A parallel transport frame field approach to soliton surfaces associated with the Betchov–Da Rios equation in four space

Abstract

UDC 514.752, 514.748

By  using the parallel transport frame field, we examine the geometric properties of a soliton surface $\Psi=\Psi(s,t)$ associated with the Betchov–Da Rios equation in four-dimensional Euclidean space. We obtain derivative formulas for the parallel transport frame field of a unit-speed $s$-parameter curve $\Psi=\Psi(s,t),$ for all $t.$ We deduce  two basic geometric invariants of the soliton surface, $k$ and $h,$ and some other important invariants, such as Gaussian curvature, mean curvature vector, and Gaussian torsion. With the aid of these, we prove a set of theorems that describe the conditions under which the soliton surface is flat, minimal, semiumbilic, or Wintgen ideal (superconformal) by using these surface invariants. In addition, we present a theorem that characterizes the curvature ellipse of the Betchov–Da Rios soliton surface with respect to the parallel transport frame field in $E^{4}.$ Finally, we construct an example of a Betchov–Da Rios soliton surface, obtain its geometric invariants, and show its projections into the three-dimensional space to illustrate our theoretical results.