Discrete Analogs of the Darboux–Egorov Metrics

A discrete analog of the Darboux–Egorov metrics is constructed and the geometry of the corresponding lattices in a Euclidean space is shown to be described by the set of functions $h_i^{\pm}(u)$, $u\in\mathbb Z^n$. A discrete analog of the Lamé equations is determined, and it is shown that these equations are necessary and sufficient for the solutions to this analog to be the rotation coefficients of the Darboux–Egorov lattice up to a gauge transformation. A scheme for the construction of explicit solutions to the discrete Lamé equations in terms of the Riemann $\theta$-functions is presented.