We introduce a new geometric structure on differentiable manifolds. A Contact Pair on a 2h+2k +2-dimensional manifold M is a pair (α, η) of Pfaffian forms of constant classes 2k + 1 and 2h + 1, respectively, whose characteristic foliations are transverse and complementary and such that α and η restrict to contact forms on the leaves of the characteristic foliations of η and α, respectively. Further differential objects are associated to Contact Pairs: two commuting Reeb vector fields, Legendrian curves on M and two Lie brackets on the set of differentiable functions on M. We give a local model and several existence theorems on nilpotent Lie groups, nilmanifolds, bundles over the circle and principal torus bundles.
We consider manifolds endowed with a contact pair structure. To such a structure are naturally associated two almost complex structures. If they are both integrable, we call the structure a normal contact pair. We generalize Morimoto's Theorem on the product of almost contact manifolds to flat bundles. We construct some examples on Boothby-Wang fibrations over contact-symplectic manifolds. In particular, these results give new methods to construct complex manifolds.
Abstract. We study the geometry of manifolds carrying symplectic pairs consisting of two closed 2-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.
We introduce the notion of contact pair structure and the corresponding associated metrics, in the same spirit of the geometry of almost contact structures. We prove that, with respect to these metrics, the integral curves of the Reeb vector fields are geodesics and that the leaves of the Reeb action are totally geodesic. Moreover, we show that, in the case of a metric contact pair with decomposable endomorphism, the characteristic foliations are orthogonal and their leaves carry induced contact metric structures.2000 Mathematics Subject Classification. Primary 53C15; Secondary 53D15, 53C12, 53D35.
We discuss a correspondence between certain contact pairs on the one hand, and certain locally conformally symplectic forms on the other. In particular, we characterize these structures through suspensions of contactomorphisms. If the contact pair is endowed with a normal metric, then the corresponding lcs form is locally conformally Kähler, and, in fact, Vaisman. This leads to classification results for normal metric contact pairs. In complex dimension two we obtain a new proof of Belgun's classification of Vaisman manifolds under the additional assumption that the Kodaira dimension is non-negative. We also produce many examples of manifolds admitting locally conformally symplectic structures but no locally conformally Kähler ones.
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