Geometry of ordinary helices in a complex projective space
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.
scientific article; zbMATH DE number 2092065