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Journals / International Electronic Journal of Geometry / 2026 / Volume: 19 Issue: 1

Timelike CGC Translation Surfaces Generated by Timelike Curves in Minkowski $3-$Space

Ahmad Ali(King Abdulaziz University, Faculty of Science)0000-0003-3747-4990
Pages
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Publication Date
2026-04-22
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

These results provide a complete local classification of timelike translation surfaces with constant Gaussian curvature in Minkowski 3-space and highlight a strong rigidity phenomenon in the timelike Lorentzian setting.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we study timelike translation surfaces with constant Gaussian curvature \textbf{(CGC)} in the three-dimensional Minkowski space.
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Key Findings & Data:Such surfaces are generated as the sum of two timelike space curves and naturally arise in the context of Lorentzian surface geometry.
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Conclusion & Impact:These results provide a complete local classification of timelike translation surfaces with constant Gaussian curvature in Minkowski 3-space and highlight a strong rigidity phenomenon in the timelike Lorentzian setting.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Timelike translation surfaces · constant Gaussian curvature · development of industries

Abstract

In this paper, we study timelike translation surfaces with constant Gaussian curvature \textbf{(CGC)} in the three-dimensional Minkowski space. Such surfaces are generated as the sum of two timelike space curves and naturally arise in the context of Lorentzian surface geometry. By employing a detailed analytic and geometric approach, we prove that any timelike translation surface with constant Gaussian curvature must be flat. As a consequence, we show that the only timelike translation surfaces satisfying this curvature condition are cylindrical surfaces. Furthermore, we establish that timelike translation surfaces with constant Gaussian curvature cannot be minimal everywhere. As a geometric characterization of the generating curves, we prove that one of the curves must necessarily be either a timelike hyperbola or a straight line. These results provide a complete local classification of timelike translation surfaces with constant Gaussian curvature in Minkowski 3-space and highlight a strong rigidity phenomenon in the timelike Lorentzian setting.

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Cite This Article

Ali, A. (2026). Timelike CGC Translation Surfaces Generated by Timelike Curves in Minkowski $3-$Space. International Electronic Journal of Geometry, Volume: 19 Issue: 1. https://doi.org/10.36890/iejg.1812842

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