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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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Journal ArticleDOI
TL;DR: In this article, the formality of the mapping torus of an orientation-preserving diffeomorphism of a manifold is studied and conditions under which such a torus has a non-zero Massey product are given.
Abstract: We study the formality of the mapping torus of an orientation-preserving diffeomorphism of a manifold. In particular, we give conditions under which a mapping torus has a non-zero Massey product. As an application we prove that there are non-formal compact co-symplectic manifolds of dimension m and with first Betti number b if and only if m = 3 and b >= 2, or m >= 5 and b >= 1. Explicit examples for each one of these cases are given.

28 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied a para-Sakian manifold whose metric g is an η-Ricci soliton (g,V ) and almost η Ricci solitons.
Abstract: In this paper, we study para-Sasakian manifold (M,g) whose metric g is an η-Ricci soliton (g,V ) and almost η-Ricci soliton. We prove that, if g is an η-Ricci soliton, then either M is Einstein and...

28 citations

Journal ArticleDOI

28 citations


Additional excerpts

  • ...(see [2, 10])....

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Journal ArticleDOI
01 Jan 2012-Filomat
TL;DR: In this article, it was shown that if (1,V, λ) is a Ricci soliton where V is collinear with the characteristic vector field ξ, then V is a constant multiple of ξ and the manifold is of constant scalar curvature provided α, β =constant.
Abstract: The object of the present paper is to study 3-dimensional trans-Sasakian manifolds admitting Ricci solitons and gradient Ricci solitons. We prove that if (1,V, λ) is a Ricci soliton where V is collinear with the characteristic vector field ξ, then V is a constant multiple of ξ and the manifold is of constant scalar curvature provided α, β =constant. Next we prove that in a 3-dimensional trans-Sasakian manifold with constant scalar curvature if 1 is a gradient Ricci soliton, then the manifold is either a β-Kenmotsu manifold or an Einstein manifold. As a consequence of this result we obtain several corollaries.

27 citations


Additional excerpts

  • ...Let M be a connected almost contact metric manifold with an almost contact metric structure(φ, ξ, η, 1), that is, φ is an (1, 1)-tensor field , ξ is a vector field , η is a 1-form and 1 is the compatible Riemannian metric such that φ2(X) = −X + η(X)ξ, η(ξ) = 1, φξ = 0, η ◦ φ = 0, 1(φX, φY) = 1(X,Y) − η(X)η(Y), 1(X, φY) = −1(φX,Y), 1(X, ξ) = η(X), for all X,Y ∈ TM ([4], [5])....

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Journal ArticleDOI
TL;DR: In this paper, it was shown that the Haydys-Witten equations can be embedded into the 4D anti-self-dual instanton equations, and the latter equations can reduce to the former by proving some vanishing theorems.
Abstract: On a five dimensional simply connected Sasaki-Einstein manifold, one can construct Yang-Mills theories coupled to matter with at least two supersymmetries. The partition function of these theories localises on the contact instantons, however the contact instanton equations are not elliptic. It turns out that these equations can be embedded into the Haydys-Witten equations (which are elliptic) in the same way the 4D anti-self-dual instanton equations are embedded in the Vafa-Witten equations. We show that under some favourable circumstances, the latter equations will reduce to the former by proving some vanishing theorems. It was also known that the Haydys-Witten equations on product manifolds $M_5=M_4\times \mathbb{R}$ arise in the context of twisting the 5D maximally supersymmetric Yang-Mills theory. In this paper, we present the construction of twisted $N=2$ Yang-Mills theory on Sasaki-Einstein manifolds, and more generally on $K$-contact manifolds. The localisation locus of this new theory thus provides a covariant version of the Haydys-Witten equation.

27 citations


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

  • ...For other equivalent definitions and proofs the reader may consult the book [32]....

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  • ...The construction is similar to the symplectic case, since dκ serves as a symplectic structure on ξ, see [32] for more details....

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