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.