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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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01 Jan 2010
TL;DR: In this paper, it was shown that every Einstein compact almost C-manifold M 2n+s whose Reeb vector fields are Killing is a C manifold.
Abstract: We prove that every Einstein compact almost C-manifold M 2n+s whose Reeb vector fields are Killing is a C-manifold. Then we extend this result considering some generalizations of the Einstein condition (η-Einstein, generalized quasi Einstein, etc.). Moreover, we find some topological properties of compact almost C-manifolds under the assumption that the Ricci tensor is transversally positive definite and the Reeb vector fields are Killing, namely we prove that the first Betti number is s and the first fundamental group is isomorphic to Z s . Finally, a splitting theorem for cosymplectic manifolds is found.

14 citations


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

  • ...[6]) that an almost contact metric manifold is cosymplectic if and only if ∇φ = 0, ∇ denoting the Levi-Civita connection....

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Journal ArticleDOI
Giulia Dileo1
TL;DR: In this paper, the authors characterize almost contact metric manifolds which are CR-integrable almost α-Kenmotsu manifolds, through the existence of a canonical linear connection, invariant under $\mathscr{D}$-homothetic deformations.
Abstract: We study $\mathscr{D}$-homothetic deformations of almost α-Kenmotsu structures. We characterize almost contact metric manifolds which are CR-integrable almost α-Kenmotsu manifolds, through the existence of a canonical linear connection, invariant under $\mathscr{D}$-homothetic deformations. If the canonical connection associated to the structure (φ, ξ, η, g) has parallel torsion and curvature, then the local geometry is completely determined by the dimension of the manifold and the spectrum of the operator h′ defined by 2αh′ = ($\mathscr{L}$ξφ) $\circ$ φ. In particular, the manifold is locally equivalent to a Lie group endowed with a left invariant almost α-Kenmotsu structure. In the case of almost α-Kenmotsu (κ, μ)′-spaces, this classification gives rise to a scalar invariant depending on the real numbers κ and α.

14 citations

Posted Content
TL;DR: In this paper, the authors introduce anti-invariant Riemannian submersions from Sasakian manifolds onto Riemanian manifold, and present necessary and sufficient conditions for such submersion to be totally geodesic and harmonic.
Abstract: We introduce anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on Sasakian manifolds. We investigate necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. We give examples of anti-invariant submersions such that characteristic vector field {\xi} is vertical or horizontal.

14 citations

Journal ArticleDOI
TL;DR: In this article, the first and second variation formulas for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds were derived and a necessary condition for the stability of a non-singular surface in a 3-manifold was obtained in terms of the pseudohermitians torsion and the Webster scalar curvature.
Abstract: We calculate the first and the second variation formula for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that can move the singular set of a C^2 surface and non-singular variation for C_H^2 surfaces. These formulas enable us to construct a stability operator for non-singular C^2 surfaces and another one for C2 (eventually singular) surfaces. Then we can obtain a necessary condition for the stability of a non-singular surface in a pseudo-hermitian 3-manifold in term of the pseudo-hermitian torsion and the Webster scalar curvature. Finally we classify complete stable surfaces in the roto-traslation group RT .

14 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that a complete K-contact manifold admitting both the Einstein-Weyl structures W± = (g, ±ω) is Sasakian.
Abstract: In this paper we study Einstein-Weyl structures in the framework of contact metric manifolds. First, we prove that a complete K-contact manifold admitting both the Einstein-Weyl structures W± = (g, ±ω) is Sasakian. Next, we show that a compact contact metric manifold admitting an Einstein-Weyl structure is either K-contact or the dual field of ω is orthogonal to the Reeb vector field, provided the Reeb vector field is an eigenvector of the Ricci operator. We also prove that a contact metric manifold admitting both the Einstein-Weyl structures and satisfying \({Q\varphi = \varphi Q}\) is either K-contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, fη) are presented.

14 citations


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

  • ...(see [4]). A Sasakian manifold is K-contact, but the converse is not true, except in dimension 3. For a K-contact manifold we have [ 1 ]...

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  • ...Conformally flat contact metric manifold has been studied by several authors (see [ 1 ], [20])....

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  • ...lξ = 0, Tr ϕ = Trh = Tr ϕh = 0, hϕ =− ϕh. For a contact metric manifold, we also have [ 1 ]...

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  • ...conformally flat M is of constant curvature 1 (see [ 1 ])....

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  • ...Because it holds for many known examples, such as K-contact manifold, tangent sphere bundle of a Riemannian manifold of constant curvature and also on some 3-dimensional contact metric manifolds (see [ 1 ])....

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