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, a homogeneous real hypersurface of type (A) or a ruled real hypersusurface in a non-flat complex space form was characterized.
Abstract: . We characterize a homogeneous real hypersurface of type (A)or a ruled real hypersurface in a non-flat complex space form, respectively. 1. IntroductionLet (Mf n (c),J,eg) be an n-dimensional complex space form with K¨ahlerianstructure (J,eg) of constant holomorphic sectional curvature c and let M be anorientable real hypersurface in Mf n (c). Then M has an almost contact metricstructure (η,φ,ξ,g) induced from (J,eg) (see Section 1). U.-H. Ki and Y. J.Suh [13] proved that are no real hypersurfaces in a non-flat complex space formsatisfying φA+Aφ = 0. From this we see that there are no almost cosymplecticor almost Kenmotsu real hypersurfaces in a non-flat complex space form (seeProposition 4 in Section 3). We put P = φA + Aφ. Then we prove that Pis invariant along the Reeb flow, that is, £ ξ P = 0 if and only if M is locallycongruent to a homogeneoushypersurface oftype (A) in P n Cor H n C(Theorem9).In Section 4, we prove that for a real hypersurface M in a non-flat com-plex space form Mf
10 citations
Cites background from "Riemannian Geometry of Contact and ..."
...For more details about the general theory of almost contact metric manifolds, we refer to [4]....
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TL;DR: In this paper, it was shown that every nearly cosymplectic manifold of dimension greater than five is a Sasakian manifold, which is a new criterion for an almost contact metric manifold.
Abstract: We prove that every nearly Sasakian manifold of dimension greater than five is Sasakian. This provides a new criterion for an almost contact metric manifold to be Sasakian. Moreover, we classify nearly cosymplectic manifolds of dimension greater than five.
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TL;DR: In this article, it was shown that the natural fibrations on 3-Sasakian manifolds and on normal complex contact metric manifolds are minima of the corrected energy of the corresponding distributions.
Abstract: In this paper we show that the natural fibrations on 3-Sasakian manifolds and on normal complex contact metric manifolds are minima of the corrected energy of the corresponding distributions.
10 citations
TL;DR: In this paper, the authors consider quasi-Einstein metrics in the framework of contact metric manifolds and prove rigidity results for (κ, μ)-spaces, and show that any complete K-contact manifold with quasi-einstein metric is compact Einstein and Sasakian.
Abstract: We consider quasi-Einstein metrics in the framework of contact metric manifolds and prove some rigidity results. First, we show that any quasi-Einstein Sasakian metric is Einstein. Next, we prove that any complete K-contact manifold with quasi-Einstein metric is compact Einstein and Sasakian. To this end, we extend these results for (κ, μ)-spaces.
10 citations
Cites background or methods from "Riemannian Geometry of Contact and ..."
...For a Sasakian manifold Q and φ commute (see [3]) and hence ∇ξ Q = 0....
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...One may consider this as an odd dimensional analogue of Goldberg conjecture, which says that a compact almost Kaehler Einstein manifold is Kaehler (see [3])....
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...Thus R(X, Y )ξ = 0, and hence M is flat in dimension 3, and in higher dimensions M is locally isometric to the trivial sphere bundle En+1 × Sn(4) (see [3])....
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...First, we note that for any contact metric manifold we have [3]: ∇Xξ = −φX − φhX....
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...Following [3] we define two self-adjoint operators h and l by h = 1 2 (£ξφ) and l = R(....
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TL;DR: In this article, a generalization of K-contact and (k,μ)-contact manifolds is studied, and it is shown that if such manifolds of dimensions ≥ 5 are conformally flat, then they have constant curvature + 1.
Abstract: We study a generalization of K-contact and (k, μ)-contact manifolds, and show that if such manifolds of dimensions ≥ 5 are conformally flat, then they have constant curvature +1. We also show under certain conditions that such manifolds admitting a non-homothetic closed conformal vector field are isometric to a unit sphere. Finally, we show that such manifolds with parallel Ricci tensor are either Einstein, or of zero $${\xi}$$
-sectional curvature.
10 citations