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Unveiling $f-$Biharmonic $θ_{α}-$Slant Curves in $\mathcal{S}-$ Space Forms

Şaban Güvenç

TL;DR

This work addresses the problem of characterizing proper $\,f$-biharmonic $\theta_{\alpha}$-slant curves within $\mathcal{S}$-space forms. It derives a key third-order tension equation and, through a detailed case analysis dependent on the ambient curvature parameters $c$ and $s$ and the alignment of $\phi T$ with the structure vectors, provides necessary and sufficient conditions expressed in terms of the Frenet curvatures $k_1,k_2,k_3$ and the function $f$. The main contribution is a comprehensive characterization theorem (with explicit ODEs for $k_1$ and relations among curvatures) and an explicit construction of a proper $f$-biharmonic $\theta_{\alpha}$-slant curve in $\mathbb{R}^{6}(-6)$. The results extend the theory of biharmonic and slant curves in Sasakian-type geometries and offer a template for building such curves in concrete spaces with prescribed geometric data.

Abstract

In this paper, we firstly provide a concise overview of $\mathcal{S}-$manifolds, $f$-biharmonicity and $θ_{α}$-slant curves. We then derive a key equation and analyze it in detail to establish the necessary and sufficient conditions for $θ_{α}$-slant curves to be $f$-biharmonic. Finally, we present an example to support our findings.

Unveiling $f-$Biharmonic $θ_{α}-$Slant Curves in $\mathcal{S}-$ Space Forms

TL;DR

This work addresses the problem of characterizing proper -biharmonic -slant curves within -space forms. It derives a key third-order tension equation and, through a detailed case analysis dependent on the ambient curvature parameters and and the alignment of with the structure vectors, provides necessary and sufficient conditions expressed in terms of the Frenet curvatures and the function . The main contribution is a comprehensive characterization theorem (with explicit ODEs for and relations among curvatures) and an explicit construction of a proper -biharmonic -slant curve in . The results extend the theory of biharmonic and slant curves in Sasakian-type geometries and offer a template for building such curves in concrete spaces with prescribed geometric data.

Abstract

In this paper, we firstly provide a concise overview of manifolds, -biharmonicity and -slant curves. We then derive a key equation and analyze it in detail to establish the necessary and sufficient conditions for -slant curves to be -biharmonic. Finally, we present an example to support our findings.
Paper Structure (6 sections, 12 theorems, 106 equations)

This paper contains 6 sections, 12 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Background and Definitions
  3. $\theta_{\alpha}-$Slant Curves in $\mathcal{S}-$Manifolds
  4. $f$-biharmonic $\theta_{\alpha}$-Slant Curves in $\mathcal{S}$-Space Forms
  5. Construction of an Example in $\mathbb{R}^{4}(-6)$
  6. Conclusions and Future Work

Key Result

Lemma 1

YeLin-2014 A curve $\gamma :(a,b)\rightarrow (M,g)$ parametrized by arclength is an $f$-biharmonic curve with a function $f:(a,b)\rightarrow (0,\infty)$ if and only if

Theorems & Definitions (18)

  • Lemma 1
  • Definition 1
  • Corollary 1
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • proof
  • Lemma 2
  • Theorem 2
  • proof
  • ...and 8 more