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, the authors define and investigate new classes of almost 3-contact metric manifolds, with two guiding ideas in mind: first, what geometric objects are best suited for capturing the key properties of almost contact metric manifold, and second, the newly defined classes should admit 'good' metric connections with skew torsion.
Abstract: In the first part, we define and investigate new classes of almost 3-contact metric manifolds, with two guiding ideas in mind: first, what geometric objects are best suited for capturing the key properties of almost 3-contact metric manifolds, and second, the newly defined classes should admit 'good' metric connections with skew torsion. In particular, we introduce the Reeb commutator function and the Reeb Killing function, we define the new classes of canonical almost 3-contact metric manifolds and of 3-$(\alpha,\delta)$-Sasaki manifolds (including as special cases 3-Sasaki manifolds, quaternionic Heisenberg groups, and many others) and prove that the latter are hypernormal, thus generalizing a seminal result by Kashiwada. We study their behaviour under a new class of deformations, called $\mathcal{H}$-homothetic deformations, and prove that they admit an underlying quaternionic contact structure, from which we deduce the Ricci curvature. For example, a 3-$(\alpha,\delta)$-Sasaki manifold is Einstein either if $\alpha=\delta$ (the 3-$\alpha$-Sasaki case) or if $\delta=(2n+3)\alpha$, where $\dim M=4n+3$.
The second part is actually devoted to finding these adapted connections. We start with a very general notion of $\varphi$-compatible connections, where $\varphi$ denotes any element of the associated sphere of almost contact structures, and make them unique by a certain extra condition, thus yielding the notion of canonical connection (they exist exactly on canonical manifolds, hence the name). For 3-$(\alpha,\delta)$-Sasaki manifolds, we compute the torsion of this connection explicitly and we prove that it is parallel, we describe the holonomy, the $
abla$-Ricci curvature, and we show that the metric cone is a HKT-manifold. In dimension 7, we construct a cocalibrated $G_2$-structure inducing the canonical connection and we prove the existence of four generalized Killing spinors.
19 citations
Cites background from "Riemannian Geometry of Contact and ..."
...for any even permutation (i, j, k) of (1, 2, 3) [Bl10]....
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...The almost contact structure is said to be normal if Nφ := [φ, φ] + dη ⊗ ξ vanishes, where [φ, φ] is the Nijenhuis torsion of φ [Bl10]....
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TL;DR: In this article, the curvature properties of normal almost contact manifolds with B-metric are studied, and relations involving scalar invariants on such manifolds are obtained.
Abstract: Curvature properties of normal almost contact manifolds with B-metric are studied. Relations involving scalar invariants on such manifolds are obtained. Necessary and sufficient conditions for a normal almost contact manifold with B-metric to be of isotropic Kahler-type are given. An example illustrating some of the obtained results is constructed on a Lie algebra.
19 citations
TL;DR: In this paper, it was shown that the metric of a (κ,μ)-almost co-Kahler manifold M2n+1 is a gradieness of a quasi-Yamabe solitons.
Abstract: We characterize almost co-Kahler manifolds with gradient Yamabe, gradient Einstein and quasi-Yamabe solitons. It is proved that if the metric of a (κ,μ)-almost co-Kahler manifold M2n+1 is a gradien...
19 citations
TL;DR: In this paper, different types of symmetries for the Tanaka-Webster connection of contact strictly pseudo-convex pseudo-Hermitian CR manifolds are studied.
Abstract: In this paper, we study different types of symmetries for the Tanaka-Webster connection of contact strictly pseudo-convex pseudo-Hermitian CR manifolds.
19 citations
TL;DR: In this article, Li et al. showed that all compact co-Kahler manifolds arise as the product of a Kahler manifold and a circle, where the covering transformations act diagonally on the manifold and are translations on the second factor.
Abstract: By the work of Li, a compact co-Kahler manifold \(M\) is a mapping torus \(K_\varphi \), where \(K\) is a Kahler manifold and \(\varphi \) is a Hermitian isometry. We show here that there is always a finite cyclic cover \(\overline{M}\) of the form \(\overline{M} \cong K \times S^1\), where \(\cong \) is equivariant diffeomorphism with respect to an action of \(S^1\) on \(M\) and the action of \(S^1\) on \(K \times S^1\) by translation on the second factor. Furthermore, the covering transformations act diagonally on \(S^1, K\) and are translations on the \(S^1\) factor. In this way, we see that, up to a finite cover, all compact co-Kahler manifolds arise as the product of a Kahler manifold and a circle.
19 citations
Cites background from "Riemannian Geometry of Contact and ..."
...Proof The normality condition implies that Lξ J = 0 (see [3]); in particular, [ξ, J X ] = J [ξ, X ] for every vector field X on M ....
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