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

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

2011-03-23

arxiv.org/abs/1103.4642v3

ISRN Geometry, vol. 2011, Article ID 879042, 12 pages, 2011

Mathematics

Differential Geometry

12 pages, title change, minor typo corrections, to appear in ISRN Geometry

Scientific paper

10.5402/2011/879042

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.

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