Discrete Differential Geometry. Integrable Structure

Discrete differential geometry is situated between differential geometry and discrete geometry. Its aim is not only to study smooth objects using discrete methods but also to look, first of all, for discrete analogues of notions and results of differential geometry. Having a discrete object, one naturally tries to pass to a limit of refinement of this object, and to find a return to differential geometry. But what is much more interesting is the influence of discrete differential geometry upon differential geometry itself. Many results of differential geometry can be much better understood, and their proofs can be simplified, when using ideas of discrete differential geometry. The role of integrable systems in differential geometry is well-known. This connection is then even clearer, relating discrete differential geometry and the theory of discrete integrable systems. Discrete differential geometry is a very young branch of geometry and this book covers many results from the last decade. It can serve as a very good introduction into contemporary research and it seems to be the first book devoted to the topic. The authors mention that they wrote this textbook for three categories of readers. The first category comprises graduate students. (The book has already been used for a one semester graduate course. It is interesting that students are not necessarily assumed to have some knowledge of differential geometry.) The second category comprises specialists in geometry and mathematical physics. The third category comprises specialists in geometry processing, computer graphics, architectural design, numerical simulations and animations. The book is well and clearly written. At the end of every chapter are exercises (solutions of some of them can be found in the appendix) and bibliographical notes.

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