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, the full classification of invariant symplectic, (almost) complex and Kahler structures, together with their paracomplex analogues, on four-dimensional pseudo-Riemannian generalized symmetric spaces was obtained.
Abstract: We obtain the full classification of invariant symplectic, (almost) complex and Kahler structures, together with their paracomplex analogues, on four-dimensional pseudo-Riemannian generalized symmetric spaces. We also apply these results to build some new examples of five-dimensional homogeneous K -contact, Sasakian, K -paracontact and para-Sasakian manifolds.
11 citations
Cites background from "Riemannian Geometry of Contact and ..."
...We may refer to [4] for further information and to [23,9] for the pseudo-Riemannian generalization....
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TL;DR: In this paper, a necessary and sufficient condition for an almost Kenmotsu 3-manifold to be conformally flat is given. But this condition is not applicable to the case of the Riemannian product.
Abstract: In this paper, by virtue of a system of partial differential equations, we give a necessary and sufficient condition for an almost Kenmotsu 3-manifold to be conformally flat. As an application, we obtain that an almost Kenmotsu 3-H-manifold with scalar curvature invariant along the Reeb vector field is conformally flat if and only if it is locally isometric to either the hyperbolic space $$\mathbb {H}^3(-1)$$
or the Riemannian product $$\mathbb {H}^{2}(-4)\times \mathbb {R}$$
. Some concrete examples verifying main results are presented.
11 citations
Posted Content•
TL;DR: In this article, the existence of an abundance of Einstein metrics on odd dimensional spheres, including exotic spheres, was discussed at the Short Program on Riemannian Geometry at the Centre de Recherche Mathematiques, Universit\'e de Montreal, during the period June 28-July 16, 2004.
Abstract: This paper is based on a talk presented by the first author at the Short Program on Riemannian Geometry that took place at the Centre de Recherche Math\'ematiques, Universit\'e de Montr\'eal, during the period June 28-July 16, 2004. It is a report on our joint work with J\'anos Koll\'ar concerning the existence of an abundance of Einstein metrics on odd dimensional spheres, including exotic spheres.
11 citations
TL;DR: In this paper, a variational interpretation of Levi harmonic morphisms is given, and a CR map f of contact (semi) Riemannian manifolds is pseudoharmonic if and only if f is Levi harmonic.
Abstract: We study Levi harmonic maps, i.e., C∞ solutions f:M→M′ to \(\tau_{\mathcal{H}} (f) \equiv \operatorname{trace}_{g} ( \varPi_{\mathcal{H}}\beta_{f} ) = 0\), where (M,η,g) is an (almost) contact (semi) Riemannian manifold, M′ is a (semi) Riemannian manifold, βf is the second fundamental form of f, and \(\varPi_{\mathcal{H}} \beta_{f}\) is the restriction of βf to the Levi distribution \({\mathcal{H}} = \operatorname{Ker}(\eta)\). Many examples are exhibited, e.g., the Hopf vector field on the unit sphere S2n+1, immersions of Brieskorn spheres, and the geodesic flow of the tangent sphere bundle over a Riemannian manifold of constant curvature 1 are Levi harmonic maps. A CR map f of contact (semi) Riemannian manifolds (with spacelike Reeb fields) is pseudoharmonic if and only if f is Levi harmonic. We give a variational interpretation of Levi harmonicity. Any Levi harmonic morphism is shown to be a Levi harmonic map.
11 citations