Table of Contents
ISRN Geometry
Volume 2011, Article ID 879042, 12 pages
http://dx.doi.org/10.5402/2011/879042
Research Article

On the Geometry of Almost ๐’ฎ -Manifolds

Department of Mathematics, University of California, Berkeley, 749 Evans Hall, No. 3840, Berkeley, CA 94720, USA

Received 3 October 2011; Accepted 20 October 2011

Academic Editors: T. Friedrich and U. Gran

Copyright © 2011 Sean Fitzpatrick. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An ๐‘“ -structure on a manifold ๐‘€ is an endomorphism field ๐œ‘ satisfying ๐œ‘ 3 + ๐œ‘ = 0 . We call an f-structure regular if the distribution ๐‘‡ = k e r ๐œ‘ 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 ๐’ฎ -structure, it determines a torus fibration of M over a symplectic manifold. When rank ๐‘‡ = 1 , this result reduces to the Boothby-Wang theorem. Unlike similar results for manifolds with ๐’ฎ -structure or ๐’ฆ -structure, we do not assume that the f-structure is normal. We also show that given an almost ๐’ฎ -structure, we obtain an associated Jacobi structure, as well as a notion of symplectization.