New Flat surfaces in $S^3$

TL;DR

This paper introduces a method for constructing flat surfaces in the 3-sphere using Ribaucour transformations, resulting in new families of complete flat surfaces with explicit examples.

Contribution

It develops a novel approach to generate flat surfaces in $S^3$ via Ribaucour transformations, expanding the known classes of such surfaces.

Findings

Constructed new families of complete flat surfaces in $S^3$

Provided explicit examples of these surfaces

Demonstrated the parameter dependence of the surfaces

Abstract

n this paper, we consider a method of constructing flat surfaces based on Ribaucour transformations in the sphere 3-space. By applying the theory to the flat torus, we obtain a families of complete flat surfaces in $S^3$ which are determined by several parameters. we provide explicit examples.