Riemannian Geometry of Contact and Symplectic Manifolds

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Riemannian Geometry of Contact and Symplectic Manifolds

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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.

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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 Article•DOI•

A New Class of Golden Riemannian Manifold

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Gherici Beldjilali

30 Jan 2020

TL;DR: In this article, a new class of almost Golden Riemannian structures and their essential examples as well as their fundamental properties are studied. And a particular type belonging to this class is investigated.

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Abstract: In this paper, we introduce a new class of almost Golden Riemannian structures and study their essential examples as well as their fundamental properties. Next, we investigate a particular type belonging to this class and we establish some basic results for Riemannian curvature tensor and the sectional curvature. Concrete examples are given.

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3 citations

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

...For more details on almost contact structures see ([3], [4], [12])....
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Journal Article•DOI•

Nearly Sasakian manifolds revisited

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Beniamino Cappelletti-Montano¹, Antonio De Nicola², Giulia Dileo³, Ivan Yudin⁴•Institutions (4)

University of Cagliari¹, University of Salerno², University of Bari³, University of Coimbra⁴

02 Apr 2019-arXiv: Differential Geometry

TL;DR: In this paper, the authors provided a self-contained and more conceptual proof of the result that an almost contact metric manifold of dimension greater than 5 is Sasakian if and only if it is nearly SSS.

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Abstract: We provide a new, self-contained and more conceptual proof of the result that an almost contact metric manifold of dimension greater than 5 is Sasakian if and only if it is nearly Sasakian.

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3 citations

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

...Let us denote by s the inclusion Σq1 into Σq1+1 defined by s(σ)(i) = σ(i − 1) + 1, i ≥ 2, s (σ) (1) = 1....
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...A relaxation of this notion was introduced by Blair, Showers and Yano in [2], under the name of nearly Sasakian manifolds, by requiring that just the symmetric part of (1) vanishes....
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...In particular one can show that an almost contact metric structure (g, φ, ξ, η) is Sasakian if and only if the covariant derivative of the endomorphism φ satisfies (∇Xφ)Y − g(X, Y )ξ + η(Y )X = 0, (1) for all vector fields X, Y ∈ Γ(TM)....
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...Then the tensor iφR̃ has the following symmetries (iφR̃)(1 + (1, 2)) = 0 (iφR̃)(1− (1, 3)(2, 4)) = 0...
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...Also (R̃ ◦ (1, 4, 3, 2))(X, Y, Z,W ) = R̃(Y, Z,W,X) = g(RY,ZW,X) = g(X,RY,ZW ) = (g ◦R)(X, Y, Z,W ),...
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Journal Article•DOI•

Biharmonic submanifolds in non-Sasakian contact metric 3-manifolds

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Michael Markellos¹, Vassilis J. Papantoniou¹•Institutions (1)

University of Patras¹

01 Mar 2011-Kodai Mathematical Journal

TL;DR: In this article, a geometric interpretation of 3-dimensional generalized (κ, μ, ν)-contact metric manifolds in terms of its Legendre curves is given, and a biharmonic anti-invariant surface with constant norm of the mean curvature vector field immersed in these spaces is studied.

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Abstract: In this paper, we characterize biharmonic Legendre curves in 3-dimensional (κ, μ, ν)-contact metric manifolds. Moreover, we give examples of Legendre geodesics in these spaces. We also give a geometric interpretation of 3-dimensional generalized (κ, μ)-contact metric manifolds in terms of its Legendre curves. Furthermore, we study biharmonic anti-invariant surfaces of 3-dimensional generalized (κ, μ)-contact metric manifolds with constant norm of the mean curvature vector field. Finally, we give examples of anti-invariant surfaces with constant norm of the mean curvature vector field immersed in these spaces.

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3 citations

Journal Article•DOI•

Real hypersurfaces with a special transversal vector field

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Zuzanna Szancer¹•Institutions (1)

University of Agriculture, Faisalabad¹

01 Jan 2012-Annales Polonici Mathematici

TL;DR: In this paper, real affine hypersurfaces of the complex space C are studied and properties of the structure determined by a J-tangent transversal vector field are proved.

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Abstract: Real affine hypersurfaces of the complex space C are studied. Some properties of the structure determined by a J-tangent transversal vector field are proved. Moreover, some generalizations of the results obtained by V. Cruceanu are given.

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3 citations

Additional excerpts

...First, we recall some definitions from [B]....
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Posted Content•

Sasakian structures on CR-manifolds

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Liviu Ornea¹, Liviu Ornea², Misha Verbitsky³•Institutions (3)

Romanian Academy¹, University of Bucharest², University of Glasgow³

06 Jun 2006-arXiv: Differential Geometry

TL;DR: In this article, the authors define a contact manifold as a quotient of a symplectic manifold by a proper, free action of $\R^{>0}$, with the symplectic form homogeneous of degree 2.

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Abstract: A contact manifold $M$ can be defined as a quotient of a symplectic manifold $X$ by a proper, free action of $\R^{>0}$, with the symplectic form homogeneous of degree 2. If $X$ is, in addition, Kaehler, and its metric is also homogeneous of degree 2, $M$ is called Sasakian. A Sasakian manifold is realized naturally as a level set of a Kaehler potential on a complex manifold, hence it is equipped with a pseudoconvex CR-structure. We show that any Sasakian manifold $M$ is CR-diffeomorphic to an $S^1$-bundle of unit vectors in a positive line bundle on a projective K\"ahler orbifold. This induces an embedding from $M$ to an algebraic cone $C$. We show that this embedding is uniquely defined by the CR-structure. Additionally, we classify the Sasakian metrics on an odd-dimensional sphere equipped with a standard CR-structure.

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3 citations

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

...1 A Riemannian manifold (M, h) of odd real dimension is called Sasakian if the metric cone C(M) = (M × R>0, t2h + dt2) is equipped with a dilatation-invariant complex structure, which makes C(M) a Kähler manifold (see [5,7])....
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