Curves orthogonal to a vector field in Euclidean spaces

A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [B.-Y. Chen, Tamkang J. Math. \textbf{48} (2017) 209--214] to any space dimension: we prove that rectifying curves are geodesics on the hypersurface of higher dimensional cones. We later use this association to characterize rectifying curves that are also slant helices in three-dimensional space as geodesics of circular cones. In addition, we consider curves that lie on a moving hyperplane normal to (i) one of the normal vector fields of the Frenet frame and to (ii) a rotation minimizing vector field along the curve. The former class is characterized in terms of the constancy of a certain vector field normal to the curve, while the latter contains spherical and plane curves. Finally, we establish a formal mapping between rectifying and spherical curves in any dimension.Comment: 12 pages; keywords: Rectifying curve, geodesic, cone, spherical curve, plane curve, slant heli

Curves and surfaces making a constant angle with a parallel transported direction in Riemannian spaces

In the last two decades, much effort has been dedicated to studying curves and surfaces according to their angle with a given direction. How- ever, most findings were obtained using a case-by-case approach, and it is often unclear what is a consequence of specificities of the ambient manifold and what could be generic. In this work, we propose a theo- retical framework to unify parts of these findings. We study curves and surfaces by prescribing the angle they make with a parallel transported vector field. We show that the characterization of Euclidean helices in terms of their curvature and torsion is also valid in any Riemannian manifold. Among other properties, we prove that surfaces making a con- stant angle with a parallel transported direction are extrinsically flat ruled surfaces. We also investigate the relation between their geodesics and the so-called slant helices. We prove that surfaces of constant angle are the rectifying surface of a slant helix, i.e., the ruled surface with rulings given by the Darboux field of the directrix. We characterize recti- fying surfaces of constant angle or, equivalently, when their geodesics are slant helices. As a corollary, we show that if every geodesic of a surface of constant angle is a slant helix, the ambient manifold is flat. Finally, we characterize surfaces in the product of a Riemannian surface with the real line making a constant angle with the vertical real direction

Curves and surfaces making a constant angle with a parallel transported direction in Riemannian spaces

In the last two decades, much effort has been dedicated to studying curves and surfaces according to their angle with a given direction. How- ever, most findings were obtained using a case-by-case approach, and it is often unclear what is a consequence of specificities of the ambient manifold and what could be generic. In this work, we propose a theo- retical framework to unify parts of these findings. We study curves and surfaces by prescribing the angle they make with a parallel transported vector field. We show that the characterization of Euclidean helices in terms of their curvature and torsion is also valid in any Riemannian manifold. Among other properties, we prove that surfaces making a con- stant angle with a parallel transported direction are extrinsically flat ruled surfaces. We also investigate the relation between their geodesics and the so-called slant helices. We prove that surfaces of constant angle are the rectifying surface of a slant helix, i.e., the ruled surface with rulings given by the Darboux field of the directrix. We characterize recti- fying surfaces of constant angle or, equivalently, when their geodesics are slant helices. As a corollary, we show that if every geodesic of a surface of constant angle is a slant helix, the ambient manifold is flat. Finally, we characterize surfaces in the product of a Riemannian surface with the real line making a constant angle with the vertical real direction