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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
TL;DR: In this article, it was shown that the normal metric contact pairs with orthogonal characteristic foliations, which are either Bochner flat or locally conformally flat, are locally isometric to the Hopf manifolds.
Abstract: We prove that the normal metric contact pairs with orthogonal characteristic foliations, which are either Bochner flat or locally conformally flat, are locally isometric to the Hopf manifolds. As a corollary we obtain the classification of locally conformally flat and Bochner-flat non-K\"ahler Vaisman manifolds.

3 citations

Posted Content
TL;DR: In this paper, the authors introduced two classes of null hypersurfaces of an indefinite Sasakian manifold, called; contact screen conformal and contact screen umbilic, and proved that they are contained in indefinite SAsakian space forms of constant $\overline{\phi}$-sectional curvature.
Abstract: We introduce two classes of null hypersurfaces of an indefinite Sasakian manifold, $(\overline{M}, \overline{\phi},\zeta, \eta)$, tangent to the characteristic vector field $\zeta$, called; {\it contact screen conformal} and {\it contact screen umbilic} null hypersurfaces. These hypersurfaces come in to fill the existing gap in screen conformal and screen totally umbilic null hypersurfaces. We prove that such hypersurfaces are contained in indefinite Sasakian space forms of constant $\overline{\phi}$-sectional curvature of $-3$.

3 citations


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

  • ...It is well-known [3] that an almost contact metric manifold (M,g) is Sasakian if and only if (∇Xφ)Y = g(X,Y )ζ − η(Y )X, ∀X,Y ∈ Γ(TM), (2....

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  • ...It is known (see Blair [3] for details) that M has a normal contact structure if Nφ + 2dη ⊗ ζ = 0, where Nφ = [φ, φ] is the Nijenhuis tensor field of φ....

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  • ...Moreover, the curvature tensor R of M satisfies (see [3])...

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  • ...It is known (see Blair [3] for details) that M has a normal contact structure if Nφ + 2dη ⊗ ζ = 0, where Nφ = [φ, φ] is the...

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  • ..., Ω is closed), then, (M,φ, ζ, η) is called a contact metric manifold [3]....

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Book ChapterDOI
01 Jan 2017
TL;DR: In this paper, the authors give an introduction to existence problems of contact structures and some historical considerations pointing important steps in the development of contact geometry are presented, and the existence of contact forms is studied in the next Section.
Abstract: The goal of this lecture is to give an introduction to existence problems of contact structures. So, in the first Section we define the notion of contact structure, as well as some specialized contact structures. We also study the rigidity and the local behavior of such a structure. Some basic problems concerning the geometry of contact manifolds are presented in Sect. 2. The existence of contact forms is studied in the next Section. Specially in the 3–dimensional case, some classical results and the new Geiges–Gonzalo theory of contact circles and contact spheres and the classification manifolds carrying such structures are presented. Some historical considerations pointing important steps in the development of contact geometry are finally presented.

3 citations

Journal ArticleDOI
TL;DR: For locally conformal symplectic foliations and contact foliations on open manifolds, the authors showed that the relation between the two types of foliations can be expressed as a regular Jacobi structure on manifolds.
Abstract: We prove \(h\)-principle for locally conformal symplectic foliations and contact foliations on open manifolds. We then interpret the results in terms of regular Jacobi structures on manifolds.

3 citations