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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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30 Apr 2017
TL;DR: In this paper, the Fischer-Marsden equation has only trivial solution on an almost CoKähler manifold of dimension greater than 3 with ξ belonging to the (κ, μ)-nullity distribution and κ < 0.
Abstract: In this paper, we characterize the solutions of the Fischer-Marsden equation Lg(λ) = 0 on an almost CoKähler manifold. We prove that the Fischer-Marsden equation has only trivial solution on almost CoKähler manifold of dimension greater than 3 with ξ belonging to the (κ, μ)-nullity distribution and κ < 0.
5 citations
01 Jan 2012
TL;DR: In this paper, Ricci pseudosymmetric and Weyl semisymmetric generalized Sasakian-space-forms have been studied for quasi-umbilical hypersurfaces.
Abstract: The object of the present paper is to study Ricci pseudosymmetric and Weyl semisymmetric generalized Sasakian-space-forms. Quasi-umbilical hypersurfaces of generalized Sasakian-space-forms have also been studied.
5 citations
TL;DR: In this paper, the curvature and torsion of the Legendre and slant curves for the Bianchi-Cartan-Vranceanu metrics are characterized through the scalar product between the normal at the curve and the vertical vector field and in the helix case they have a proper (nonharmonic) mean curvature vector field.
Abstract: We study Legendre and slant curves for Bianchi-Cartan-Vranceanu metrics. These curves are characterized through the scalar product between the normal at the curve and the vertical vector field and in the helix case they have a proper (non-harmonic) mean curvature vector field. The general expression of the curvature and torsion of these curves and the associated Lancret invariant (for the slant case) are computed as well as the corresponding variant for some particular cases. The slant (particularly Legendre) curves which are helices are completely determined.
5 citations
Cites methods from "Riemannian Geometry of Contact and ..."
...Next, following [5], page 164, we recall the notion of a Frenet curve in a (2n+ 1)dimensional manifold: Let m be an integer with 1 6 m 6 2n + 1....
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...We choose this subject for two reasons: 1) the rôle of Legendre curves in almost contact geometry is remarkable and wellknown; in [5] the reader finds an excellent survey on these curves, 2) although the literature on Legendre curves is rich ([4], [6], [8], [17], [21], [26], [27]), slant curves have been studied until now only for the Sasakian geometry in [13], for the contact pseudo-Hermitian geometry in [15], for the f -Kenmotsu geometry in [11], in normal almost contact geometry in [9] and for warped products in [10]....
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TL;DR: In this article, the notion of radical transversal screen semi-slant light-like submanifolds of indefinite Sasakian manifolds was introduced and sufficient conditions for foliations determined by above distributions to be totally geodesic were obtained.
Abstract: In this paper, we introduce the notion of radical transversal screen semi-slant lightlike submanifolds of indefinite Sasakianmanifolds giving some non-trivial examples and give characterization theorem of such submanifolds. Integrability conditions of distributions D1, D2 and RadTM on radical transversal screen semi-slant lightlike submanifolds of indefinite Sasakian manifolds have been obtained. Further we obtain necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic. We also study mixed geodesic radical transversal screen semi-slant lightlike submanifolds of indefinite Sasakian manifolds.
5 citations
TL;DR: In this paper, the authors studied the geometric properties of almost contact metric submersions involving symplectic manifolds and showed that the structures of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are related to (1, 2)-symplectic structures.
Abstract: In this paper, we discuss some geometric properties of almost contact metric submersions involving symplectic manifolds. We show that the structures of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are related to (1, 2)-symplectic structures. For horizontally submersions of contact CR-submanifolds of quasi-K-cosymplectic and quasi-Kenmotsu manifolds, we study the principal characteristics and prove that their total spaces are CR-product. Curvature properties between curvatures of quasi-K-cosymplectic and quasi-Kenmotsu manifolds and the base spaces of such submersions are also established. We finally prove that, under a certain condition, the contact CR-submanifold of a quasi Kenmotsu manifold is locally a product of a totally geodesic leaf of an integrable horizontal distribution and a curve tangent to the normal distribution.
5 citations
Cites background from "Riemannian Geometry of Contact and ..."
...Symplectic and almost contact manifolds were treated in [4]....
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