Geometric Invariants and Evolution of RM Hasimoto Surfaces in Minkowski 3-Space E 1 3

Emad Solouma (), Sayed Saber, Marin Marin and Haci Mehmet Baskonus
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Emad Solouma: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
Sayed Saber: Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha 65779, Saudi Arabia
Marin Marin: Department of Mathematics and Computer Science, Transilvania, University of Brasov, 500036 Brasov, Romania
Haci Mehmet Baskonus: Department of Mathematics and Science Education, Harran University, Sanliurfa 63050, Turkey

Mathematics, 2025, vol. 13, issue 21, 1-16

Abstract: Research on surfaces generated by curves plays a central role in linking differential geometry with physical applications, especially following Hasimoto’s transformation and the development of Hasimoto-inspired surface models. In this work, we introduce a new class of such surfaces, referred to as RM Hasimoto surfaces, constructed by employing the rotation-minimizing (RM) Darboux frame along both timelike and spacelike curves in Minkowski 3-space E 1 3 . In contrast to the classical Hasimoto surfaces defined via the Frenet or standard Darboux frames, the RM approach eliminates torsional difficulties and reduces redundant rotational effects. This leads to more straightforward expressions for the first and second fundamental forms, as well as for the Gaussian and mean curvatures, and facilitates a clear classification of key parameter curves. Furthermore, we establish the associated evolution equations, analyze the resulting geometric invariants, and present explicit examples based on timelike and spacelike generating curves. The findings show that adopting the RM Darboux frame provides greater transparency in Lorentzian surface geometry, yielding sharper characterizations and offering new perspectives on relativistic vortex filaments, magnetic field structures, and soliton behavior. Thus, the RM framework opens a promising direction for both theoretical studies and practical applications of surface geometry in Minkowski space.

Keywords: Minkowski geometry; Hasimoto surface; vortex filament equation; RM Darboux frame (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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