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•

Lengths of contact isotopies and extensions of the hofer metric

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Augustin Banyaga¹, Paul Donato²•Institutions (2)

Pennsylvania State University¹, University of Provence²

20 Jul 2006-Annals of Global Analysis and Geometry

TL;DR: In this paper, the Hofer metric on Ham(M) has a right-invariant (but not left invariant) extension to the identity component in the groups of strictly contact diffeomorphisms of a compact regular contact manifold.

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Abstract: Using the Hofer metric, we construct, under a certain condition, a bi-invariant distance on the identity component in the group of strictly contact diffeomorphisms of a compact regular contact manifold. We also show that the Hofer metric on Ham(M) has a right-invariant (but not left invariant) extension to the identity component in the groups of symplectic diffeomorphisms of certain symplectic manifolds.

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30 citations

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

...We refer to [ 4 ] for the basic notions of contact geometry recalled below....
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...AUGUSTIN BANYAGA AND PAUL DONATO Moreover α is connection form on M [ 4 ]....
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Journal Article•DOI•

Almost Kenmotsu manifolds with conformal Reeb foliation

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Anna Maria Pastore, Vincenzo Saltarelli

01 Nov 2011-Bulletin of The Belgian Mathematical Society-simon Stevin

TL;DR: In this paper, the authors consider almost Kenmotsu manifolds with conformal Reeb foliation and prove that such a foliation produces harmonic morphisms, study the $k$-nullity distributions and discuss the isometrical immersion of such a manifold as hypersurface in a real space form of constant curvature.

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Abstract: We consider almost Kenmotsu manifolds with conformal Reeb foliation. We prove that such a foliation produces harmonic morphisms, we study the $k$-nullity distributions and we discuss the isometrical immersion of such a manifold $M$ as hypersurface in a real space form $\widetilde{M}(c)$ of constant curvature $c$ proving that $c \leq -1$ and, if $c<-1$, $M$ is totally umbilical, Kenmotsu and locally isometric to the hyperbolic space of constant curvature $-1$. Finally, the Einstein and $\eta$-Einstein conditions are discussed.

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30 citations

Journal Article•DOI•

Ricci Solitons on Almost Co-Kähler Manifolds

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Yaning Wang¹•Institutions (1)

Henan Normal University¹

01 Dec 2019-Canadian Mathematical Bulletin

TL;DR: In this paper, it was shown that if an almost co-Kahler manifold of dimension greater than three satisfying rigid motions of the Minkowski 2-space can be shown to have a rigid motion.

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Abstract: In this paper, we prove that if an almost co-Kahler manifold of dimension greater than three satisfying of rigid motions of the Minkowski 2-space.

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30 citations

Journal Article•DOI•

Biharmonic legendre curves in sasakian space forms

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Dorel Fetcu

31 Mar 2008-Journal of The Korean Mathematical Society

TL;DR: Biharmonic Legendre curves in a Sasakian space form are studied in this article, and a non-existence result in a 7-dimensional 3-Sasakian manifold is obtained.

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Abstract: Biharmonic Legendre curves in a Sasakian space form are studied. A non-existence result in a 7-dimensional 3-Sasakian manifold is obtained. Explicit formulas for some biharmonic Legendre curves in the 7-sphere are given.

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30 citations

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

...It is proved that every 3-contact structure is 3-Sasakian, (see [3])....
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...(ii) If n = 1 then χ2 = 1 and g(N1, φT ) = ±1, (see [3])....
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...In [3] it is proved that an almost contact metric structure (φ, ξ, η, g) is Sasakian if and only if (∇Xφ)Y = g(X, Y )ξ − η(Y )X, where ∇ is the Levi-Civita connection of g....
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...Concerning the Sasakian manifolds and the Legendre curves let us recall some notions and results as they are presented in [3]....
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...If the tensor field S, of type (1,2), defined by S = Nφ+2dη⊗ξ, where Nφ(X,Y ) = [φX,φY ]−φ[φX, Y ]−φ[X, φY ]+φ(2)[X, Y ], is the Nijenhuis tensor field of φ, vanishes, then the almost contact structure is said to be normal (for more details see [3])....
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Journal Article•DOI•

Almost f-Cosymplectic Manifolds

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Nesip Aktan¹, Mustafa Yildirim¹, Cengizhan Murathan²•Institutions (2)

Düzce University¹, Uludağ University²

01 May 2014-Mediterranean Journal of Mathematics

TL;DR: In this paper, a new class of contact manifolds called almost f-cosymplectic manifolds are studied and several tensor conditions are studied for such type of manifolds.

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Abstract: The purpose of this paper is to study a new class of contact manifolds. Such manifolds are called almost f-cosymplectic manifolds. Several tensor conditions are studied for such type of manifolds. We conclude our results with two examples of almost f-cosymplectic manifolds.

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30 citations