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

In the present paper, we define and study $C$-parallel and $C$-proper slant curves of $S$-manifolds. We prove that a curve $γ$ 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 $γ$ 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 $\mathbb{R}^{2m+s}(-3s).$

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