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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
Martin Wolf1
TL;DR: In this paper, a twisted supersymmetric Yang-Mills theory on the five-dimensional sphere using localisation techniques is presented, and the twistor construction of this equation when formulated on K-contact manifolds and the discussion of its integrability properties is discussed.
Abstract: Recently, Kallen and Zabzine computed the partition function of a twisted supersymmetric Yang-Mills theory on the five-dimensional sphere using localisation techniques. Key to their construction is a five-dimensional generalisation of the instanton equation to which they refer as the contact instanton equation. Subject of this article is the twistor construction of this equation when formulated on K-contact manifolds and the discussion of its integrability properties. We also present certain extensions to higher dimensions and supersymmetric generalisations.

16 citations

Journal ArticleDOI
01 Jan 2019-Filomat
TL;DR: In this paper, it was shown that in a quasi-Sasakian 3-manifold admitting Ricci soliton, the structure function is a constant, which is the same as in the present paper.
Abstract: The object of the present paper is to prove that in a quasi-Sasakian 3-manifold admitting ?-Ricci soliton, the structure function ? is a constant. As a consequence we obtain several important results.

16 citations


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

  • ...Let M be a (2n + 1)-dimensional connected differentiable manifold endowed with an almost contact metric structure (φ, ξ, η, 1), where φ is a tensor field of type (1, 1), ξ is a vector field, η is a 1-form and 1 is the Riemannian metric on M such that ([4], [5]) φ2X = −X + η(X)ξ, (5)...

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Journal ArticleDOI
TL;DR: In this article, an odd-dimensional version of the Goldberg conjecture was formulated and proved by using an orbifold analogue of Sekigawa's arguments in [8], and an approximation argument of K -contact structures with quasiregular ones.
Abstract: An odd-dimensional version of the Goldberg conjecture was formulated and proved in [5], by using an orbifold analogue of Sekigawa's arguments in [8], and an approximation argument of K -contact structures with quasiregular ones. We provide here another proof of this result. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

16 citations


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

  • ...Since Ric = 2ng, it follows that (g, ξ′) is K–contact (see [2], Theorem 7....

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Journal ArticleDOI
TL;DR: In this paper, the authors obtained a necessary condition for a three dimensional invariant submanifold of a Kenmotsu manifold to be totally geodesic, where S, R are the Ricci tensor and curvature tensor respectively and α is the second fundamental form.
Abstract: The object of the present paper is to obtain a necessary condition for a three dimensional invariant submanifold of a Kenmotsu manifold to be totally geodesic. Besides this we study an invariant submanifold of Kenmotsu manifolds satisfying Q(α, R) = 0 and Q(S, α) = 0, where S, R are the Ricci tensor and curvature tensor respectively and α is the second fundamental form. Finally, we construct an example to verify our results.

16 citations

Journal ArticleDOI
30 Mar 2015-Filomat
TL;DR: The notion of F-geodesics was introduced in this paper, which generalizes the magnetic curves, and implicitly the geodesics, by using any (1, 1)-tensor field on the manifold (in particular the electro-magnetic field or the Lorentz force).
Abstract: The notion of F-geodesic, which is slightly different from that of F-planar curve (see [14], [15]), generalizes the magnetic curves, and implicitly the geodesics, by using any (1,1)-tensor field on the manifold (in particular the electro-magnetic field or the Lorentz force). We give several examples of F-geodesics and the characterizations of the F-geodesics w.r.t. Vranceanu connections on foliated manifolds and adapted connections on almost contact manifolds. We generalize the classical projective transformation, holomorphic-projective transformation and C-projective transformation, by considering a pair of symmetric connections which have the same F-geodesics. We deal with the transformation between such two connections, called F-projective transformation. We obtain a Weyl type tensor field, invariant under any F-projective transformation, on a 1-codimensional foliation.

16 citations


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

  • ...[7], [9]), as well as the almost para-contact ones are used in physics (see [4], [20])....

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  • ...special connections, namely Vranceanu connections (see [6], [30]) on foliated manifolds and adapted connections (see [15]) on almost contact manifolds (see [7])....

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  • ...(23) From (23) one may easily deduce that φξ = 0 and η ◦ φ = 0 (see [7])....

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