that the four lines ( l , k, l i j , l j ) in R 3 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 l : Zm -> {lines in M3}, there exists a two-parameter family of discrete A-nets

(/, IP) : Zm —> {contact elements in R3} such that, for every u G Zm, the line I belongs to the contact element (/, IP), that is, passes through the point f and lies in the plane IP. Such a discrete A-net is uniquely determined by prescribing a contact element (/, CP)(0 ) con­ taining the line 1(0 ).

Proof. This is a translation of the content of Theorem 3.19 into the lan­ guage of Pliicker line geometry. □ 3.10. E x ercises 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

/l? fz , fzi / 4 ? and choose a point fij on each side (f i f j ) - Then the four spheres n S jk i through {fi, fi j , fik> fu ) intersect at one point / i 234- Prove this theorem using the 4D-consistency of circular nets.

3.2. Prove that a quadrilateral (/i, f 2l fz , f 4) in C is circular if and only if its cross-ratio is real: 9 (/i ,/2 ,/3 ,/4 ) =

/2 - /3

/4 - / l

€ R.

3.3. Let the quadrilateral ( /i, f 2, fz , f 4) 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{ f i , f 2, h , f 4) < 0 . 3.4. Consider an elementary hexahedron of a circular net. Prove that one can choose the cross-ratios q \ , . . ., 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 corre­ sponding 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 i , f i j , 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 RP , 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 QC 3.11. The following three theorems constitute the beginning of an infinite sequence (Clifford’s chain of theorems). C l i f f o r d ’s f i r s t th eo rem . Let Ci, C 2 , C3 , C 4 be four circles in general position in a plane, with a common point / . Let fij be the second intersection point of the circles Ci and Cj. Let Cijk denote the circle through fi j , f i k , f j k • Then the four circles C m , C 124, C 134, C234 all pass through one point /1234; cf. Figure 3.14.

122

3. Discretization Principles . Nets in Quadrics

Figure 3.14. Clifford’s first theorem.

C lif f o r d ’s seco n d th eo rem . Let C i,. . . , C5 be five circles in general position in a plane, with a common point / . Then the five points / 1234, / 1235, / 12 4 5 , / 13 4 5 , /2345 all lie on one circle CV2345 C lifford ’s third theorem . Let C i,. . . , Cq be six circles in general position in a plane, with a common point / . Then the six circles CV2345, Cl 2346>Ci2356, Ci2456, C 13456, ^23456 all pass through one point / l 23456* Prove these theorems, by restricting Cox’s chain of theorems to a sphere. 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 P i , . . . , P 4 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 (P 1P 2) and (-P3P4 ). Hint: Consider a pencil (oneparameter family) of conics through P i , . . . , P 4 . Comment: If P i , . . . , P 4 lie on a surface, then the conic approximates the intersection of the surface with

3.11. Bibliographical notes

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 ) £ IR3 : x 2 + y 2 —z 2 = 1 } 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 u i,u>2 >^3, Q. m — 2 : basic 2 D system . Knowing two coordinate curves f \ ^ 2 of a two-dimensional T-net / : Z2 —> Q allows one to extend the net / to the

127

4. Special Classes of Discrete Surfaces

128

whole of Z2. The induction step consists in computing r\T2f = f + a i 2 (r2/ - n / ) ,

where the coefficient a \ 2 (attached to every elementary square of Z2) is determined by the condition r \ r 2f G Q, provided / , t i / , r2f G Q. A simple computation using the formula ( / + a i 2 (r2f - r i /) , / + a\ 2 (r2f - n f ) ) = k0 shows that this condition is equivalent to ( f , n f - r 2f )

k0 - ( r i/,r 2/ )

'

This elementary construction step, i.e., finding the fourth vertex of an ele­ mentary quadrilateral from the known three vertices, is symbolically repre­ sented 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. /2#

Q fl2

Figure 4.1. 2D system on an elementary quadrilateral.

m > 3 : consistency. Turning to the case m > 3, we see that one can prescribe all coordinate lines of a T-net n in Q, i.e., /f^ . for all 1 < i < 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 / , Tif and Tj f are known. In order for this induction process to work without contradictions, equations (

4

.

