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 Z2. 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). Structure of this book. The structure of this book follows the logic of this Introduction. We start in Chapter 1 with an overview of some classical results from surface theory, focusing on transformations of surfaces. The brief presentation in this chapter is oriented towards the specialists already familiar with the differential geometry of surfaces. The geometries 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.

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