1.9. Bibliographical notes 29 detailed study of conjugate nets in arbitrary quadric is in Tzitz´ eica (1924). The surface theory in the framework of M¨ obius, Laguerre and Lie geometry was developed by Blaschke (1929). Permutability theorem for Ribaucour transformations (part one of Theorem 1.20) is due to Bianchi (1923). Eisen- hart type permutability theorem for Ribaucour transformations (part two of Theorem 1.20) was found in Ganzha-Tsarev (1996). Orthogonal coordinate systems from the viewpoint of the theory of inte- grable systems were investigated in Zakharov (1998). Algebro-geometric or- thogonal coordinate systems were constructed by Krichever (1997). A survey of integrable systems in Lie geometry is given in Ferapontov (2000b). Spinor frames for orthogonal nets were introduced in Bobenko-Hertrich-Jeromin (2001). A modern textbook on the M¨ obius surface theory including the theory of orthogonal nets is Hertrich-Jeromin (2003). A Lie-geometric de- scription of Ribaucour transformations is given in Burstall-Hertrich-Jeromin (2006). Section 1.5: Principally parametrized sphere congruences. This topic was rather popular in the classical literature see, e.g., Darboux (1914– 27), Coolidge (1916), Eisenhart (1923), but is not well presented in the modern literature. Section 1.6: K-surfaces and their transformations. Surfaces with constant negative Gaussian curvature (pseudospherical surfaces) played an important role in the theory of surfaces and their transformations. The sine- Gordon equation is the oldest integrable equation. It goes back at least to Bour (1862) and Bonnet (1867). The B¨ acklund transformation was found by B¨ acklund (1884) the permutability theorem is due to Bianchi (1892). For a modern presentation, generalizations and description in terms of loop groups see Rogers-Schief (2002) and Terng-Uhlenbeck (2000). Section 1.7: Isothermic surfaces and their transformations. The classical period of the theory of isothermic surfaces is summarized in Dar- boux (1914–27) and Bianchi (1923). In particular, the Darboux transfor- mations for isothermic surfaces as well as the characterization of isothermic surfaces as Moutard nets in the light cone from Theorem 1.32 are due to Darboux. The permutability of Darboux transformations was established by Bianchi. The constant cross-ratio property in the permutability theo- rem was shown by Demoulin (1910). Moutard nets in general quadrics were investigated in Tzitz´ eica (1924). Isothermic surfaces played an important role in the development of the modern integrable differential geometry. A relation of the classical the- ory of isothermic surfaces to the theory of integrable systems was found in Cie´ sli´ nski-Goldstein-Sym (1995). A spinor zero-curvature representation for isothermic surfaces was found in Bobenko-Pinkall (1996b). A relation of

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