Helicoidal flat surfaces in the 3-sphere

TL;DR

This paper classifies helicoidal flat surfaces in the 3-sphere using fundamental forms and angle functions, employing Bianchi-Spivak construction and representation techniques.

Contribution

It provides a complete classification of helicoidal flat surfaces in -sphere, linking their geometry to fundamental forms and angle functions.

Findings

Classification based on fundamental forms and angle functions.

Use of Bianchi-Spivak construction for flat surfaces.

Representation of constant angle surfaces in -sphere.

Abstract

In this paper, helicoidal flat surfaces in the $3$-dimensional sphere $\mathbb{S}^3$ are considered. A complete classification of such surfaces is given in terms of their first and second fundamental forms and by linear solutions of the corresponding angle function. The classification is obtained by using the Bianchi-Spivak construction for flat surfaces and a representation for constant angle surfaces in $\mathbb{S}^3$.