4 1. Classical Differential Geometry Rotation coefficients satisfy a closed system of differential equations, which follow from (1.2) upon substitution (1.6): (1.10) ∂iγkj = γkiγij , i = j = k = i. Eqs. (1.10), known as the Darboux system, can be regarded as compatibility conditions of the linear differential equations (1.8). Observe an important difference between the two descriptions of conju- gate nets: while the functions cij describe the local geometry of a net, this is not the case for the rotation coefficients γij. Indeed, to define the latter, one needs first to find gi as solutions of differential equations (1.4). 1.1.3. Transformations of conjugate nets. The most general class of transformations of conjugate nets was introduced by Jonas and Eisenhart. Definition 1.2. (Fundamental transformation) Two m-dimensional conjugate nets f, f + : Rm RN are said to be related by a fundamental transformation (F-transformation) if at every point u Rm of the domain and for each 1 i m the three vectors ∂if, ∂if + and δf = f + f are coplanar. The net f + is called an F-transform of the net f. This definition yields that F-transformations are described by the fol- lowing (linear) differential equations: (1.11) ∂if + = ai∂if + bi(f + f). Of course, the functions ai, bi : Rm R must satisfy (nonlinear) differential equations, which express the compatibility of (1.11) with (1.1): ∂iaj = (ai aj)cij + bi(aj 1), (1.12) ∂ibj = c+bj ij + c+bi ji bjbi, (1.13) ajcij + = aicij + bi(aj 1). (1.14) The following data determine an F-transform f + of a given conjugate net f uniquely: (F1) a point f + (0) (F2) the values of ai, bi on the coordinate axes Bi for 1 i m, i.e., 2m smooth real-valued functions ai B i , bi B i of one variable. Observe a remarkable conceptual similarity between Definitions 1.1 and 1.2. Indeed, one can interpret the condition of Definition 1.1 as planarity of infinitesimal quadrilaterals (f(u), f(u + i ei), f(u + i ei + j ej), f(u + j ej)), while the condition of Definition 1.2 can be interpreted as planarity of in- finitesimally narrow quadrilaterals (f(u), f(u + i ei), f + (u + i ei), f + (u)).
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