Riemannian Geometry of Contact and Symplectic Manifolds

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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.

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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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A note on Almost Riemann Soliton and gradient almost Riemann soliton.

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Krishnendu De, Uday Chand De

24 Aug 2020-arXiv: Differential Geometry

TL;DR: In this paper, it was shown that if the metric of a non-cosymplectic normal almost contact metric manifold is Riemann soliton with divergence-free potential vector field (Z), then the manifold is quasi-Sakian and is of constant sectional curvature -$\lambda.

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Abstract: The quest of the offering article is to investigate \emph{almost Riemann soliton} and \emph{gradient almost Riemann soliton} in a non-cosymplectic normal almost contact metric manifold $M^3$. Before all else, it is proved that if the metric of $M^3$ is Riemann soliton with divergence-free potential vector field $Z$, then the manifold is quasi-Sasakian and is of constant sectional curvature -$\lambda$, provided $\alpha,\beta =$ constant. Other than this, it is shown that if the metric of $M^3$ is \emph{ARS} and $Z$ is pointwise collinear with $\xi $ and has constant divergence, then $Z$ is a constant multiple of $\xi $ and the \emph{ARS} reduces to a Riemann soliton, provided $\alpha,\;\beta =$constant. Additionally, it is established that if $M^3$ with $\alpha,\; \beta =$ constant admits a gradient \emph{ARS} $(\gamma,\xi,\lambda)$, then the manifold is either quasi-Sasakian or is of constant sectional curvature $-(\alpha^2-\beta^2)$. At long last, we develop an example of $M^3$ conceding a Riemann soliton.

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Cites background from "Riemannian Geometry of Contact and ..."

...After fulfilling the condition, the structure J is integrable, M(η, ξ, φ) is said to be normal (see, [2],[3])....
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The twistor space of a quaternionic contact manifold

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Johann Davidov¹, Stefan Ivanov², Ivan Minchev³•Institutions (3)

Bulgarian Academy of Sciences¹, Sofia University², University of Marburg³

24 Oct 2010-arXiv: Differential Geometry

TL;DR: In this paper, the Ricci curvature of the Biquard connection commutes with the endomorphisms in the quaternionic structure of the contact distribution, and it is shown that the CR structure on the twistor space of a quaternion contact structure is normal.

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Abstract: We show that the CR structure on the twistor space of a quaternionic contact structure described by Biquard is normal if and only if the Ricci curvature of the Biquard connection commutes with the endomorphisms in the quaternionic structure of the contact distribution.

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7 citations

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Gray Curvature Identities for Almost Contact Metric Manifolds

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Raluca Mocanu, Marian Ioan Munteanu

18 Jun 2007-arXiv: Differential Geometry

TL;DR: In this article, the authors studied the Gray curvature identities for the class of almost hermitian manifolds and showed that these identities can be satisfied by an almost contact manifold.

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Abstract: The aim of this research is the study of Gray curvature identities, introduced by Alfred Gray in \cite{kn:Gra76} for the class of almost hermitian manifolds. As known till now, there is no equivalent for the class of almost contact manifolds. For this purpose we use the Boohby-Wang fibration and the warped manifolds construction in order to establish which identities could be satisfied by an almost contact manifold.

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Journal Article•DOI•

A note on almost contact riemannian 3-manifolds ii

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Jun-ichi Inoguchi

31 Jan 2017-Bulletin of The Korean Mathematical Society

TL;DR: In this paper, the authors investigated curvatures of normal almost contact Riemannian 3-manifolds and showed that they are of constant scalar curvature −1.

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Abstract: We investigate curvatures of normal almost contact Riemannian 3-manifolds. In particular, we show that Kenmotsu 3-manifolds of constant scalar curvature are of constant curvature −1.

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Journal Article•DOI•

The Local Structure of Generalized Contact Bundles

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Jonas Schnitzer¹, Luca Vitagliano¹•Institutions (1)

University of Salerno¹

22 Nov 2017-arXiv: Differential Geometry

TL;DR: In this paper, the authors prove a local splitting theorem similar to those appearing in Poisson geometry for generalized contact bundles, where a generalized contact bundle is either the product of a contact and a complex manifold or a symplectic manifold equipped with an integrable complex structure on the gauge algebroid of the trivial line bundle.

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Abstract: Generalized contact bundles are odd dimensional analogues of generalized complex manifolds. They have been introduced recently and very little is known about them. In this paper we study their local structure. Specifically, we prove a local splitting theorem similar to those appearing in Poisson geometry. In particular, in a neighborhood of a regular point, a generalized contact bundle is either the product of a contact and a complex manifold or the product of a symplectic manifold and a manifold equipped with an integrable complex structure on the gauge algebroid of the trivial line bundle.

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Cites background from "Riemannian Geometry of Contact and ..."

...n of Φ [22]. One can actually show that the ﬁrst condition in (A.3) implies the other ones [7, Section 6.1] (see also [22]). An almost contact structure (Φ,ξ,η) such that NΦ +dη⊗ξ= 0 is called normal [7]. So normal almost contact structures provide examples of complex structures on the Atiyah algebroid (of the trivial line bundle), and, in turn, of generalized contact structures of complex type. It t...
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... contact structure on a manifold M is a triple (Φ,ξ,η), where Φ : TM→TM is a (1,1)-tensor, ξis a vector ﬁeld, and ηis a 1-form on Msuch that Φ2 = −id +η⊗ξ, Φ(ξ) = 0, η◦Φ = 0, and η(ξ) = 1. See, e.g., [7] for more details. The idea behind this deﬁnition is that an almost contact structure is the odd-dimensional analogue of an almost complex structure. We believe that THE LOCAL STRUCTURE OF GENERALIZED...
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...treme they encompass line bundles equipped with an integrable complex structure on their gauge algebroid. In turn, such line bundles are intrinsic models for so called normal almost contact manifolds [7]. In our opinion, generalized contact bundles have an advantage over previous proposals of a generalized contact geometry: they have a ﬁrm conceptual basis in the so called homogenization scheme [40],...
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Riemannian Geometry of Contact and Symplectic Manifolds

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Cites background from "Riemannian Geometry of Contact and ..."

Cites background from "Riemannian Geometry of Contact and ..."

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