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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Nullity conditions in paracontact geometry

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B. Cappelletti Montano¹, I. Küpeli Erken², Cengizhan Murathan²•Institutions (2)

University of Cagliari¹, Uludağ University²

01 Dec 2012-Differential Geometry and Its Applications

TL;DR: In this paper, a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition (1.2) for some real numbers κ ˜ and μ ˜ ).

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Abstract: The paper is a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition (1.2), for some real numbers κ ˜ and μ ˜ ). This class of pseudo-Riemannian manifolds, which includes para-Sasakian manifolds, was recently defined in Cappelletti Montano (2010) [13] . In this paper we show in fact that there is a kind of duality between those manifolds and contact metric ( κ , μ ) -spaces. In particular, we prove that, under some natural assumption, any such paracontact metric manifold admits a compatible contact metric ( κ , μ ) -structure (eventually Sasakian). Moreover, we prove that the nullity condition is invariant under D -homothetic deformations and determines the whole curvature tensor field completely. Finally non-trivial examples in any dimension are presented and the many differences with the contact metric case, due to the non-positive definiteness of the metric, are discussed.

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60 citations

Posted Content•

Structure theorem for compact Vaisman manifolds

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Liviu Ornea¹, Misha Verbitsky²•Institutions (2)

University of Bucharest¹, University of Glasgow²

18 May 2003-arXiv: Differential Geometry

TL;DR: A locally conformally Kaehler (l.c.k.) manifold admits a holomorphic flow acting by non-trivial homotheties on the manifold.

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Abstract: A locally conformally Kaehler (l.c.K.) manifold is a complex manifold admitting a Kaehler covering $\tilde M$, with each deck transformation acting by Kaehler homotheties. A compact l.c.K. manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on $\tilde M$. We prove a structure theorem for compact Vaisman manifolds. Every compact Vaisman manifold M is fibered over a circle, the fibers are Sasakian, the fibration is locally trivial, and M is reconstructed as a Riemannian suspension from the Sasakian structure on the fibers and the monodromy automorphism induced by this fibration. This construction is canonical and functorial in both directions.

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59 citations

Additional excerpts

...To describe examples of Vaisman manifolds we need to recall the notion of Sasakian manifold (see [Bl] and [BG1] for a survey and references on Sasakian geometry)....
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Journal Article•DOI•

Killing-Yano tensors and some applications

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O. P. Santillan

31 Jul 2011-arXiv: High Energy Physics - Theory

TL;DR: In this paper, the role of the Killing and Yano tensors for studying the geodesic motion of the particle and the superparticle in a curved background is reviewed, and the Papadopoulos list for G structures is reproduced by studying the torsion types these structures admit.

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Abstract: The role of Killing and Killing-Yano tensors for studying the geodesic motion of the particle and the superparticle in a curved background is reviewed. Additionally the Papadopoulos list [74] for Killing-Yano tensors in G structures is reproduced by studying the torsion types these structures admit. The Papadopoulos list deals with groups G appearing in the Berger classification, and we enlarge the list by considering additional G structures which are not of the Berger type. Possible applications of these results in the study of supersymmetric particle actions and in the AdS/CFT correspondence are outlined.

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59 citations

Journal Article•DOI•

Magnetic curves in Sasakian manifolds

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Simona-Luiza Druţă-Romaniuc¹, Jun-ichi Inoguchi², Marian Ioan Munteanu¹, Ana Irina Nistor¹•Institutions (2)

Alexandru Ioan Cuza University¹, University of Tsukuba²

06 Jan 2021-Journal of Nonlinear Mathematical Physics

TL;DR: In this paper, the authors classified the magnetic trajectories corresponding to contact magnetic fields in Sasakian manifolds of arbitrary dimension and proved that the codimension of the magnetic curve may be reduced to 2.

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Abstract: In this paper we classify the magnetic trajectories corresponding to contact magnetic fields in Sasakian manifolds of arbitrary dimension. Moreover, when the ambient is a Sasakian space form, we prove that the codimension of the curve may be reduced to 2. This means that the magnetic curve lies on a 3-dimensional Sasakian space form, embedded as a totally geodesic submanifold of the Sasakian space form of dimension (2n+1).

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57 citations

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

...Nevertheless, in this paper we use an equivalent definition for Sasakian manifolds, according to [11], which is more appropriate for our study....
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...Extending this theory to curves in a Riemannian manifold (M,g) of arbitrary dimension, let us recall the notion of Frenet curve of osculating order r, where r ≥ 1, according to [11]....
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Journal Article•DOI•

Existence of isoperimetric regions in contact sub-Riemannian manifolds

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Matteo Galli¹, Manuel Ritoré¹•Institutions (1)

University of Granada¹

15 Jan 2013-Journal of Mathematical Analysis and Applications

TL;DR: In this paper, the existence of regions minimizing perimeter under a volume constraint in contact sub-Riemannian manifolds whose quotient by the group of contact transformations preserving the sub-riemannians metric is compact is proved.

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56 citations

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

... in the last equality we have used |T| = 1. Since we trivially have T,DTT = 0, we get DT T = 0, as we claimed. A usual class deﬁned in contact geometry is the one of contact Riemannian manifolds, see [5], [33]. Given a contact manifold, one can assure the existence of a Riemannian metric g and an (1,1)-tensor ﬁeld J so that (2.3) g(T,X) = ω(X), 2g(X,J(Y)) = dω(X,Y ), J2(X) = −X + ω(X)T. The structure...
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...we state and prove some properties of the isoperimetric proﬁle. Finally, in section 6 we prove our main result, Theorem 6.1, on existence of isoperimetric regions. 2. Preliminaries A contact manifold [5] is a C∞ manifold M2n+1 of odd dimension so that there is an one-form ω such that dω is non-degenerate when restricted to H := ker(ω). Since dω(X,Y) = X(ω(Y )) −Y(ω(X)) −ω([X,Y]), the horizontal distr...
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Riemannian Geometry of Contact and Symplectic Manifolds

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