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The geometry of symplectic pairs. (English) Zbl 1088.53017

A symplectic pair on a smooth manifold \(M\) is a pair of non-trivial closed two-forms \(\omega_1\), \(\omega_2\) of constant and complementary ranks, for which \(\omega_1\) restricts as a symplectic form to the leaves of the kernel foliation of \(\omega_2\), and vice versa. The authors show how a variation of the classical Boothby-Wang construction allows one to construct contact-symplectic pairs from symplectic pairs for which \(\omega_1\) represents an integral cohomology class in \(M\), and contact pairs from contact-symplectic pairs in which the leafwise symplectic form represents an integral cohomology class. Several constructions of symplectic pairs are given. Riemannian metrics compatible with a symplectic pair are studied.
To construct examples of symplectic pairs the authors study flat bundles with symplectic total holonomy, the Gompf sum of closed symplectic manifolds, and four-dimensional Thurston geometries. They clarify the metric properties of symplectic pairs.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57R17 Symplectic and contact topology in high or arbitrary dimension
57R30 Foliations in differential topology; geometric theory
53C12 Foliations (differential geometric aspects)
53D35 Global theory of symplectic and contact manifolds
58A17 Pfaffian systems
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References:

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