Unveiling $f-$Biharmonic $\theta_{\alpha }-$Slant Curves in $\mathcal{S}-$ Space Forms

TL;DR

This paper investigates the conditions under which certain special curves called $ heta_{eta}$-slant curves in $ ext{S}$-space forms are $f$-biharmonic, providing theoretical criteria and an illustrative example.

Contribution

It establishes necessary and sufficient conditions for $ heta_{eta}$-slant curves to be $f$-biharmonic in $ ext{S}$-space forms, expanding understanding of biharmonic curves in these manifolds.

Findings

Derived a key equation characterizing $f$-biharmonic $ heta_{eta}$-slant curves.

Established necessary and sufficient conditions for such curves to be $f$-biharmonic.

Provided an example illustrating the theoretical results.

Abstract

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