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 paper, the authors studied the geometry of almost contact pseudo-metric manifold in terms of tensor fields, emphasizing analogies and differences with respect to the contact metric case.
Abstract: We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $$h:=\frac{1}{2}\pounds _\xi \varphi $$
and $$\ell := R(\cdot ,\xi )\xi $$
, emphasizing analogies and differences with respect to the contact metric case. Certain identities involving $$\xi $$
-sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost CR structure $$(\mathcal {H}(M), J, \theta )$$
corresponding to almost contact pseudo-metric manifold M to be CR manifold. Finally, we prove that a contact pseudo-metric manifold $$(M, \varphi ,\xi ,\eta ,g)$$
is Sasakian pseudo-metric if and only if the corresponding nondegenerate almost CR structure $$(\mathcal {H}(M), J)$$
is integrable and J is parallel along $$\xi $$
with respect to the Bott partial connection.
4 citations
TL;DR: In this paper, the authors considered the notion of almost analytic form in a unifying setting for both almost complex and almost paracomplex geometries and used a global formalism, which yields, in addition to generalizations of the main results of the previously known almost complex case, a relationship with the Frolicher-Nijenhuis theory.
Abstract: The goal of this paper is to consider the notion of almost analytic form in a unifying setting for both almost complex and almost paracomplex geometries. We use a global formalism, which yields, in addition to generalizations of the main results of the previously known almost complex case, a relationship with the Frolicher-Nijenhuis theory. A cohomology of almost analytic forms is also introduced and studied as well as deformations of almost analytic forms with pairs of almost analytic functions.
4 citations
Cites background from "Riemannian Geometry of Contact and ..."
...6) For ε = −1 we get the almost contact geometry [2], while for ε = +1 we have the almost paracontact geometry [19]....
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TL;DR: In this paper, the authors studied generalized Sasakian space form M(f1, f2, f3) when the Reeb vector field of the almost contact metric structure is Killing and the Ricci tensor satisfies Einstein-like conditions.
Abstract: We study generalized Sasakian space form M(f1, f2, f3) when (i) the Reeb vector field of the almost contact metric structure is Killing, (ii) the Ricci tensor satisfies Einstein-like conditions and (iii) the fundamental 2-form of the almost contact metric structure is a twistor form.
4 citations
TL;DR: In this article, a manifold M locally modelled on this geometry is said to be a spherical Q C-C manifold, where G is a three-dimensional connected Lie group which acts smoothly and almost freely on M, preserving the geometric structure.
Abstract: The (4n+3)-dimensional sphere S4n+3 can be viewed as the boundary of the quaternionic hyperbolic space HHn+1 and the group PSp(n+1,1) of quaternionic hyperbolic isometries extends to a real analytic transitive action on S4n+3. We call the pair (PSp(n+1,1),S4n+3) a spherical Q C–C geometry. A manifold M locally modelled on this geometry is said to be a spherical Q C–C manifold. We shall classify all pairs (G,M) where G is a three-dimensional connected Lie group which acts smoothly and almost freely on a compact spherical Q C–C manifold M, preserving the geometric structure. As an application, we shall determine all compact 3-pseudo-Sasakian manifolds admitting spherical Q C–C structures.
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TL;DR: The main theorem in this article states that if such examples exist, they are not D-homothetic to other locally φ-symmetric contact metric spaces, such as the (κ, μ)-spaces.
Abstract: We present a step towards the solution of an open problem in contact Riemannian geometry: whether there exists an example of non-Sasakian (strongly) locally φ-symmetric spaces other than the so-called (κ, μ)-spaces. The main theorem in the present paper says that if such examples exist, they are not D-homothetic to other locally φ-symmetric contact metric spaces.
4 citations