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
Citations
More filters
Journal Article•
TL;DR: In this paper, contact Riemannian invariants for almost contact metric manifolds analogues to those invariants introduced in [5, 6] were defined and sharp inequalities between these invariants and the squared mean curvature were established.
Abstract: . First, we define some contact Riemannian invariants for almost
contact metric manifolds analogues to those invariants introduced in [5, 6].
We then establish sharp inequalities between these contact Riemannian invariants
and the squared mean curvature for almost contact Riemannian
manifolds in a real space form. We also investigate almost contact Riemannian
submanifolds which verify the equality case of the inequalities.
Examples of contact Riemannian submanifolds satisfying the equality case
are provided as well.
9 citations
TL;DR: In this article, it is shown that the answer is positive for locally homogeneous contact metric manifolds, i.e., every strongly locally φ-symmetric contact metric space is a π-space.
Abstract: It is an open question whether every strongly locally $\\varphi$-symmetric contact metric space is a $(\\kappa,\\mu)$-space. We show that the answer is positive for locally homogeneous contact metric manifolds.
9 citations
Cites background from "Riemannian Geometry of Contact and ..."
...We refer to [2] for a more detailed treatment....
[...]
...Finally, we use the equality ∇ξφ = 0, which holds for every contact metric space ([2]), together with (8)6 to obtain 0 = (∇ξφ)Z = (Tξ · φ)Z = Tξ (φZ) − φ(Tξ Z)....
[...]
01 Jan 2011
TL;DR: In this article, the Da-homothetic deformations of generalized (•;")-space forms were studied, and it was shown that the deformed spaces are again generalized in dimension 3, but not in general, although a slight change in their deflunction would make them so.
Abstract: We study the Da-homothetic deformations of generalized (•;")- space forms. We prove that the deformed spaces are again generalized (•;")-space forms in dimension 3, but not in general, although a slight change in their deflnition would make them so. We give inflnitely many examples of generalized (•;")-space forms of dimension 3.
9 citations
Cites background from "Riemannian Geometry of Contact and ..."
...For more background on almost contact metric manifolds, we recommend the reference [7]....
[...]
...It is well known that on a contact metric manifold (M, φ, ξ, η, g), the tensor h, defined by 2h = Lξφ, is symmetric and satisfies the following relations [7]:...
[...]
25 Dec 2015
TL;DR: In this article, it was shown that a 3D normal almost paracontact metric manifold is φ-projectively flat if and only if it is an Einstein manifold for α, β =const.
Abstract: The aim of present paper is to investigate 3-dimensional ξ-projectively flat and φ-projectively flat normal almost paracontact metric manifolds. As a first step, we proved that if the 3dimensional normal almost paracontact metric manifold is ξ-projectively flat then ∆β = 0. If additionally β is constant then the manifold is β-para-Sasakian. Later, we proved that a 3-dimensional normal almost paracontact metric manifold is φ-projectively flat if and only if it is an Einstein manifold for α, β =const. Finally, we constructed an example to illustrate the results obtained in previous sections.
9 citations
Cites background from "Riemannian Geometry of Contact and ..."
...Similarly as in the class of almost contact metric manifolds [4], a normal almost paracontact metric manifold will be called para-Sasakian if F = dη [10] and quasi-para-Sasakian if dF = 0....
[...]
Posted Content•
TL;DR: In this paper, the authors define a new class of LCK-manifolds called LCK manifolds with potential, which is closed under small deformations, and show that any LCK manifold M with potential admits a covering which can be compactified to a Stein variety by adding one point.
Abstract: A locally conformally K\"ahler (LCK) manifold $M$ is one which is covered by a K\"ahler manifold $\tilde M$ with the deck transform group acting conformally on $\tilde M$. If $M$ admits a holomorphic flow, acting on $\tilde M$ conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vaisman manifolds is stable under small deformations. We define a new class of LCK-manifolds, called LCK manifolds with potential, which is closed under small deformations. All Vaisman manifolds are LCK with potential. We show that an LCK-manifold with potential admits a covering which can be compactified to a Stein variety by adding one point. This is used to show that any LCK manifold M with potential, $\dim M > 2$, can be embedded to a Hopf manifold, thus improving on similar results for Vaisman. manifolds.
9 citations
Cites methods from "Riemannian Geometry of Contact and ..."
...equence of Theorem 3.6 we can improve the immersion result for Sasakian manifolds in [OV2, §6]. For the sake of completeness, we first recall some necessary facts about Sasakian manifolds (we refer to [Bl] and to recent papers by Ch. Boyer, K. Galicki et.al. for examples). 10 LCK manifolds with potential L. Ornea and M. Verbitsky, 14 July 2004 A Riemannian manifold (N,h) of odd real dimension is Sasaki...
[...]