Discrete differential geometry. Integrable structure. (English) Zbl 1158.53001

Graduate Studies in Mathematics 98. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4700-8/hbk). xxiv, 404 p. (2008).
Discrete differential geometry aims at establishing discrete equivalents of notions and methods of smooth surface theory. A straightforward way to discretize differential geometry would be to take the analytic description in terms of differential equations and to apply standard methods of numerical analysis, but discrete differential geometry is concerned not with the mere discretization of the equations but with the whole theory. 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. The aim of this book is to provide a systematic presentation of current achievements in this field. It gives a comprehensive presentation of the results of discrete differential geometry of parametrized surfaces and coordinate systems along with its relation to integrable systems.
The book consists of 9 chapters. Chapter 1 is devoted to an overview of some classical results of surface theory without proofs. In Chapter 2 the authors define and investigate discrete analogs of the most fundamental objects of projective differential geometry. In Chapter 3, following Klein’s Erlangen Program, they show that the nets and congruences defined in Chapter 2 can be restricted to quadrics. Imposing simultaneously several constraints on conjugate nets, one comes to special classes of surfaces, which are the subject of Chapter 4. In Chapter 5 the authors 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. It is shown that the discrete nets in Chapters 2, 3 and 4 approximate the corresponding geometries in Chapter 1 and simultaneously their transformations. In Chapter 6 the concept of multidimensional consistency is formulated as a defining principle of integrability. Basic features of integrable systems such as the zero curvature representation and Bäcklund-Darboux transformations as well as a complete list of \(2\)-dimensional integrable systems are derived from the consistency principle. These ideas are applied to discrete complex analysis in Chapters 7 and 8, where the reader can find, e.g., Laplace operators on graphs and discrete harmonic and holomorphic functions. Chapter 9 gives a brief introduction to projective geometry and the geometries of Lie, Möbius, Laguerre and Plücker for the reader’s convenience.
The intended readership of this book is threefold. It can be used as a textbook on discrete differential geometry. It is also written for specialists in geometry and mathematical physics. The third group for which this book is intended are specialists of such areas as geometry processing, computer graphics, architectural design, numerical simulations and animation. The book will surely be welcome by all these three groups.


53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53A04 Curves in Euclidean and related spaces
53A05 Surfaces in Euclidean and related spaces
52B70 Polyhedral manifolds