On the geometry of almost $\mathcal{S}$-manifolds

An $f$-structure on a manifold $M$ is an endomorphism field $\varphi$ satisfying ${\varphi }^3+\varphi =0$. We call an $f$-structure {\em regular} if the distribution $T=\ker \varphi$ is involutive and regular, in the sense of Palais. We show that when a regular $f$-structure on a compact manifold $M$ is an almost $\mathrm{\S }$-structure, as defined by Duggal, Ianus, and Pastore, it determines a torus fibration of $M$ over a symplectic manifold. When $\text{rank}T=1$, this result reduces to the Boothby-Wang theorem. Unlike similar results due to Blair-Ludden-Yano and Soare, we do not assume that the $f$-structure is normal. We also show that given an almost $\mathcal{S}$-structure, we obtain an associated Jacobi structure, as well as a notion of symplectization.