2 1. Classical Differential Geometry context, and the nonuniqueness of the fourth net in these theorems reflects this. The natural setting for permutability is given in the second part, where the permutability is associated with an octuple of nets, depicted as vertices of a combinatorial cube, so that the eighth net is uniquely determined by the other seven (Eisenhart hexahedron). Our discrete philosophy makes the origin of such permutability theorems quite transparent. A few remarks on notation. We denote independent variables of a net f : Rm RN by u = (u1, . . . , um) Rm, and we set ∂i = ∂/∂ui. All nets are supposed to be sufficiently smooth, so that all the required partial derivatives exist. We write Bi 1 ...is = u Rm : ui = 0 for i = i1, . . . , is for s-dimensional coordinate planes (coordinate axes, if s = 1). 1.1. Conjugate nets 1.1.1. Notion of conjugate nets. We always suppose that the dimension of the ambient space N 3. Definition 1.1. (Conjugate net) A map f : Rm RN is called an m-dimensional conjugate net in RN if at every u Rm and for all pairs 1 i = j m we have ∂i∂jf span(∂if, ∂jf). Two-dimensional nets (m = 2) are nothing but parametrized surfaces. A parametrization of a surface in the three-space (m = 2, N = 3) is a conjugate net if its second fundamental form is diagonal. For a generic surface in the three-space, infinitely many such parametrizations can be found. A generic surface in the four-space carries an essentially unique conjugate net (uniqueness is understood here up to reparametrizations of coordinate lines). In higher-dimensional spaces such a parametrization does not need to exist at all (that is, only special surfaces of codimension 2 support conjugate nets). From Definition 1.1 it follows that the conjugate nets are described by the (linear) differential equations (1.1) ∂i∂jf = cji∂if + cij∂jf, i = j, with some functions cij : Rm R. Compatibility of these equations, i.e. the requirement ∂i(∂j∂kf) = ∂j(∂i∂kf), is expressed by the following system of (nonlinear) differential equations: (1.2) ∂icjk = cijcjk + cjicik cjkcik, i = j = k = i. Note that the latter equations for the coefficients cij do not contain f any- more. The system (1.1), (1.2) is hyperbolic (see Chapter 5) the following
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