Book•
Riemannian Geometry of Contact and Symplectic Manifolds
08 Jan 2002-
TL;DR: In this article, the authors describe a complex geometry model of Symplectic Manifolds with principal S1-bundles and Tangent Sphere Bundles, as well as a negative Xi-sectional Curvature.
Abstract: Preface * 1. Symplectic Manifolds * 2. Principal S1-bundles * 3. Contact Manifolds * 4. Associated Metrics * 5. Integral Submanifolds and Contact Transformations * 6. Sasakian and Cosymplectic Manifolds * 7. Curvature of Contact Metric Manifolds * 8. Submanifolds of Kahler and Sasakian Manifolds * 9. Tangent Bundles and Tangent Sphere Bundles * 10. Curvature Functionals and Spaces of Associated Metrics * 11. Negative Xi-sectional Curvature * 12. Complex Contact Manifolds * 13. Additional Topics in Complex Geometry * 14. 3-Sasakian Manifolds * Bibliography * Subject Index * Author Index
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TL;DR: In this article, the formality of the mapping torus of an orientation-preserving diffeomorphism of a manifold is studied and conditions under which such a torus has a non-zero Massey product are given.
Abstract: We study the formality of the mapping torus of an orientation-preserving diffeomorphism of a manifold. In particular, we give conditions under which a mapping torus has a non-zero Massey product. As an application we prove that there are non-formal compact co-symplectic manifolds of dimension $m$ and with first Betti number $b$ if and only if $m=3$ and $b \geq 2$, or $m \geq 5$ and $b \geq 1$. Explicit examples for each one of these cases are given.
21 citations
Cites background from "Riemannian Geometry of Contact and ..."
...In the literature, co-symplectic manifolds are often called almost cosymplectic, while co-Kähler manifolds are called cosymplectic (see [3, 5, 7, 16])....
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TL;DR: In this paper, it was shown that any contact metric manifold whose Reeb vector field belongs to the $(\kappa,\mu)$-nullity distribution canonically carries an almost bi-paracontact structure.
Abstract: We study almost bi-paracontact structures on contact manifolds. We prove that if an almost bi-paracontact structure is defined on a contact manifold $(M,\eta)$, then under some natural assumptions of integrability, $M$ carries two transverse bi-Legendrian structures. Conversely, if two transverse bi-Legendrian structures are defined on a contact manifold, then $M$ admits an almost bi-paracontact structure. We define a canonical connection on an almost bi-paracontact manifold and we study its curvature properties, which resemble those of the Obata connection of an anti-hypercomplex (or complex-product) manifold. Further, we prove that any contact metric manifold whose Reeb vector field belongs to the $(\kappa,\mu)$-nullity distribution canonically carries an almost bi-paracontact structure and we apply the previous results to the theory of contact metric $(\kappa,\mu)$-spaces.
21 citations
Additional excerpts
...2) it follows that φξ = 0, η◦φ = 0 and the (1, 1)-tensor field φ has constant rank 2n ([3])....
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TL;DR: In this article, the authors proved that there exists no proper doubly warped product contact CR-submanifolds in trans-Sasakian manifolds, which is a line of research similar to that concerning Sasakian geometry as the odd dimensional version of Kaehlerian geometry.
Abstract: Warped product CR-submanifolds in Kaehlerian manifolds were intensively studied only since 2001 after the impulse given by B.Y. Chen. Immediately after, another line of research, similar to that concerning Sasakian geometry as the odd dimensional version of Kaehlerian geometry, was developed, namely warped product contact CR-submanifolds in Sasakian manifolds. In this note we proved that there exists no proper doubly warped product contact CR-submanifolds in trans-Sasakian manifolds.
21 citations
TL;DR: In this article, the authors generalize Morimoto's Theorem on the product of almost contact manifolds to flat bundles and construct some examples on Boothby-Wang fibrations over contact-symplectic manifolds.
Abstract: 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.
21 citations
TL;DR: In this article, it was shown that the Reeb vector field of an almost cosymplectic three-manifold is minimal if and only if it is an eigenvector of the Ricci operator.
Abstract: We show that the Reeb vector field of an almost cosymplectic three-manifold is minimal if and only if it is an eigenvector of the Ricci operator. Then, we show that Reeb vector field ξ of an almost cosymplectic three-manifold M is minimal if and only if M is (κ, μ, ν)-space on an open dense subset. After, using the notion of strongly normal unit vector field introduced in [8], we study the minimality of ξ for an almost cosymplectic (2n + 1)-manifold. Finally, we classify a special class of almost cosymplectic three-manifold whose Reeb vector field is minimal.
20 citations