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 paper, it was shown that the Reeb vector field ξ of a K-contact manifold defines a harmonic map for any Riemannian natural metric and is a special metric of the Kaluza-Klein type.
Abstract: Let (M, g) be a Riemannian manifold and T1 M its unit tangent sphere bundle. Minimality and harmonicity of unit vector fields have been extensively studied by considering on T1M the Sasaki metric . This metric, and other well-known Riemannian metrics on T1 M, are particular examples of Riemannian natural metrics. In this paper we equip T1 M with a Riemannian natural metric and in particular with a natural contact metric structure. Then, we study the minimality for Reeb vector fields of contact metric manifolds and of quasi-umbilical hypersurfaces of a Kahler manifold. Several explicit examples are given. In particular, the Reeb vector field ξ of a K-contact manifold is minimal for any that belongs to a family depending on two parameters of metrics of the Kaluza–Klein type. Next, we show that the Reeb vector field ξ of a K-contact manifold defines a harmonic map for any Riemannian natural metric . Besides this, if the Reeb vector ξ of an almost contact metric manifold is a CR map then the induced almost CR structure on M is strictly pseudoconvex and ξ is a pseudo-Hermitian map; if in addition ξ is geodesic then is a harmonic map. Moreover, the Reeb vector field ξ of a contact metric manifold is a CR map iff ξ is Killing and is a special metric of the Kaluza–Klein type. Finally, in the final section, we obtain that there is a family of strictly pseudoconvex CR structures on T1S2n+1 depending on one parameter, for which a Hopf vector field ξ determines a pseudo-harmonic map (in the sense of Barletta–Dragomir–Urakawa [8]) from S2n+1 to T1S2n+1.
13 citations
Cites background from "Riemannian Geometry of Contact and ..."
...Under this assumption, it is easy to check that η̃ is homothetic — with homothety factor r — to the classical contact form on T1M (see, for example, [11] for a definition), and consequently, η̃ is a contact form....
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...It is well-known that the Reeb vector field of a contact manifold plays a fundamental role in the study of the Riemannian geometry of a contact metric manifold [11,13]....
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...We refer to [11] for more information about contact Riemannian geometry....
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Posted Content•
TL;DR: In this paper, it was shown that the structure of the set of recurrent directions in the unit normal bundle of a Riemannian manifold can be used to obtain general quantitative improvements for the Laplace norm of eigenfunctions.
Abstract: Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{\phi_\lambda \}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-\Delta_g\phi_\lambda =\lambda^2 \phi_\lambda$. Given a smooth submanifold $H \subset M$ of codimension $k\geq 1$, we find conditions on the pair $(M,H)$, even when $H=\{x\}$, for which $$ \Big|\int_H\phi_\lambda d\sigma_H\Big|=O\Big(\frac{\lambda^{\frac{k-1}{2}}}{\sqrt{\log \lambda}}\Big)\qquad \text{or}\qquad |\phi_\lambda(x)|=O\Big(\frac{\lambda ^{\frac{n-1}{2}}}{\sqrt{\log \lambda}}\Big), $$ as $\lambda\to \infty$. These conditions require no global assumption on the manifold $M$ and instead relate to the structure of the set of recurrent directions in the unit normal bundle to $H$. Our results extend all previously known conditions guaranteeing improvements on averages, including those on sup-norms. For example, we show that if $(M,g)$ is a surface with Anosov geodesic flow, then there are logarithmically improved averages for any $H\subset M$. We also find weaker conditions than having no conjugate points which guarantee $\sqrt{\log \lambda}$ improvements for the $L^\infty$ norm of eigenfunctions. Our results are obtained using geodesic beam techniques, which yield a mechanism for obtaining general quantitative improvements for averages and sup-norms.
13 citations
01 Jan 2004
TL;DR: In this article, a classification of (•, ε)-manifolds whose concircular curvature tensors Z and Ricci tensors S satisfy Z ε = 0.
Abstract: We give a classification of (•;„)-manifolds, whose concircular curvature tensor Z and Ricci tensor S satisfy Z (»;X) ¢ S = 0.
13 citations
TL;DR: In this article, it was shown that if an integrable contact pseudo-metric manifold of dimension 2n + 1, n ≥ 2, has constant sectional curvature, then the structure is Sasakian and \({\kappa=\varepsilon=g(\xi,\xi)}) is the Reeb vector field.
Abstract: In this paper, we show that if an integrable contact pseudo-metric manifold of dimension 2n + 1, n ≥ 2, has constant sectional curvature \({\kappa}\), then the structure is Sasakian and \({\kappa=\varepsilon=g(\xi,\xi)}\), where \({\xi}\) is the Reeb vector field. We note that the notion of contact pseudo-metric structure is equivalent to the notion of non-degenerate almost CR manifold, then an equivalent statement of this result holds in terms of CR geometry. Moreover, we study the pseudohermitian torsion \({\tau}\) of a non-degenerate almost CR manifold.
13 citations
TL;DR: In this paper, the curvature of contact semi-Riemannian manifold is characterized in terms of curvature, and it is shown that any conformally flat -contact SRSM is Sasakian and of constant sectional curvature.
Abstract: In this paper we characterize -contact semi-Riemannian manifolds and Sasakian semi-Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat -contact semi-Riemannian manifold is Sasakian and of constant sectional curvature , where denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a -contact Lorentzian manifold.
13 citations
Cites background from "Riemannian Geometry of Contact and ..."
...The recent monographs [2, 5] give a wide and detailed overview of the results obtained in this framework....
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