On completeness of the Ribaucour transformations for triply orthogonal curvilinear coordinate systems in R^3

TL;DR

This paper proves the local density of solutions for a (2+1)-dimensional integrable system describing triply orthogonal curvilinear coordinates in R^3, using Ribaucour transformations, extending classical differential geometry methods.

Contribution

It establishes the completeness of Ribaucour transformations in generating solutions for the integrable system of triply orthogonal coordinates in R^3.

Findings

Proves local density of solutions via Ribaucour transformations.

Extends classical differential geometry to a (2+1)-dimensional integrable system.

Provides a method to generate all solutions from initial data.

Abstract

In this paper we solve positively the problem of (local) density of solutions of the (2+1)-dimentional integrable system describing triply orthogonal curvilinear coordinates in R^3 (a (2+1)-dimensional generalization of the 3-wave system) obtainable from a given initial solution with consecutive B\"acklund transformations (called Ribaucour transformations in classical differential geometry) in the space of all solutions of the system in question.