Normal curves using the Bishop frame in Minkowski 3-space

Abstract

The geometric characterization of curves with zero curvature points presents inherent limitations when employing the classical Frenet frame. This research investigates the mathematical properties of normal curves in $E^3_1$ (Minkowski 3-space) through the application of the Bishop frame. The study systematically analyzes the geometric and topological properties of both spacelike and timelike normal curves, establishing rigorous mathematical conditions necessary and sufficient for their classification within the Bishop frame formalism. We derive fundamental theorems characterizing these curves and examine their differential geometric invariants. The investigation extends to the analysis of curvature relationships, parallel transport properties, and the behavior of normal curves under the Bishop frame parametrization. Our findings contribute to the theoretical framework of curve theory in pseudo-Riemannian geometry, particularly in spaces with indefinite metrics, demonstrating the robustness of the Bishop frame approach for characterizing curves where traditional Frenet analysis becomes singular.