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http://dx.doi.org/10.4134/JKMS.j210119

CURVES ORTHOGONAL TO A VECTOR FIELD IN EUCLIDEAN SPACES  

da Silva, Luiz C.B. (Department of Physics of Complex Systems Weizmann Institute of Science)
Ferreira, Gilson S. Jr. (Department of Mathematics Federal Rural University of Pernambuco)
Publication Information
Journal of the Korean Mathematical Society / v.58, no.6, 2021 , pp. 1485-1500 More about this Journal
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. 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;
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