Chapter 1 Classical Differential Geometry In this chapter we discuss some classical results of the differential geome- try of nets (parametrized surfaces and coordinate systems) in RN , mainly concentrated around the topics of transformations of nets and of their per- mutability properties. This classical area was very popular in the differential geometry of the 19th and of the first quarter of the 20th century, and is well documented in the fundamental treatises by Bianchi, Darboux, Eisenhart and others. Our presentation mainly follows these classical treatments, of course with modifications which reflect our present points of view. We do not trace back the exact origin of the concrete classical results: often enough this turns out to be a complicated task in the history of mathematics, which still waits for its competent investigation. For the classes of nets described by essentially two-dimensional systems (special classes of surfaces such as surfaces with a constant negative Gaussian curvature or isothermic surfaces), the permutability theorems, mainly due to Bianchi, are dealing with a quadruple of surfaces (depicted as vertices of a so-called Bianchi quadrilateral). Given three surfaces of such a quadruple, the fourth one is uniquely defined see Theorems 1.27 and 1.31. For the classes of nets described by essentially three-dimensional systems (conjugate nets Moutard nets asymptotic line parametrized surfaces or- thogonal nets, including curvature line parametrized surfaces), the situation is somewhat different. The corresponding permutability theorems (Theo- rems 1.3, 1.10, 1.15, and 1.20) consist of two parts. The first part of each theorem presents the traditional view and deals with Bianchi quadrilater- als. In our opinion, this is not the proper setting in the three-dimensional 1
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