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

• Datum: 2026
• Hauptverfasser: Altın, Mustafa, Kazan, Ahmet
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2026
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/9337
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

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.