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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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3 citations


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

  • ...1) is a consequence of the other conditions [4, 5]....

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Posted Content
23 Jun 2020
TL;DR: In this paper, it was shown that a Legendre transformation is a mere change of symplectic polarization from the point of view of contact geometry and it is not possible to find a class of metric tensors which fulfills two properties: on the one hand, to be polarization independent, and on the other, to induce a Hessian metric into the corresponding Legendre submanifolds.
Abstract: In this work we show that a Legendre transformation is nothing but a mere change of symplectic polarization from the point of view of contact geometry Then, we construct a set of Riemannian and pseudo-Riemannian metrics on a contact manifold by introducing almost contact and para-contact structures and we analyze their isometries We show that it is not possible to find a class of metric tensors which fulfills two properties: on the one hand, to be polarization independent ie the Legendre transformations are the corresponding isometries and, on the other, that it induces a Hessian metric into the corresponding Legendre submanifolds This second property is motivated by the well known Riemannian structures of the geometric description of thermodynamics which are based on Hessian metrics on the space of equilibrium states and whose properties are related to the fluctuations of the system We find that to define a Riemannian structure with such properties it is necessary to abandon the idea of an associated metric to an almost contact or para-contact structure We find that even extending the contact metric structure of the thermodynamic phase space the thermodynamic desiderata cannot be fulfilled

3 citations


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

  • ...A similar structure arises in almost contact manifolds [25] (resp....

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  • ...We say that a metric tensor is a compatible metric if it satisfies [25, 28] g (φX, φY ) = g(X,Y )− η(X)η(Y ) (...

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  • ...In the literature, the vector field satisfying (8) is called the Reeb vector field [25]....

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  • ...Almost Contact and Almost para-contact Structures An almost contact structure is a triplet (η, ξ, φ) consisting of a contact 1-form η, its corresponding Reeb vector field ξ and an automorphism φ : TT −→ TT such that [25] φ(2) = φ ◦ φ = −1+ η ⊗ ξ with φ(ξ) = 0 and η ◦ φ = 0....

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  • ...Thus, around each point p ∈ T there is a local set of coordinates {w, q, pa} in which the the 1-form η is written as [25] η = dw − n ∑...

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01 Jan 2011
TL;DR: In this paper, Baikousssis and Verstlraelen gave a complete characterization of surfaces with paralel second fundamental form in 3-dimensional Bianchi-Cartan-Vranceanu spaces (BCV).
Abstract: C. Baikousssis, D.E. Blair[1] made a study of Legendre curves in contact metric manifolds. J. I. Inoguchi, T. Kumamoto, N. Ohsugi, and Y. Suyama[2] studied fundamental properties of Heisenberg 3-spaces. M. Belkhelfa, I.E. Hirica, R. Rosca, L. Verstlraelen[3] obtained a complete characterization of surfaces with paralel second fundamental form in 3-dimensional Bianchi-Cartan-Vranceanu spaces(BCV). In this paper, making use of method in paper of C. Baikousssis, D.E. Blair and M. Belkhelfa, I.E. Hirica, R. Rosca, L. Verstlraelen we obtained helices and their characterizations in BCV-Sasakian spaces such that the circular helices in BCV−Sasakian space correspond to the circles in E 3 , the circular helices in Eucliden space correspond to the circular helices in BCV−Sasakian space and these helices are nongeodesical BCV− Legendre curves. We have seen calculable that the covariant derivative of vector field Y with respect to vector field X without christoffel symbols in BCV-Sasakian spaces. Also we obtained more general curvature κ and torsion τ of a curve γ in BCV-Sasakian spaces.

3 citations


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

  • ...Moreover, if the tensor N (1) : χ(M)× χ(M) −→ χ(M) (X, Y ) −→ N (X, Y ) = [φ, φ](X, Y ) + 2dη(X, Y )ξ on the Sasakian manifold (M, η, ξ, φ, g) vanishes then the tensor N (1) is called Sasakian tensor and the contact manifold (M, η, ξ, φ, g) is called Sasakian manifold [3]....

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  • ...1−dimensional integral submanifold of Dm is called a Legendre curve [3]....

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  • ...Verstlraelen[3] obtained a complete characterization of surfaces with paralel second fundamental form in 3-dimensional Bianchi-Cartan-Vranceanu spaces(BCV)....

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Journal ArticleDOI
TL;DR: In this article, the authors studied the closed Einstein-Weyl structure on compact K-contact manifolds and proved that a compact k-contact manifold admitting a closed EWE structure is a Sasakian manifold.
Abstract: We study closed Einstein-Weyl structure on compact K-contact manifolds and prove that a compact K-contact manifold admitting a closed Einstein-Weyl structure is Einstein and Sasakian.

3 citations


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

  • ...The following formulas are valid for a K-contact (Sasakian) manifold ( see [1]):...

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  • ...For details about contact metric manifolds we refer to [1]....

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Posted Content
28 Apr 2020
TL;DR: In this paper, it was shown that if a $(2n + 1)$-dimensiinal $(k,\mu)'$-almost Kenmotsu manifold admits the Ricci soliton, then the manifold is locally isometric to a Ricci flat manifold.
Abstract: The goal of this paper is to characterize a class of almost Kenmotsu manifolds admitting $\ast$-conformal Ricci soliton. It is shown that if a $(2n + 1)$-dimensiinal $(k,\mu)'$-almost Kenmotsu manifold $M$ admits $\ast$-conformal Ricci soliton, then the manifold $M$ is $\ast$-Ricci flat and locally isometric to $\mathbb{H}^{n+1}(-4) \times \mathbb{R}^n$. The result is also verified by an example.

3 citations