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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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TL;DR: Trans-S-manifolds as mentioned in this paper is a general class of metric f-Manifolds, which includes S- manifolds, C-MANIFolds, s-th Sasakian manifold and generalized Kenmotsu manifold.
Abstract: We introduce a new general class of metric f-manifolds which we call (nearly) trans-S-manifolds and includes S- manifolds, C-manifolds, s-th Sasakian manifolds and generalized Kenmotsu manifold studied previously. We prove their main properties and we present many examples which justify their study.

7 citations


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

  • ...Introduction In complex geometry, the relationships between the different classes of manifolds can be summarize in the well known diagram by Blair [3]:...

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  • ...In this context, D.E. Blair [2] defined K-manifolds (and particular cases of S-manifolds and C-manifolds)....

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  • ...The above theorem generalizes the result given by D.E. Blair and J.A. Oubiña in [4] for trans-Sasakian manifolds....

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  • ...In complex geometry, the relationships between the different classes of manifolds can be summarize in the well known diagram by Blair [3]: Complex metric // Hermitian dΩ=0 // Kaehler Almost Complex [J,J ]=0 OO metric // Almost Hermitian [J,J ]=0 OO dΩ=0 // ∇J=0 88 r r r r r r r r r r r Almost Kaehler [J,J ]=0 OO And the same for contact geometry: Normal Almost Contact metric // Normal Almost Contact Metric Φ=dη // Sasakian Almost Contact normal OO metric // Almost Contact Metric normal OO Φ=dη // (1) 66 ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ Contact Metric normal OO 2010 Mathematics Subject Classification....

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Journal ArticleDOI
TL;DR: This article improves the first Chen inequality for Legendrian submanifolds in Sasakian space forms, which was established in 1997 for C-totally real submanIFolds in Bigfoot space forms.
Abstract: Legendrian submanifolds in Sasakian space forms play an important role in contact geometry. Defever et al. (Boll Unione Mat Ital B 7(11):365–374, 1997) established the first Chen inequality for C-totally real submanifolds in Sasakian space forms. In this article, we improve this first Chen inequality for Legendrian submanifolds in Sasakian space forms.

7 citations


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

  • ...As examples of Sasakian space forms we mention R2m+1 and S2m+1 with standard Sasakian structures (see [1,9])....

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09 Dec 2015
TL;DR: In this paper, the authors make the first contribution to investigate under which conditions normal almost paracontact metric manifold of dimension 3 has cyclic parallel Ricci tensor, η-parallel R-Ricci tensors, Ricci-semisymmetry and locally ϕ-symmetry.
Abstract: In this study, we make the first contribution to investigate under which conditions normal almost paracontact metric manifold of dimension 3 has cyclic parallel Ricci tensor, η-parallel Ricci tensor, Ricci-semisymmetry and locally ϕ-symmetry. Finally, an example of a 3-dimensional normal almost paracontact metric manifold which is locally ϕ-symmetric and has cyclic parallel Ricci tensor is presented.

7 citations


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

  • ...Similarly to the class of almost contact metric manifolds [4], a normal almost paracontact metric manifold will be called para-Sasakian if F = dη [11] and quasi-para-Sasakian if dF = 0....

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Journal ArticleDOI
TL;DR: In this paper, the authors studied the class of metrics which are invariant along the infinitesimal generators of Legendre transformations and showed that the vanishing of the scalar curvature for this class results in a further differential equation for the metric function which is not compatible with the Legendre invariance constraint.
Abstract: The work within the Geometrothermodynamics programme rests upon the metric structure for the thermodynamic phase-space. Such structure exhibits discrete Legendre symmetry. In this work, we study the class of metrics which are invariant along the infinitesimal generators of Legendre transformations. We solve the Legendre-Killing equation for a $K$-contact general metric. We consider the case with two thermodynamic degrees of freedom, i.e. when the dimension of the thermodynamic phase-space is five. For the generic form of contact metrics, the solution of the Legendre-Killing system is unique, with the sole restriction that the only independent metric function -- $\Omega$ -- should be dragged along the orbits of the Legendre generator. We revisit the ideal gas in the light of this class of metrics. Imposing the vanishing of the scalar curvature for this system results in a further differential equation for the metric function $\Omega$ which is not compatible with the Legendre invariance constraint. This result does not allow us to use the regular interpretation of the curvature scalar as a measure of thermodynamic interaction for this particular class.

7 citations


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

  • ...Following the standard formulation of contact metric manifolds [11], a contact metric is typically written in blocks....

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Journal ArticleDOI
15 Oct 2020
TL;DR: In this paper, some soliton types on a quasi-Sasakian 3-manifold with respect to the Schouten-van Kampen connection were studied.
Abstract: In this paper we study some soliton types on a quasi-Sasakian 3-manifold with respect to the Schouten-van Kampen connection.

7 citations


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

  • ...A connected differentiable manifold M of dimension (2n+ 1) is called an almost contact metric manifold, if there exist tensor fields φ, ξ, η on M of types (1, 1), (1, 0), (0, 1), respectively, such that [2, 3, 35] φ(2) = −I + η ⊗ ξ, η(ξ) = 1, (2....

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