xxii Introduction According to Klein’s Erlangen Program, the classical geometries (Eu- clidean, spherical, hyperbolic, obius, Pl¨ ucker, 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 Pl¨ ucker 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 B¨acklund-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 holomor- phic 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, obius, Laguerre and Pl¨ ucker. We also include a number of classical incidence theorems rel- evant to discrete differential geometry. How to read this book. Different audiences (see the Preface) should read this book differently, as suggested in Figure 0.3. Namely, Chapter 1 on 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
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