MDPI - Publisher of Open Access Journals

21 pages, 1506 KB

Open AccessArticle

A Unified Rotation-Minimizing Darboux Framework for Curves and Relativistic Ruled Surfaces in Minkowski Three-Space

by Mona Bin-Asfour, Ghaliah Alhamzi, Emad Solouma and Sayed Saber

Axioms 2026, 15(3), 207; https://doi.org/10.3390/axioms15030207 - 11 Mar 2026

Cited by 1 | Viewed by 355

We propose a comprehensive rotation-minimizing (RM) Darboux framework for the study of curve theory and relativistic ruled surfaces in Minkowski three-space

E_{1}^{3}

. The construction merges the adaptability of the classical Darboux frame to surface geometry with the reduced rotational behavior [...] Read more.

We propose a comprehensive rotation-minimizing (RM) Darboux framework for the study of curve theory and relativistic ruled surfaces in Minkowski three-space

E_{1}^{3}

. The construction merges the adaptability of the classical Darboux frame to surface geometry with the reduced rotational behavior characteristic of RM frames, yielding a natural geometric description of curves in a Lorentzian environment. For unit speed non-null curves, the governing equations of the RM Darboux frame are derived, and precise connections between the RM curvature functions and the classical Frenet and Darboux invariants are obtained, thereby elucidating the geometric significance of RM curvatures in Lorentzian geometry. Within this setting, multiple classes of ruled surfaces are generated using RM Darboux frame vector fields. Necessary and sufficient conditions for developability, minimality, and flatness are formulated exclusively in terms of RM curvature quantities. The role of the causal character of the generating curve is analyzed in detail, revealing distinct geometric behaviors for space-like and time-like cases. These findings indicate that the RM Darboux framework constitutes a flexible and effective approach for modeling curve-induced surface geometries in Minkowski space, with potential relevance to relativistic kinematics, world sheet constructions, and geometric problems arising in mathematical physics. Full article

(This article belongs to the Special Issue Theory and Applications: Differential Geometry)

► Show Figures

Figure 1

16 pages, 722 KB

Open AccessArticle

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

E_{1}^{3}

by Emad Solouma, Sayed Saber, Marin Marin and Haci Mehmet Baskonus

Mathematics 2025, 13(21), 3420; https://doi.org/10.3390/math13213420 - 27 Oct 2025

Cited by 3 | Viewed by 600

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Analysis on Differentiable Manifolds)

► Show Figures

Figure 1

14 pages, 710 KB

Open AccessArticle

Exploring Harmonic Evolute Geometries Derived from Tubular Surfaces in Minkowski 3-Space Using the RM Darboux Frame

by Emad Solouma, Sayed Saber and Haci Mehmet Baskonus

Mathematics 2025, 13(15), 2329; https://doi.org/10.3390/math13152329 - 22 Jul 2025

Cited by 5 | Viewed by 816

Abstract

In this study, We explore for Minkowski 3-space

E_{1}^{3}

In this study, We explore for Minkowski 3-space

E_{1}^{3}

harmonic surfaces’ geometric features by employing a common tangent vector field along a curve situated on the surface. Our analysis is grounded in the rotation minimizing (RM) Darboux frame, which offers a robust alternative to the classical Frenet frame particularly valuable in the Lorentzian setting, where singularities frequently arise. The RM Darboux frame, tailored to curves lying on surfaces, enables the expression of fundamental invariants such as geodesic curvature, normal curvature, and geodesic torsion. We derive specific conditions that characterize harmonic surfaces based on these invariants. We also clarify the connection between the components of the RM Darboux frame and thesurface’s mean curvature vector. This formulation provides fresh perspectives on the classification and intrinsic structure of harmonic surfaces within Minkowski geometry. To support our findings, we present several illustrative examples that demonstrate the applicability and strength of the RM Darboux approach in Lorentzian differential geometry. Full article

(This article belongs to the Special Issue Differential Geometric Structures and Their Applications)

► Show Figures

Figure 1

Search Results (3)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (3)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI