Sequences of Levy Transformations and Multi-Wro\'nski Determinant Solutions of the Darboux System

TL;DR

This paper investigates Levy transformations applied to the Darboux system of conjugate nets, deriving explicit multi-Wronski determinant formulas for geometric quantities after multiple transformations.

Contribution

It introduces explicit multi-Wronski determinant formulas for the Darboux system's geometric data after sequences of Levy transformations, extending previous results.

Findings

Closed-form formulas for tangent vectors and coefficients in terms of multi-Wronski determinants

Demonstrates the effect of Levy transformations in multidimensional conjugate nets

Provides a systematic approach to generate solutions of the Darboux system

Abstract

Sequences of Levy transformations for the Darboux system of conjugates nets in multidimensions are studied. We show that after a suitable number of Levy transformations, with at least a Levy transformation in each direction, we get closed formulae in terms of multi-Wro\'nski determinants. These formulae are for the tangent vectors, Lam\`e coefficients, rotation coefficients and points of the surface.