1)

TiTjf

-

/

=

Oij(Tjf -

Tif),

Oi j =

j

—

- \ Ti J , T j 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 / and fi one determines all fij uniquely. After that, one has, in principle, three different ways to determine / 123, from three squares adjacent to this point; see Figure 4.2. These three values for /123 have to coincide, independently of initial conditions.

D efinition 4.1. (3D consistency) A 2 D system is called 3D consistent if it can be imposed on all two-dimensional faces of an elementary cube of Z3.

4.1. Discrete Moutard nets in quadrics

129

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 di­ m ensions) Any 3D consistent discrete 2 D 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. □ Theorem 4.3. (3D consistency o f 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 equa­ tion. Indeed, after finding / 12, f 23 and / 13, one can construct /123 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 ad­ jacent to / 123. Therefore, these quadrilaterals satisfy our 2 D system. □ To formulate the next important property of T-nets in quadrics often used in the sequel, the following definition will be convenient.

D efinition 4.4. (Labelling of edges) A system of functions ai defined on the edges of Zm parallel to the coordinate axes *Bi is called a labelling of edges if these functions satisfy

(4.2)

TiOij — Otj,

i 7^ J,

4. Special Classes of Discrete Surfaces

130

i.e., if in every elementary square the opposite edges carry equal values of the corresponding a *.

If one assigns the value of oti on the edge (a, u + ei) to the lattice point u E Zm, then the labelling property is expressed as a % — ai(ui) for i — 1 , . . . , rn. To determine a labelling, one can prescribe it on the coordinate axes Bi. T h eorem 4.5. (L abelling p rop erty for T -n ets in quadrics) For a T-net f : Zm —> Q, the functions (4.3)

ai = ( f, Ti f ),

defined on the edges of 7*m parallel to Bi, have the labelling property (4.2).

P roof. It follows directly from (4.1) that ( Ti T j f , T j f ) = { T i f , f ) ,

(Ti Tj f, T i f ) =

(Tjf, f ) ,

which is equivalent to (4.2).

□

With the notation (4.3), the expression in (4.1) for the coefficients of the discrete Moutard equations takes the form , Oi•i CV . v

o

(4-4)

a ij =

1

7'

7\ *

^0 - i n f , Tjf)

4.2. D isc r e te K -n ets 4.2.1. N o tio n o f a d iscrete K -n et. In discretizing K-surfaces and their transformations, we take as a starting point the characterization given in Theorem 1.24. D efin ition 4.6. (D iscrete K -n et) A discrete A-net f : Zm —> M3 is called an m-dimensional discrete K-net if for any elementary quadrilateral i f , nf , Ti Tj f , Tj f ) , (4.5)

\TiT jf - Tjf \ = \ n f - f\

and

\ Ti Tj f - Ti f \ = \ T j f - f \ ;

in other words , if the functions Pi = \5if\, defined on the edges parallel to the coordinate axes Bi for i — 1 , . . . , m, have the labelling property (depend on Ui only).

The property (4.5) of a net / is known as the Chebyshev property , so a quadrilateral { f , u f , TiTjf, Tj f) satisfying (4.5) can be called a Chebyshev quadrilateral. Thus, a Chebyshev quadrilateral can be considered as a par­ allelogram bent in space along one of its diagonals. L em m a 4.7. (C h eb y sh ev quadrilateral) A Chebyshev quadrilateral is symmetric under the 180° rotation about the line through the midpoints of its diagonals.

4.2. Discrete K-nets

131

F igure 4.3. A discrete K-surface

Proof. Let 0 \ and 0 2 denote the midpoints of the diagonals [/, TiTjf] and [Tif, 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 congru­ ent triangles A ( /, Ti f, TiTjf) and A (/, Tj f , n T j f ), the point 0 \ is equidistant from Tif and T j f , and therefore belongs to the plane through 0 2 orthogonal to [ nf, Tjf]. Hence, the line (0 \ 0 2 ) is also orthogonal to [Tif, Tjf]. For similar reasons, this line is orthogonal to the second diagonal as well. □

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 : Zm —►M3 of a discrete K-net f : Zm —> M3 takes values, possibly upon a black-white rescaling, in some sphere S 2 C K3, thus being proportional to the Gauss map.

