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 slant curve gamma in an S-manifold of order r >= 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 R2m+s(-3s).