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

Emad Solouma (), Sayed Saber 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
Haci Mehmet Baskonus: Department of Mathematics and Science Education, Harran University, Sanliurfa 63050, Turkey

Mathematics, 2025, vol. 13, issue 15, 1-14

Abstract: 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.

Keywords: Minkowski geometry; harmonic surface; tubular surface; RM Darboux frame (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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