Geometry of ordinary helices in a complex projective space
type: Item
instance of: scholarly article
title: Geometry of ordinary helices in a complex projective space
zbMATH Open document ID: 1063.53022
author: Toshiaki Adachi
author: Sadahiro Maeda
author: Seiichi Udagawa
published in: Hokkaido Mathematical Journal
publication date: 2004-08-16T00:00:00Z
review text: An ordinary helix \( \gamma \) in a Riemannian manifold is defined to be a normal curve such that, for an orthonormal triplet \( V_{1} := \gamma',V_{2},V_{3} \) of vector fields along \( \gamma \), the Frenet equations are satisfied with constant positive coefficients \( \kappa, \tau \). While in real Riemannian space forms there is a one-to-one correspondence between these pairs \( (\kappa, \tau) \) and the set of congruence classes of ordinary helices under the isometry group, this ceases to hold in more general situations, e.g. already in Riemannian symmetric spaces of rank one. Continuing a former study of \textit{S. Maeda} and \textit{T. Adachi} [Proc. Am. Math. Soc. 125, No. 4, 1197--1202 (1997; Zbl 0876.53045)], two more main results are presented in the paper: In the complex projective plane \( \mathbb{C}P^{2}(4)\), using the Hopf fibration \( \pi: S^{5}(1) \to \mathbb{C}P^{2}(4) \), a closed ordinary helix with six self-intersection points not generated by any Killing vector field is constructed. In the class of Riemannian homogeneous spaces, given positive constants \( \kappa, \tau \), every ordinary helix of curvature \( \kappa \) and torsion \( \tau \) is generated by some Killing vector field iff the space has constant curvature.
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 2092065
zbMATH Keywords: helix
zbMATH Keywords: real space form
zbMATH Keywords: symmetric space of rank 1
zbMATH Keywords: homogeneous space
zbMATH Keywords: Killing vector field
reviewed by: Rolf Walter
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1285766002
OpenAlex ID: W1973972186
DOI: 10.14492/HOKMJ/1285766002
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description: scientific article; zbMATH DE number 2092065