## Algebraic-geometric $$n$$-orthogonal curvilinear coordinate systems and solutions of the associativity equations.(English. Russian original)Zbl 1004.37052

Funct. Anal. Appl. 31, No. 1, 25-39 (1997); translation from Funkts. Anal. Prilozh. 31, No. 1, 32-50 (1997).
From the introduction: The problem of constructing $$n$$-orthogonal curvilinear coordinate systems, or flat diagonal metrics $ds^2=\sum^n_{i=1} H^2_i(u) (du^i)^2, \quad u=(u^1,\dots, u^n),\tag{1}$ for more than a century since the famous work of Dupin and Binet published in 1810 was one of the most important problems of differential geometry. Treated as a classification problem, it was mainly solved in the beginning of the 20th century. The crucial contribution here was due to G. Darboux.
In the beginning of the 1980s, it was found that this classical problem has deep connections with the modern theory of integrable quasilinear hydrodynamic type systems in $$(1+1)$$-dimensions. This theory was proposed by B. Dubrovin and S. Novikov as a Hamiltonian theory of the averaged (Whitham) equations for periodic solutions of integrable soliton equations in $$(1+1)$$-dimensions. Later, it was noticed that the classification of Egorov metrics, i.e., flat diagonal metrics such that $\partial_j H^2_j= \partial_iH^2_j, \quad\partial_i =\partial/ \partial u^i,\tag{2}$ is equivalent to the classification problem for massive topological field theories. It was shown that locally the general solution of the Lamé equations $\partial_k \beta_{ij}= \beta_{ik}\beta_{kj}, \quad i\neq j\neq k, \tag{3}$
$\partial_i \beta_{ij}+ \partial_j \beta_{ji}+ \sum_{m\neq i,j} \beta_{mi} \beta_{mj}= 0, \quad i\neq j,\tag{4}$ for the rotation coefficients $\beta_{ij}= \partial_i H_j/H_i,\quad i\neq j,\tag{5}$ depends on $$n(n-1)/2$$ arbitrary functions of two variables. The system (1.3), (1.4) is equivalent to the vanishing of all a priori nontrivial components of the curvature tensor.
Quite recently, solutions of (3) and (4) have been constructed by V. Zakharov with the help of the “dressing procedure” within the framework of the inverse problem method. The main goal of this paper is not merely constructing finite-gap or algebraic-geometric solutions of the Lamé equations (3), (4) but proposing a scheme that simultaneously solves the complete system i.e., gives both the Lamé coefficients $$H_i$$ and the flat coordinates $$x^i(u)$$.
We use the method of Baker-Akhiezer functions, for constructing algebro-geometrical solutions to the Lamé equations. These solutions are explicitly expressed in terms of the theta functions of auxiliary algebraic curves.

### MSC:

 37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions 58D29 Moduli problems for topological structures 14H55 Riemann surfaces; Weierstrass points; gap sequences 35Q58 Other completely integrable PDE (MSC2000)
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### References:

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