Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the geometry of almost $\mathcal{S}$-manifolds   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.

16
The categories of this paper are: stat
18
The categories of this paper are: econ
19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph