28 1. Classical Differential Geometry 1.9. Bibliographical notes Achievements of the classical period of the differential geometry of surfaces and their transformations are documented in the treatises by Darboux (1910, 1914–27), Bianchi (1923) and Eisenhart (1909, 1923). These books cover huge material and are indispensable sources for a detailed treatment of the special geometries of this chapter. Section 1.1: Conjugate nets and their transformations. The clas- sical description of multidimensional conjugate nets, given in Section 1.1.2, can be found in Darboux (1914–27). The fundamental transformations of conjugate nets, given in Section 1.1.4, as well as the permutability theo- rem for F-transformations (part one of Theorem 1.3) are due to Eisenhart (1923) and Jonas (1915). The first instance of the Eisenhart hexahedron (part two of Theorem 1.3) we were able to localize is the “extended theorem of permutability” for conjugate nets in Eisenhart (1923, §24). In the modern literature on integrable systems, the Darboux system (1.10) is known as the n-wave equation see Novikov-Manakov-Pitaevskii- Zakharov (1984). Section 1.2: Koenigs and Moutard nets and their transformations. Classically, Koenigs nets were known as nets with equal Laplace invariants. Their geometry was studied, among others, by Koenigs (1891, 1892a,b), Darboux (1914–27), Tzitz´ eica (1924). An exhaustive treatment of nets with dependent Laplace invariants is in Bol (1967). For the classical formulation of the Moutard transformation see, e.g., Moutard (1878), and for its geometric interpretation as Koenigs transfor- mation see Koenigs (1891). The two- and three-dimensional permutability theorems for Koenigs transformations are due to Eisenhart (1923). In terms of Moutard transformations this was formulated in Ganzha-Tsarev (1996) and Nimmo-Schief (1997). Section 1.3: Asymptotic nets and their transformations. For the description of asymptotic nets, W-congruences and their Weingarten trans- formations in terms of Lelieuvre normals see the classical books by Dar- boux (1914–27), Bianchi (1923) and Eisenhart (1923) or, for example, Lane (1942). The two-dimensional permutability of Weingarten transformations (part one of Theorem 1.15) is due to Bianchi (1923). For the projective interpretation of the Lelieuvre normals, see Konopel- chenko-Pinkall (2000). A survey on integrable systems in projective differ- ential geometry based on asymptotic line parametrization is in Ferapontov (2000a). Section 1.4: Orthogonal nets and their transformations. A funda- mental monograph on orthogonal coordinate systems is Darboux (1910). A
Previous Page Next Page