Anti-torqued slant helices and Torqued Curves in Riemannian manifolds
Muhittin Evren Aydin, Adela Mihai, Cihan Özgür
TL;DR
This work defines and analyzes anti-torqued slant helices and torqued curves on Riemannian manifolds, using axis vector fields to constrain the principal normal via constant-angle conditions. The authors develop a Frenet-/frame-based differential framework for general-order curves, derive a trio of case-based classifications, and present a system of invariant-differential equations that governs such curves. They provide concrete 3D and Euclidean examples (e.g., logarithmic spirals, loxodromes, rectifying curves) to illustrate the theory and show how concircular and anti-torqued/torqued structures influence curvature ratios and geodesic properties. The results generalize known helices and rectifying curves to broad Riemannian settings, offering explicit conditions and ODEs to characterize these special curves with potential geometric and physical applications.
Abstract
In this paper, we introduce the notion of an anti-torqued slant helix in a Riemannian manifold, defined as a curve whose principal vector field makes a constant angle with an anti-torqued vector field globally defined on the ambient manifold. We characterize and classify such curves through systems of differential equations involving their invariants. Several illustrative examples are also provided. Finally, we study torqued curves, defined as curves for which the inner product function of the principal vector field and a torqued vector field along the curve is a given constant.