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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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Citations
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Journal ArticleDOI
TL;DR: N(k)-quasi Einstein manifolds are introduced and studied in this article, where the authors introduce the concept of quasi-Einstein manifolds and study the N(k-quasi Eiffel manifold.
Abstract: N(k)-quasi Einstein manifolds are introduced and studied.

43 citations


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

  • ...A proper η-Einstein contact metric manifold ([1], [5]) is a natural example of a quasi Einstein manifold....

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Journal ArticleDOI
TL;DR: In this paper, the authors completely describe paracontact metric three-manifolds whose Reeb vector field satisfies the Ricci soliton equation, and correct the main result in [1], concerning three-dimensional normal parAContact Ricci Solitons.

42 citations


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

  • ...By the classic Theorem of Darboux (see [2], p....

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Journal ArticleDOI
TL;DR: In this article, the authors considered almost Kenmotsu manifolds (M 2 n + 1, φ, ξ, η, g ) with a parallel tensor φ = h ○ φ, 2h being the Lie derivative of φ with respect to the Reeb vector field.
Abstract: We consider almost Kenmotsu manifolds ( M 2 n + 1 , φ , ξ , η , g ) with η-parallel tensor h ′ = h ○ φ , 2h being the Lie derivative of the structure tensor φ with respect to the Reeb vector field ξ. We describe the Riemannian geometry of an integral submanifold of the distribution orthogonal to ξ, characterizing the CR-integrability of the structure. Under the additional condition ∇ ξ h ′ = 0 , the almost Kenmotsu manifold is locally a warped product. Finally, some lightlike structures on M 2 n + 1 are introduced and studied.

42 citations

Journal ArticleDOI
TL;DR: In this article, the authors introduce a systematic study of contact structures with pseudo-Riemannian associated metrics, emphasizing analogies and differences with respect to the Riemannians case.
Abstract: We introduce a systematic study of contact structures with pseudo-Riemannian associated metrics, emphasizing analogies and differences with respect to the Riemannian case. In particular, we classify contact pseudo-metric manifolds of constant sectional curvature, three-dimensional locally symmetric contact pseudo-metric manifolds and three-dimensional homogeneous contact Lorentzian manifolds.

42 citations


Cites background or methods or result from "Riemannian Geometry of Contact and ..."

  • ...It is well known that a three-dimensional K -contact Riemannian manifold is Sasakian (see for example [4])....

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  • ...Following the argument exposed in Chapter 6 of [4], we consider M2n+1 × R and, denoting by (X, f d dt ) an arbitrary vector field on such manifold, the almost complex structure defined by...

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  • ...In particular, the Blair’s result (Theorem A of [3]) that K -contact Riemannian manifolds of dimension 2n + 1 * Corresponding author....

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  • ...If we consider the diffeomorphism f : R3 → R3, (x, y, z) → (x1, x2, x3) = (z cos x − y sin x,−z sin x − y cos x,−x), then ( f −1)∗η = η0 and ( f −1)∗ g0 = ḡ0R , where η0 = (1)2 (cos x3 dx1 + sin x3 dx2) and ḡ0R = (1)4 ∑ i dx 2 i [4]....

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  • ...It is well known that ξ = 2∂z and η = (1)2 (dz − y dx) in local Darboux coordinates (see for example [4])....

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Journal ArticleDOI
TL;DR: In this paper, the authors generalize the methods of (1-3) to 5-dimensional Riemannian manifolds M and study the relations between the geometry of M and the number of solu- tions to a generalized Killing spinor equation obtained from a 5D supergravity.
Abstract: In this note we generalize the methods of (1-3) to 5-dimensional Riemannian manifolds M. We study the relations between the geometry of M and the number of solu- tions to a generalized Killing spinor equation obtained from a 5-dimensional supergravity. The existence of 1 pair of solutions is related to almost contact metric structures. We also discuss special cases related to M = S 1 × M4, which leads to M being foliated by sub- manifolds with special properties, such as Quaternion-Kahler. When there are 2 pairs of solutions, the closure of the isometry sub-algebra generated by the solutions requires M to be S 3 or T 3 -fibration over a Riemann surface. 4 pairs of solutions pin down the geometry of M to very few possibilities. Finally, we propose a new supersymmetric theory for N = 1 vector multiplet on K-contact manifold admitting solutions to the Killing spinor equation.

42 citations


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

  • ...24) Together with the vector field sR and 1-form sκ, φλ defines an almost contact structure on M [22] (see also Appendix D)....

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