A Geometric Application of Soliton Surfaces associated with the Betchov-Da Rios Equation using an Extended Darboux Frame Field in $E^{4}$

TL;DR

This paper explores the geometry of soliton surfaces linked to the Betchov-Da Rios equation using an extended Darboux frame in four-dimensional space, deriving invariants and characterizations like flatness and minimality.

Contribution

It introduces a novel geometric analysis of Betchov-Da Rios soliton surfaces via an extended Darboux frame, including curvature properties and special surface classifications.

Findings

Derived derivative formulas for the extended Darboux frame.

Computed geometric invariants such as curvature and torsion.

Characterized surface properties like flatness and minimality.

Abstract

In this paper, for a soliton surface $\Omega=\Omega(u,v)$ associated with the Betchov-Da Rios equation, we obtain the derivative formulas of an extended Darboux frame field of a unit speed curve $u$-parameter curve $\Omega=\Omega(u,v)$ for all $v$. Also, we get the geometric invariants $k$ and $h$ of the soliton surface $\Omega=\Omega(u,v)$ and we obtain the Gaussian curvature, mean curvature vector and Gaussian torsion of $\Omega$. We give some important geometric characterizations such as flatness, minimality and semi-umbilicaly with the aid of these invariants. Additionally, we study the curvature ellipse of the Betchov-Da Rios soliton surface and Wintgen ideal (superconformal) Betchov-Da Rios soliton surface with respect to an extended Darboux frame field. Finally, we construct an application for the Betchov-Da Rios soliton surface with the aid of an extended Darboux frame field.