4. Special Classes of Discrete Surfaces

132

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

(4.6)

cos a; = (Tin, n)

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

(4.7)

ct ■

7 i = \8 in\ = 2 s i n y

and

fa = \Sif\ = |sina*|.

Proof. The definition of K-nets is equivalent to the following conditions for the Lelieuvre normals: ITiTjU X TjU I = ITin X n|,

|TiTjU X Tin I = ITjU X n|.

Because of the symmetry formulated in Lemma 4.7 (which clearly yields the rotational symmetry also for the directions of normal vectors), we derive: \TiTjn\ ' \tju \ = \Tin\ • \n\,

\TiTjn\ • fau\ = \Tjn\ • \n\.

As a consequence, |TiTjn| = \n\,

\Tifi\ = \Tjn\.

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 a T-net n in the unit sphere §2, described by the equations ( a q\

(4.8)

( \ TiTjU - n = aij(Tjn - Tin),

(n, T{n —T j n ) = --------------. 1 - {TiU, TjU)

Due to Theorem 4.5, the edge functions cosa^ = f an, n) depend on Ui only, and therefore — \8 i n \2 = 2(1 —cosc^) = 4sin 2 (a;/2) also depend only on Ui. Define the discrete A-net / : Zm —►R 3 by (2.62). Then Pi ~ \ $i f \2 = 1 — f a n 5ti ) 2 = 1 —cos2 ai = sin2 a^,

which proves that (4.5) is fulfilled.

□

According to Theorem 4.8, the discrete K-nets / (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 (Ka ) the values of n on the coordinate axes 3 i for 1 < i < m, i.e., m discrete curves n \ ^ in § 2 through a common point n( 0 ).

4.2. Discrete K-nets

133

4.2.2. Backlund transform ation. D efinition 4.9. (D iscrete Backlund transform ation) Two discrete Knets / , / + : Z771 —>R 3 with corresponding edges of equal length, \Tif+ - f + \ = \ n f - f \ ,

i = 1 , . . . , m,

are related by a Backlund transformation if they are related by a Weingarten transformation and the distance | / + —f\ is constant, i.e., does not depend on u G Zm. The net / + is called a Backlund transform of f .

Comparing this with Definition 4.6, we see that the net F : Zm x { 0 , 1 } —> R 3 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 /+ of a given m-dimensional discrete K-net / , or, equivalently, a Moutard transform n+ of the Gauss map n, one can prescribe the following data: (Ba ) the value of n+ (0) G §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. (Perm utability of discrete Backlund transforma­ tions) Let f be a discrete K-net, and let f ^ and f ® be two of its Backlund transforms. Then there exists a unique discrete K-net f ( 12^ which is simul­ taneously a Backlund transform of f ^ and of f^2\ The points of the fourth surface f ^ lie in the intersection of the tangent planes to f ^ and to / ( 2) at the corresponding points, and are uniquely defined by the properties 1/ ( 12) _ / ( 1)| = 1/ ( 2) _ / | and | / d 2) _ / ( 2>| = |/(D _ / | , or, in terms of the Gauss maps, (n^^n^12^) — ( n , n ^ ) and (n^2\ n ^ ) = (n ,n ^ ) . The four Gauss maps are related by the discrete Moutard equation with the minus sign, that is, nS12^ —n is parallel to n ^ —n^2\

4.2.3. H irota equation. For a convenient analytic description of discrete K-nets and their Gauss maps, we will use the following matrix formalism. The space M3 can be identified with the Lie algebra su( 2 ), (4'9)

( x 2~ “ x,

" T r ‘) £ ! “(2)

“

( * i.* 2 , « ) T £ R 3.

The vector product in R 3 and the matrix commutator in su( 2 ) correspond as follows: [x,y] = 2 x x y . This isomorphism makes it unnecessary to distinguish between vectors in R 3 and matrices in su(2). In other words, we use the following basis of the

4. Special Classes of Discrete Surfaces

134

linear space su( 2 ):

o') = -"'■