ИСТИНА

We are developing the idea of using an algebraic geometrical approach to classical differential geometry problems. This idea was presented in works of I. M. Krichever, A. E. Mironov and I. A. Taimanov. They described orthogonal curvilinear coordinate systems in terms of theta-functions of algebraic curves. It was also shown how this construction could be applied to the associativity equations and Frobenius manifolds. In this work we present results on Ribaucour transformations of orthogonal nets. Ribaucour transformations naturally occured in the field of discrete integrable systems. We introduce an algebraic geometrical construction to obtain any number of smooth orthogonal nets that are Ribaucour transformations of the initial orthogonal net. This construction also allows us to illustrate the permutability property. Starting with algebraic geometric data we obtain a family of orthogonal nets and prove that some pairs of them are Ribaucour pairs. We show how to get all the formulae expressed in terms of elementary functions choosing special set of algebraic geometrical data.