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

TL;DR

This paper explores the geometry of almost $ ext{S}$-structures on manifolds, showing they induce torus fibrations over symplectic manifolds and establishing connections to Jacobi structures without assuming normality.

Contribution

It demonstrates that regular almost $ ext{S}$-structures on compact manifolds lead to torus fibrations over symplectic bases, extending previous results without the normality assumption.

Findings

Regular almost $ ext{S}$-structures induce torus fibrations.

Connection established between almost $ ext{S}$-structures and Jacobi structures.

Generalization of Boothby-Wang theorem to higher ranks.

Abstract

An $f$-structure on a manifold $M$ is an endomorphism field $\phi$ satisfying $\phi^3+\phi=0$. We call an $f$-structure {\em regular} if the distribution $T=\ker\phi$ 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 $\S$-structure, as defined by Duggal, Ianus, and Pastore, it determines a torus fibration of $M$ over a symplectic manifold. When $\rank T = 1$, 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.