Stationary-Angle Conditions and Bertrand Offsets in Timelike-Ruled Surfaces

In this study, we introduce the concept of stationary-angle timelike-ruled surfaces and examine their geometric properties, particularly in relation to their Bertrand offsets. A timelike-ruled surface is generated by the motion of a straight ruling along a striction curve, and its structure is analyzed using the Blaschke and Darboux frames. We derive key geometric invariants, including spherical curvature, geodesic curvature, normal curvature, and geodesic torsion. Additionally, we establish the conditions under which the striction curve of a timelike-ruled surface behaves as a geodesic, an asymptotic curve, or a curvature line. Special cases, such as timelike-tangential developables and timelike-cones, are also discussed. Using curvature-axis analysis, we develop a higher-order contact framework to better understand the behavior of these surfaces. Finally, we investigate the Bertrand offsets of stationary-angle timelike-ruled surfaces, proving that they preserve a stationary angle between their rulings and determining the necessary conditions for their existence. This work enhances the understanding of differential geometry in Lorentzian spaces and provides new insights into ruled surfaces in Minkowski space.MSC2020 Classification: 53A04, 53A05, 53A17