Timelike line congruences via surface theory in Minkowski 3-space

Abstract

Line congruences are crucial in classical geometry, particularly in relating one surface to another through families of lines. These correspondences are most valuable when they preserve key geometric features of the original surface. A line congruence, understood as a two-parameter family of lines, can itself be viewed as a surface within the space of lines. This paper focuses on timelike line congruences, using the Study map to explore their geometry within Minkowski 3-space. By interpreting a timelike line congruence as a region on the hyperbolic dual unit sphere, we connect surface theory with the geometry of these congruences. We introduce the first and second fundamental forms to establish conditions for when a timelike surface is developable and to study its differential properties. Applying Blaschke’s moving frame technique, we derive curvature formulas and provide Minkowski analogs of classical results for ruled surfaces within the congruence. Specifically, we extend known Euclidean results, including a Minkowski version of Plücker’s conoid. We also derive Dupin’s indicatrix for timelike line congruences, offering a classification based on curvature invariants. In addition, we construct the Liouville formula within this framework and discuss its geometric implications for closed timelike ruled surfaces contained in a timelike line congruence. To highlight the practical outcomes of our approach, we provide several illustrative models.