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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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17 Aug 2008

10 citations


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

  • ...Since characteristic surfaces are defined via contact fields, we begin by providing a useful characterization of contact vector fields (see also [41] and [13] p....

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Journal ArticleDOI
TL;DR: In this article, the notion of a slant light-like submanifold of an indefinite cosymplectic manifold was introduced and necessary and sufficient conditions for the existence of such a sub-manifolds were derived.
Abstract: In this paper, we introduce the notion of a slant lightlike submanifold of an indefinite Cosymplectic manifold. We provide a nontrivial example and obtain necessary and sufficient conditions for the existence of a slant lightlike submanifold. Also, we give an example of a minimal slant lightlike submanifold of \({R^{9}_{2}}\) and prove some characterization Theorems.

10 citations


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

  • ...An indefinite almost contact metric manifold M is called an indefinite Cosymplectic manifold if [ 2 ],...

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Journal ArticleDOI
TL;DR: In this paper, a Riemannian manifold with a compatible f-structure which admits a parallelizable kernel is considered, and relations between (almost) K, K, C, S and transverse almost Hermitian structures on the foliated manifolds are considered.
Abstract: We consider a Riemannian manifold with a compatible f-structure which admits a parallelizable kernel. With some additional integrability conditions it is called (almost) \({\mathcal{K}}, {\mathcal{C}}, {\mathcal{S}}\)-manifold and is a natural generalization of the (almost) contact metric and the Sasakian manifolds. There are presented various methods of constructing examples of such manifolds. There are used structures on the principal bundles and the pull-back bundles. Then there are considered relations between (almost) \({\mathcal{K}}, {\mathcal{C}}, {\mathcal{S}}\)-manifolds and transverse almost Hermitian structures on the foliated manifolds.

10 citations


Additional excerpts

  • ...[2, 3]....

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Journal ArticleDOI
TL;DR: In this paper, the authors consider the case where the potential vector field is collinear with the characteristic vector field on an open set of manifolds and show that the potential field is equal to the soliton vector field.
Abstract: In this paper, we consider $*$-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if the metric of a Kenmotsu manifold $M$ is a $*$-Ricci soliton, then soliton constant $\lambda$ is zero. For 3-dimensional case, if $M$ admits a $*$-Ricci soliton, then we show that $M$ is of constant sectional curvature -1. Next, we show that if $M$ admits a $*$-Ricci soliton whose potential vector field is collinear with the characteristic vector field $\xi$, then $M$ is Einstein and soliton vector field is equal to $\xi$. Finally, we prove that if $g$ is a gradient almost $*$-Ricci soliton, then either $M$ is Einstein or the potential vector field is collinear with the characteristic vector field on an open set of $M$. We verify our result by constructing examples for both $*$-Ricci soliton and gradient almost $*$-Ricci soliton.

10 citations

Journal ArticleDOI
27 Feb 2017-Filomat
TL;DR: In this paper, the authors considered f-biharmonic Legendre curves in Sasakian space forms and found Curvature characterizations of these types of curves in four cases.
Abstract: We consider f-biharmonic Legendre curves in Sasakian space forms. We find curvature characterizations of these types of curves in four cases.

10 citations


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

  • ...A 1-dimensional integral submanifold of a Sasakian manifold (M2m+1, φ, ξ, η, 1) is called a Legendre curve of M [3]....

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  • ...In order to obtain explicit examples, we will first need to recall some notions about the Sasakian space form R2m+1(−3) [3]: Let us consider M = R2m+1 with the standard coordinate functions ( x1, ....

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  • ...form a 1-orthonormal basis and the Levi-Civita connection is calculated as ∇Xi X j = ∇Xm+i Xm+ j = 0,∇Xi Xm+ j = δi jξ,∇Xm+i X j = −δi jξ, ∇Xiξ = ∇ξXi = −Xm+i,∇Xm+iξ = ∇ξXm+i = Xi, (see [3])....

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