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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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TL;DR: In this article, a family of Riemannian structures on the tangent bundle (TM,G) was investigated and a direct correlation between the locally decomposable property of TM,G and the locally flatness of manifold (M,g) was found.
Abstract: Starting from the g-natural Riemannian metric G on the tangent bundle TM of a
Riemannian manifold (M,g), we construct a family of the Golden Riemannian
structures ? on the tangent bundle (TM,G). Then we investigate the
integrability of such Golden Riemannian structures on the tangent bundle TM
and show that there is a direct correlation between the locally decomposable
property of (TM,?,G) and the locally flatness of manifold (M,g).
6 citations
TL;DR: In this paper, 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-Kahler Vaisman manifolds.
6 citations
TL;DR: The conjecture of Blair as mentioned in this paper that there are no nonflat Riemannian metrics of nonpositive curvature associated with a contact structure has been proved for a certain class of contact structures on closed 3-dimensional manifolds.
Abstract: The conjecture of Blair says that there are no nonflat Riemannian metrics of nonpositive curvature associated with a contact structure. We prove this conjecture for a certain class of contact structures on closed 3-dimensional manifolds and construct a local counterexample.
6 citations
TL;DR: It is demonstrated that geometric computing in the Siegel-Klein disk allows one to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in theSiegel-Poincare disk model, and to approximate fast and numerically the SGA distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.
Abstract: We study the Hilbert geometry induced by the Siegel disk domain, an open bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel-Klein disk model to differentiate it with the classical Siegel upper plane and disk domains. In the Siegel-Klein disk, geodesics are by construction always unique and Euclidean straight, allowing one to design efficient geometric algorithms and data-structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel-Poincare disk and in the Siegel-Klein disk: We demonstrate that geometric computing in the Siegel-Klein disk allows one (i) to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in the Siegel-Poincare disk model, and (ii) to approximate fast and numerically the Siegel-Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.
6 citations
Cites background from "Riemannian Geometry of Contact and ..."
...Nowadays, symplectic geometry is mainly understood as the study of symplectic manifolds [19] which are even-dimensional differentiable manifolds equipped with a closed and nondegenerate differential 2-form ω, called the symplectic form, studied in geometric mechanics....
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01 Jan 2016
TL;DR: Wang et al. as mentioned in this paper showed that the spectrum of a compact Riemannian manifold with isolated conical singularities is stable if the scalar curvature of the cross section of conical neighborhood is greater than 2.
Abstract: Author(s): Wang, Changliang | Advisor(s): Dai, Xianzhe | Abstract: In this thesis, we study linear stability of Einstein metrics and develop the theory of Perelman's $\lambda$-functional on compact manifolds with isolated conical singularities. The thesis consists of two parts. In the first part, inspired by works in \cite{DWW05}, \cite{GHP03}, and \cite{Wan91}, by using a Bochner type argument, we prove that complete Riemannian manifolds with non-zero imaginary Killing spinors are stable, and provide a stability condition for Riemannian manifolds with non-zero real Killing spinors in terms of a twisted Dirac operator. Regular Sasaki-Einstein manifolds are essentially principal circle bundles over K quot;{a}hler-Einstein manifolds. We prove that if the base space of a regular Sasaki-Einstein manifold is a product of at least two K quot;{a}hler-Einstein manifolds, then the regular Sasaki-Einstein manifold is unstable. More generally, we show that Einstein metrics on principal torus bundles constructed in \cite{WZ90} are unstable, if the base spaces are products of at least two K quot;{a}hler-Einstein manifolds.In the second part, we prove that the spectrum of $-4\Delta+R$ consists of discrete eigenvalues with finite multiplicities on a compact Riemannian manifold of dimension $n$ with a single conical singularity, if the scalar curvature of cross section of conical neighborhood is greater than $n-2$. Moreover, we obtain an asymptotic behavior for eigenfunctions near the singularity. As a consequence of these spectrum properties, we extend the theory of Perelman's $\lambda$-functional on smooth compact manifolds to compact manifolds with isolated conical singularities.
6 citations