Curves orthogonal to a vector field in Euclidean spaces

• Published in: Journal of the Korean Mathematical Society pp. 1485 - 1500
• Main Authors: Luiz C. B. da Silva, Gilson S. Ferreira Jr
• Format: Journal Article
• Language: English
• Published: 대한수학회 01.11.2021
• Subjects: 수학
• ISSN: 0304-9914; 2234-3008
• DOI: 10.4134/JKMS.j210119

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 [Chen 2017, Tamkang J. Math. {\bf 48}, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. 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 curves in an $(m+2)$-dimensional space and spherical curves in an $(m+1)$-dimensional space. KCI Citation Count: 0