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

## TL;DR

This paper explores special classes of slant curves in $S$-manifolds, characterizing their properties and providing examples, thereby advancing the understanding of geometric structures in these manifolds.

## Contribution

It introduces and analyzes $C$-parallel and $C$-proper slant curves in $S$-manifolds, establishing conditions for their classification and providing explicit examples.

## Key findings

- Characterization of $C$-parallel and $C$-proper slant curves in $S$-manifolds.
- Equivalence conditions linking these curves to non-Legendre and Legendre helices.
- Explicit examples of such curves in $ ext{R}^{2m+s}(-3s)$.

## 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 $\mathbb{R}^{2m+s}(-3s).$