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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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TL;DR: In this article, the authors give more evidence of the existence of minimal metrics, by presenting several ex-plicit examples, and also provide many continuous families of symplectic, complex and hypercomplex nilpotent Lie groups.
Abstract: A left invariant metric on a nilpotent Lie group is called minimal, if it minimizes the norm of the Ricci tensor among all left invariant metrics with the same scalar curvature. Such metrics are unique up to isometry and scaling and the groups admitting a minimal metric are precisely the nilradicals of (standard) Einstein solvmanifolds. If N is endowed with an invariant sym- plectic, complex or hypercomplex structure, then minimal compatible metrics are also unique up to isometry and scaling. The aim of this paper is to give more evidence of the existence of minimal metrics, by presenting several ex- plicit examples. This also provides many continuous families of symplectic, complex and hypercomplex nilpotent Lie groups. A list of all known examples of Einstein solvmanifolds is also given.

27 citations

Journal ArticleDOI
TL;DR: In this article, a general half-BPS A-type boundary condition for compact 3-manifolds with boundary was formulated for complex supersymmetric field theories on compact 3D manifolds.
Abstract: General half-BPS A-type boundary conditions are formulated for $$ \mathcal{N} $$ = 2 supersymmetric field theories on compact 3-manifolds with boundary. We observe that under suitable conditions manifolds of the real A-type admitting two complex supersymmetries (related by charge conjugation) possess, besides a contact structure, a natural integrable toric foliation. A boundary, or a general co-dimension-1 defect, can be inserted along any leaf of this preferred foliation to produce manifolds with boundary that have the topology of a solid torus. We show that supersymmetric field theories on such manifolds can be endowed with half-BPS A-type boundary conditions. We specify the natural curved space generalization of the A-type projection of bulk supersymmetries and analyze the resulting A-type boundary conditions in generic 3d non-linear sigma models and YM/CS-matter theories.

27 citations

Journal Article
TL;DR: In this article, it was proved that a locally φ -recurrent K-meansu manifold is the Robertson-Walker spacetime, and a concrete example of a three-dimensional K-Mean manifold is given.
Abstract: The object of this paper is to study φ -recurrent Kenmotsu manifolds. Also three-dimensional locally φ - recurrent Kenmotsu manifolds have been considered. Among others it is proved that a locally φ -recurrent Kenmotsu spacetime is the Robertson-Walker spacetime. Finally we give a concrete example of a three- dimensional Kenmotsu manifold.

27 citations

Journal ArticleDOI
TL;DR: The Ricci-flat Ricci solitons with the potential vector fields pointwise collinear with the Reeb vector fields on K-almost coKahler manifolds were studied in this article.
Abstract: Let M be a compact almost coKahler manifold. If the metric g of M is a Ricci soliton and the potential vector field is pointwise collinear with the Reeb vector field, then we prove that M is Ricci-flat and coKahler and the soliton g is steady. This generalizes a Goldberg-like conjecture for coKahler manifolds obtained by Cappelletti-Montano and Pastore, namely any compact Einstein K-almost coKahler manifold is coKahler. Without the assumption of compactness, Ricci solitons with the potential vector fields pointwise collinear with the Reeb vector fields on K-almost coKahler manifolds are also studied. Moreover, we prove that there exist no gradient Ricci solitons on proper \({(\kappa, \mu)}\)-almost coKahler manifolds.

27 citations

Posted Content
TL;DR: In this article, the notions of pointwise slant submanifolds and pointwise semi-slant sun-mansifolds of an almost contact metric manifold were introduced.
Abstract: As a generalization of slant submanifolds and semi-slant submanifolds, we introduce the notions of pointwise slant submanifolds and pointwise semi-slant sunmanifolds of an almost contact metric manifold. We obtain a characterization at each notion, investigate the topological properties of pointwise slant submanifolds, and give some examples of them. We also consider some distributions on cosymplectic, Sasakian, Kenmotsu manifolds and deal with some properties of warped product pointwise semi-slant submanifolds. Finally, we give some inequalities for the squared norm of the second fundamental form in terms of a warping function and a semi-slant function for warped product submanifolds of cosymplectic, Sasakian, Kenmotsu manifolds.

27 citations