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, it was shown that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍn+1(−4)×ℝn or the potential vector field is strict infinitesimal contact transformation.
Abstract: Abstract Let (M, g) be a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold of dimension 2n + 1. In this paper, we prove that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍn+1(−4)×ℝn or the potential vector field is strict infinitesimal contact transformation. Moreover, two concrete examples of (κ, μ)′-almost Kenmotsu 3-manifolds admitting a Killing vector field and strict infinitesimal contact transformation are given.
24 citations
TL;DR: The Riemann curvature tensor of (κ, μ, ν)-spaces when they have almost cosymplectic and almost Kenmotsu structures was studied in this article.
Abstract: We study the Riemann curvature tensor of (κ, μ, ν)-spaces when they have almost cosymplectic and almost Kenmotsu structures, giving its writing explicitly. This leads to the definition and study of a natural generalisation of the contact metric (κ, μ, ν)-spaces. We present examples or obstruction results of these spaces in all possible cases.
24 citations
TL;DR: In this paper, almost contact B-metric structures on 3-dimensional Lie groups considered as smooth manifolds are investigated and some geometric characteristics of these manifolds in all basic classes are established.
Abstract: The object of investigations are almost contact B-metric structures on 3-dimensional Lie groups considered as smooth manifolds. There are established the existence and some geometric characteristics of these manifolds in all basic classes. An example is given as a support of obtained results.
24 citations
Posted Content•
TL;DR: In this paper, the authors studied the existence of 3-dimensional almost cosymplectic (kappa,{\mu,{
u})-spaces in all dimensions, and proved that for dimensions greater than three, π, ρ,φ are not necessary constant smooth functions such that df^{\eta} = 0.
Abstract: Main interest of the present paper is to investigate the almost {\alpha}-cosymplectic manifolds for which the characteristic vector field of the almost {\alpha}-cosymplectic structure satisfies a specific ({\kappa},{\mu},{
u})-nullity condition. This condition is invariant under D-homothetic deformation of the almost cosymplectic ({\kappa},{\mu},{
u})-spaces in all dimensions. Also, we prove that for dimensions greater than three, {\kappa},{\mu},{
u} are not necessary constant smooth functions such that df^{\eta}=0. Then the existence of the three-dimensional case of almost cosymplectic ({\kappa},{\mu},{
u})-spaces are studied. Finally, we construct an appropriate example of such manifolds.
24 citations
Additional excerpts
...ing φ2 = −I +η ⊗ξ and η(ξ) = 1, where I : TM2n+1 → TM2n+1 is the identity mapping. From the definition it follows also that φξ = 0, η ◦ φ = 0 and that the (1,1)-tensor field φ has constant rank 2n (see [4]). An almost contact manifold (M2n+1,φ,ξ,η) is said to be normal when the tensor field N = [φ,φ] + 2dη ⊗ ξ vanishes identically, [φ,φ] denoting the Nijenhuis tensor of φ. It is known that any almost co...
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TL;DR: In this article, a complete description of all nonminimal biminimal Legendrian submanifolds in Sasakian space forms which are Legendrian H-umbilical is given.
Abstract: A biminimal submanifold is a critical point of the bienergy functional for any normal variation. We give a complete description of all nonminimal biminimal Legendrian submanifolds in Sasakian space forms which are Legendrian H-umbilical.
24 citations
Cites background from "Riemannian Geometry of Contact and ..."
...[4]), Legendrian H -umbilical submanifolds are the simplest Legendrian submanifolds satisfying (a) and (b)....
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...For more details about contact metric manifolds and their submanifolds, see [4]....
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