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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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Journal ArticleDOI
04 Aug 2016-Symmetry
TL;DR: This work investigates the class of left invariant almost contact metric structures on corresponding Lie groups and determines certain classes that a five-dimensional nilpotent Lie group cannot be equipped with.
Abstract: We study almost contact metric structures on 5-dimensional nilpotent Lie algebras and investigate the class of left invariant almost contact metric structures on corresponding Lie groups. We determine certain classes that a five-dimensional nilpotent Lie group can not be equipped with.

9 citations


Cites background from "Riemannian Geometry of Contact and ..."

  • ...For example, the trivial class for which∇Φ = 0 [8], corresponds to the class of cosymplectic (called co-Kähler by some authors) manifolds, C1 is the class of nearly-K-cosymplectic manifolds, etc....

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  • ...Throughout the paper, the definitions in and [7,8] will be followed....

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Journal ArticleDOI
TL;DR: In this article, it was shown that periodic contact magnetic curves in SL(2,R) can be quantized in the set of rational numbers, and that they project to horocycles in H2(-4).
Abstract: We investigate contact magnetic curves in the real special linear group of degree 2. They are geodesics of the Hopf tubes over the projection curve. We prove that periodic contact magnetic curves in SL(2,R) can be quantized in the set of rational numbers. Finally, we study contact homogeneous magnetic trajectories in SL(2,R) and show that they project to horocycles in H2(-4).

9 citations

Journal ArticleDOI
TL;DR: In this article, a convexity theorem for the twisted Hamiltonian torus on strict lcs manifolds was proved. But this was based on the minimal presentation of the toric Vaisman manifold.

9 citations

Journal ArticleDOI
TL;DR: In this paper, the equations of motion for a J-trajectory in the product space R × S 3 were derived and the contact angle for the projection on S 3 of a J -traveto was obtained.

9 citations

Posted Content
TL;DR: In this article, the authors studied the characteristic connection of the cone construction for Riemannian manifolds with skew torsion, and established the explicit correspondence between classes of metric almost contact structures on $M$ and almost hermitian classes on $\bar M$, resp.
Abstract: This paper is devoted to the systematic investigation of the cone construction for Riemannian $G$ manifolds M, endowed with an invariant metric connection with skew torsion $ abla^c$, a `characteristic connection'. We show how to define a $\bar G$ structure on the cone $\bar M=M\x \R^+$ with a cone metric, and we prove that a Killing spinor with torsion on $M$ induces a spinor on $\bar M$ that is parallel w.\,r.\,t. the characteristic connection of the $\bar G$ structure. We establish the explicit correspondence between classes of metric almost contact structures on $M$ and almost hermitian classes on $\bar M$, resp. between classes of $G_2$ structures on $M$ and $\Spin(7)$ structures on $\bar M$. Examples illustrate how this `cone correspondence with torsion' works in practice.

9 citations