C-parallel and C-proper Slant Curves of S-manifolds

Abstract

In the present paper, we define and study C-parallel and C-proper slant
curves of S-manifolds. We prove that a curve $\gamma $ in an S-manifold of order $r\geq 3,$ under certain conditions, is C-parallel or C-parallel in the normal bundle if and only if it is a non-Legendre slant
helix or Legendre helix, respectively. Moreover, under certain conditions,
we show that $\gamma $ is C-proper or C-proper in the normal bundle if and only if it is a non-Legendre slant curve or Legendre curve,
respectively. We also give two examples of such curves in $\mathbb{R}^{2m+s}(-3s).$