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•

Hard Lefschetz Theorem for Vaisman manifolds

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Beniamino Cappelletti-Montano¹, Antonio De Nicola², Juan Carlos Marrero³, Ivan Yudin⁴•Institutions (4)

University of Cagliari¹, University of Salerno², University of La Laguna³, University of Coimbra⁴

16 Oct 2015-arXiv: Differential Geometry

TL;DR: In this article, the authors established a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds with respect to the Lee vector field.

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Abstract: We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) manifold of the first kind. We prove that the two notions are equivalent if there exists a Riemannian metric such that the Lee vector field is unitary and parallel and its metric dual $1$-form coincides with the Lee $1$-form. Finally, we discuss several examples of compact l.c.s. manifolds of the first kind which do not admit compatible Vaisman metrics.

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4 citations

Posted Content•

On special types of minimal and totally geodesic unit vector fields

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Alexander Yampolsky

30 Sep 2005-arXiv: Differential Geometry

TL;DR: In this article, a new equation with respect to a unit vector field on Riemannian manifold $M^n$ such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasaki metric was presented.

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Abstract: We present a new equation with respect to a unit vector field on Riemannian manifold $M^n$ such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasaki metric and apply it to some classes of unit vector fields. We introduce a class of covariantly normal unit vector fields and prove that within this class the Hopf vector field is a unique global one with totally geodesic property. For the wider class of geodesic unit vector fields on a sphere we give a new necessary and sufficient condition to generate a totally geodesic submanifold in $T_1S^n$.

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4 citations

Additional excerpts

...In a case of a general almost contact metric manifold (M̃ , ξ̃, η̃, φ̃, g̃) the following definition is known [7]....
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Foliations and contact structures

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Hamidou Dathe, Philippe Rukimbira

01 Jan 2004

TL;DR: In this paper, a notion of linear deformation of codimension one foliations into contact structures is introduced and a condition for a foliation defined by a closed nonsingular 1-form to be linearly early deformable into contact structure is given.

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Abstract: We introduce a notion of linear deformation of codimension one foliations into contact structures and describe some foliations which deform instantly into contact structures and some which do not Restricting ourselves to closed smooth manifolds, we obtain a neces- sary and su‰cient condition for a foliation defined by a closed nonsingular 1-form to be lin- early deformable into contact structures

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4 citations

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

...A contact manifold M with contact form a, Reeb vector field Z, partial almost complex operator J and contact metric g is said to be Kcontact [1] if Z is Killing relative to g. Suppose a0 is a nonsingular harmonic (relative to g) 1-form, then a0 is basic relative to the flow of Z [3]....
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...A contact manifold M with contact form a, Reeb vector field Z, partial almost complex operator J and contact metric g is said to be Kcontact [1] if Z is Killing relative to g....
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...Therefore, if a is a Kcontact form with contact metric g, then each Zt is a Killing vector field with respect to g....
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...In general, a contact structure on a 2nþ 1-dimensional manifold is a rank 2n tangent subbundle which is locally determined by contact forms (see Blair’s book [1] for more details about contact structures)....
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Journal Article•DOI•

Locally homogeneous aspherical Sasaki manifolds

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Oliver Baues¹, Yoshinobu Kamishima²•Institutions (2)

University of Fribourg¹, Josai University²

01 Jun 2020-Differential Geometry and Its Applications

TL;DR: In this paper, it was shown that a compact locally homogeneous aspherical Sasaki manifold is always quasi-regular, that is, Γ \ G/H is an S 1 -Seifert bundle over a locally homogenous aspherial Kahler orbifold.

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Abstract: Let G / H be a contractible homogeneous Sasaki manifold. A compact locally homogeneous aspherical Sasaki manifold Γ \ G / H is by definition a quotient of G / H by a discrete uniform subgroup Γ ≤ G . We show that a compact locally homogeneous aspherical Sasaki manifold is always quasi-regular, that is, Γ \ G / H is an S 1 -Seifert bundle over a locally homogeneous aspherical Kahler orbifold. We discuss the structure of the isometry group Isom ( G / H ) for a Sasaki metric of G / H in relation with the pseudo-Hermitian group Psh ( G / H ) for the Sasaki structure of G / H . We show that a Sasaki Lie group G, when Γ \ G is a compact locally homogeneous aspherical Sasaki manifold, is either the universal covering group of SL ( 2 , R ) or a modification of a Heisenberg nilpotent Lie group with its natural Sasaki structure. We also show that any compact regular aspherical Sasaki manifold with solvable fundamental group is finitely covered by a Heisenberg manifold and its Sasaki structure may be deformed to a locally homogeneous one. In addition, we classify all aspherical Sasaki homogeneous spaces for semisimple Lie groups.

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4 citations

Journal Article•DOI•

On the classification of almost contact metric manifolds

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Francisco Martín Cabrera¹•Institutions (1)

University of La Laguna¹

01 Jun 2019-Differential Geometry and Its Applications

TL;DR: In this article, the existence of 132 Chinea and Gonzalez-Davila types of almost contact metric structures is proved on connected manifolds of dimension higher than three, which is a consequence of some interrelations among components of the intrinsic torsion of an almost contact distance metric structure.

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Abstract: On connected manifolds of dimension higher than three, the non-existence of 132 Chinea and Gonzalez-Davila types of almost contact metric structures is proved. This is a consequence of some interrelations among components of the intrinsic torsion of an almost contact metric structure. Such interrelations allow to describe the exterior derivatives of some relevant forms in the context of almost contact metric geometry.

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4 citations

Additional excerpts

...Some of these classes are referred to, by diverse authors [1, 2], as: {ξ = 0} = cosymplectic manifolds or integrable almost contact metric structure, C1 = nearly-K-cosymplectic manifolds, C5 = α-Kenmotsu manifolds, C6 = α-Sasakian manifolds, C5 ⊕ C6 = trans-Sasakian manifolds, C2 ⊕ C9 = almost cosymplectic manifolds, C6 ⊕ C7 = quasi-Sasakian manifolds, C1 ⊕ C5 ⊕ C6 = nearly-trans-Sasakian manifolds, C1 ⊕ C2 ⊕ C9 ⊕ C10 = quasi-K-cosymplectic manifolds, C3 ⊕ C4 ⊕ C5 ⊕ C6 ⊕ C7 ⊕ C8 = normal manifolds, C3 ⊕ C4 ⊕ C5 ⊕ C8 = integrable almost contact structure, etc....
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Riemannian Geometry of Contact and Symplectic Manifolds

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