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

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.