Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2021; 58(6): 1485-1500

Online first article October 8, 2021      Printed November 1, 2021

https://doi.org/10.4134/JKMS.j210119

Copyright © The Korean Mathematical Society.

Curves orthogonal to a vector field in Euclidean spaces

Luiz C. B. da~Silva, Gilson S. Ferreira~Jr.

Weizmann Institute of Science; Federal Rural University of Pernambuco

Abstract

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.

Keywords: Rectifying curve, geodesic, cone, spherical curve, plane curve, slant helix

MSC numbers: 53A04, 53A05, 53C22

Supported by: L. C. B. da Silva would like to thank the financial support provided by the Mor a Miriam Rozen Gerber fellowship for Brazilian postdocs.