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 unchen, 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. for differential geometers for those interested in integrable systems 6 2,3,4,9 7,8 1 5 graduate course Figure 0.3. A suggestion for the focus on chapters, depending on the readers background. Acknowledgements. 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 Schr¨ oder, Boris Springborn, Sergey Tsarev, Alexander Veselov, unter 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
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