Book•
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 Schouten-van Kampen connection associated to an almost contact or paracontact metric structure was studied and some classes of almost (para) contact metric manifolds were characterized.
Abstract: We study the Schouten-van Kampen connection associated to an almost contact or paracontact metric structure. With the help of such a con- nection, some classes of almost (para) contact metric manifolds are character- ized. Certain curvature properties of this connection are found.
17 citations
TL;DR: Some optimal inequalities involving the intrinsic scalar curvature and the extrinsic Casorati curvature of submanifolds in a generalized complex space form and a generalized Sasakian space form with a semi-symmetric non-metric connection are proved.
Abstract: In this paper, we prove some optimal inequalities involving the intrinsic scalar curvature and the extrinsic Casorati curvature of submanifolds in a generalized complex space form with a semi-symmetric non-metric connection and a generalized Sasakian space form with a semi-symmetric non-metric connection Moreover, we show that in both cases, the equalities hold if and only if submanifolds are invariantly quasi-umbilical
17 citations
TL;DR: In this paper, it was shown that all g-natural contact metric structures on a two-point homogeneous space are homogeneous contact, and the converse was also proved for metrics of Kaluza-Klein type.
Abstract: We prove that all g-natural contact metric structures on a two-point homogeneous space are homogeneous contact. The converse is also proved for metrics of Kaluza–Klein type. We also show that if (M,g) is an Einstein manifold and is a Riemannian g-natural metric on T1M of Kaluza–Klein type, then is H-contact if and only if (M,g) is 2-stein, so proving that the main result of Chun et al. [‘H-contact unit tangent sphere bundles of Einstein manifolds’, Q. J. Math., to appear. DOI: 10.1093/qmath/hap025] is invariant under a two-parameter deformation of the standard contact metric structure on T1M. Moreover, we completely characterize Riemannian manifolds admitting two distinct H-contact g-natural contact metric structures, with associated metric of Kaluza–Klein type.
17 citations
Cites background from "Riemannian Geometry of Contact and ..."
...[9] Homogeneous and H -contact unit tangent sphere bundles 331...
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...When this condition holds, η̃ is homothetic, with homothety factor r , to the classical contact form on T1M (see, for example, [9] for a definition), and consequently, η̃ is again a contact form....
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TL;DR: In this paper, it was shown that Willett's reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter, since a symplectic manifold is prequantizable iff the symplectic form is integral.
Abstract: We introduce a new method to perform reduction of contact manifolds that extends Willett's and Albert's results. To carry out our reduction procedure all we need is a complete Jacobi map J: M → Γ 0 from a contact manifold to a Jacobi manifold. This naturally generates the action of the contact groupoid of Γ 0 on M, and we show that the quotients of fibers J -1 (x) by suitable Lie subgroups Γ x are either contact or locally conformal symplectic manifolds with structures induced by the one on M. We show that Willett's reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter. Since a symplectic manifold is prequantizable iff the symplectic form is integral, this explains why Willett's reduction can be performed only at distinguished points. As an application we obtain Kostant's prequantizations of coadjoint orbits. Finally we present several examples where we obtain classical contact manifolds as reduced spaces.
17 citations
TL;DR: In this article, the authors define and study both slant light-like submanifolds and screen slant-light-like subsets of an indefinite Sasakian manifold.
Abstract: In this paper, we define and study both slant lightlike submanifolds and screen slant lightlike submanifolds of an indefinite Sasakian manifold. We provide non-trivial examples and obtain necessary and sufficient conditions for the existence of a slant lightlike submanifold.
17 citations
Cites background from "Riemannian Geometry of Contact and ..."
...An odd dimensional semi-Riemannian manifold (M̄, 1̄) is called a contact metric manifold [4] if there exists a (1, 1) tensor field φ, a vector field V, called the characteristic vector field, and its 1-form η satisfying 1̄(φX, φY) = 1̄(X,Y) − η(X)η(Y), 1̄(V,V) = , (1....
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...We say that M has a normal contact structure if Nφ + d η⊗ ξ = 0,where Nφ is the Nijenhuis tensor field of φ [4]....
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