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, an integrable, non-degenerate codimension 3-subbundle on a (4n + 3)-manifold M whose fiber supports the structure of 4n-dimensional quaternionic vector space is studied.
Abstract: We study an integrable, nondegenerate codimension 3-subbundle $${\mathcal{D}}$$
on a (4n + 3)-manifold M whose fiber supports the structure of 4n-dimensional quaternionic vector space It is thought of as a generalization of quaternionic CR structure We single out an $${\mathfrak{s}}{\mathfrak{p}}(1)$$
-valued 1-form ω locally on a neighborhood U such that $${\rm Null}\omega = \mathcal D|U$$
and construct the curvature invariant on (M, ω) whose vanishing gives a uniformization to flat quaternionic CR geometry The invariant obtained on M has the same formula as that of pseudo-quaternionic Kahler 4n-manifolds From this viewpoint, we exhibit a quaternionic analogue of Chern-Moser’s CR structure
38 citations
TL;DR: In this article, a compact contact Ricci soliton with a potential vector field V collinear with the Reeb vector field, is shown to be the equivalent of an Eigenvector.
Abstract: We show that a compact contact Ricci soliton with a potential vector field V collinear with the Reeb vector field, is Einstein. We also show that a homogeneous H-contact gradient Ricci soliton is locally isometric to En+1 × Sn(4). Finally we obtain conditions so that the horizontal and tangential lifts of a vector field on the base manifold may be potential vector fields of a Ricci soliton on the unit tangent bundle.
38 citations
TL;DR: In this article, the authors consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field and introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the hypo and half-flat evolution equations in higher dimensions.
Abstract: We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the (nearly) hypo and half-flat evolution equations in higher dimensions. We use these equations to classify Einstein-Sasaki 5-manifolds of cohomogeneity one.
38 citations
TL;DR: In this article, generalized Sasakian structures are defined for Riemannian metrics and generalized Cauchy-Riemann structures are constructed from a pair of generalized F-structures.
Abstract: A generalized F-structure is a complex, isotropic subbundle E of $${T_cM \oplus T^*_cM}$$
(
$$T_cM = TM \otimes_{{\mathbb{R}}} {\mathbb{C}}$$
and the metric is defined by pairing) such that $$E \cap \bar{E}^{\perp} = 0$$
. If E is also closed by the Courant bracket, E is a generalized CRF-structure. We show that a generalized F-structure is equivalent with a skew-symmetric endomorphism Φ of $$TM \oplus T^*M$$
that satisfies the condition Φ3 + Φ = 0 and we express the CRF-condition by means of the Courant-Nijenhuis torsion of Φ. The structures that we consider are generalizations of the F-structures defined by Yano and of the CR (Cauchy-Riemann) structures. We construct generalized CRF-structures from: a classical F-structure, a pair $$({\mathcal{V}}, \sigma)$$
where $${\mathcal{V}}$$
is an integrable subbundle of TM and σ is a 2-form on M, a generalized, normal, almost contact structure of codimension h. We show that a generalized complex structure on a manifold M induces generalized CRF-structures into some submanifolds $$M \subseteq \tilde{M}$$
. Finally, we consider compatible, generalized, Riemannian metrics and we define generalized CRFK-structures that extend the generalized Kahler structures and are equivalent with quadruples (γ, F
+, F
−, ψ), where (γ, F
±) are classical, metric CRF-structures, ψ is a 2-form and some conditions expressible in terms of the exterior differential d
ψ and the γ-Levi-Civita covariant derivatives ∇ F
± hold. If d
ψ = 0, the conditions reduce to the existence of two partially Kahler reductions of the metric γ. The paper ends by an Appendix where we define and characterize generalized Sasakian structures.
38 citations
TL;DR: In this article, the fundamental relation between Jacobi structures and the classical Spencer operator coming from the theory of PDEs is revealed, and a direct and much simpler/geometric approach to the integrability of Jacobi structure is provided.
38 citations