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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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Journal ArticleDOI
TL;DR: In this paper, an almost contact metric structure is parametrized by a section σ of an associated homogeneous fiber bundle, and conditions for σ to be a harmonic section, and a harmonic map, are studied.
Abstract: An almost contact metric structure is parametrized by a section σ of an associated homogeneous fibre bundle, and conditions for σ to be a harmonic section, and a harmonic map, are studied These involve the characteristic vector field ξ, and the almost complex structure in the contact subbundle Several examples are given where the harmonic section equations for σ reduce to those for ξ, regarded as a section of the unit tangent bundle These include trans-Sasakian structures On the other hand, there are examples where ξ is harmonic but σ is not a harmonic section Many examples arise by considering hypersurfaces of almost Hermitian manifolds, with the induced almost contact structure, and comparing the harmonic section equations for both structures

23 citations

Posted Content
TL;DR: In this paper, a selection of results on locally conformally Kahler geometry published after 1997 are presented, and the proofs are mainly sketched, some of them are even omitted.
Abstract: I present a selection of results on locally conformally Kahler geometry published after 1997. The proofs are mainly sketched, some of them are even omitted. Several open problems are indicated in the end.

23 citations


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

  • ...Y KAHLER MANIFOLDS 5¨ I denote by F the foliation generated by B and JB. Note also that the leaves of the foliation generated by the nullity of the Lee form carry an induced α-Sasakian structure (see [Blair] as concerns metric contact manifolds) with JB as characteristic (Reeb) vector field. If these foliations are quasi-regular, then the leaf spaces are, respectively, a Ka¨hler and a Sasakian orbifold. T...

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Journal ArticleDOI
TL;DR: In this article, real hypersurfaces in complex projective spaces whose structure Jacobi operator is Lie parallel are classified as hypersurface in the projective projective space, where the Jacobi operators are Lie parallel.
Abstract: We classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Lie parallel in

23 citations


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

  • ...Then (φ, ξ, η, g) is an almost contact metric structure on M , see [1]....

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Journal ArticleDOI
TL;DR: In this paper, it was shown that 3-quasi-Sasakian manifolds have a rank-based classification and a splitting theorem for these manifolds assuming the integrability of one of the almost product structures.
Abstract: In the present article we carry on a systematic study of 3-quasi-Sasakian manifolds. In particular, we prove that the three Reeb vector fields generate an involutive distribution determining a canonical totally geodesic and Riemannian foliation. Locally, the leaves of this foliation turn out to be Lie groups: either the orthogonal group or an abelian one. We show that 3-quasi-Sasakian manifolds have a well-defined rank, obtaining a rank-based classification. Furthermore, we prove a splitting theorem for these manifolds assuming the integrability of one of the almost product structures. Finally, we show that the vertical distribution is a minimum of the corrected energy.

23 citations


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

  • ...It is proven in [2] that the distribution generated by the Reeb vector fields ξ1, ξ2, ξ3 is integrable, defining a canonical totally geodesic and Rie- The online version of the original article can be found under doi:10.1007/s10455-007-9093-5....

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Journal ArticleDOI
09 May 2016
TL;DR: In this paper, it was shown that the Ricci tensor is semisymmetric if and only if either it is locally isometric to the Riemannian product, or if the generator is either an Einstein or a Kenmotsu manifold.
Abstract: In this paper, we prove that a \((k,\mu )'\)-almost Kenmotsu manifold \(M^{2n+1}\) is \(\xi \)-Ricci semisymmetric if and only if either it is Einstein or it is locally isometric to the Riemannian product \({\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n\). Some results on generalized \((k,\mu )\) and \((k,\mu )'\)-almost Kenmotsu manifolds satisfying some conditions related to the cyclic-parallelism and \(\eta \)-parallelism of Ricci tensor are also obtained.

23 citations