2025/07/02 by Mustafa Altın, Altin, Mustafa, Ahmet Kazan +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2507.06248
openalex publication_date 2025/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the present paper, the geometric properties of a soliton surface Ψ=Ψ(s,t) associated with the Betchov-Da Rios (B-DR) equation using the parallel transport frame field in four-dimensional Euclidean space are examined. We obtain the derivative formulas for the parallel transport frame field of a unit-speed s-parameter curve Ψ=Ψ(s,t), for all t. We obtain the soliton surface's two basic geometric invariants, k and h, and some other important invariants such as Gaussian curvature, mean curvature vector and Gaussian torsion. With the aid of these, a set of theorems describing the conditions in which the soliton surface is flat, minimal, semi-umbilic or Wintgen ideal (superconformal) are proved using these surface invariants. Also, we give a theorem which characterizes the curvature ellipse of the B-DR soliton surface with respect to the parallel transport frame field in E4. Finally, we construct an example of a B-DR soliton surface, obtain its geometric invariants and showed its projections into three-dimensional space to demonstrate our theoretical understanding.