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 authors studied the geometric properties of the base manifold for the unit tangent bundle satisfying the η-Einstein condition with the canonical contact metric structure, and they proved that the manifold is a 4-dimensional Einstein manifold if and only if the space of constant sectional curvature 1 or 2.
Abstract: We study the geometric properties of the base manifold for the unit tangent bundle satisfying the η-Einstein condition with the canonical contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is η-Einstein manifold if and only if the base manifold is the space of constant sectional curvature 1 or 2.
11 citations
Cites background from "Riemannian Geometry of Contact and ..."
...First, we give some preliminaries on a contact metric manifold. We refer to [ 2 ] for more details....
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TL;DR: For a fat sub-Riemannian structure of 3-Sasakian manifolds, the authors of as discussed by the authors introduced three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li.
Abstract: For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet-Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider the sub-Riemannian structure of 3-Sasakian manifolds, for which we provide explicit curvature formulas. We prove that any complete 3-Sasakian structure of dimension 4d + 3, with d > 1, has sub-Riemannian diameter bounded by π. When d = 1, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure of the quaternionic Hopf fibrations on the 4d+3 dimensional sphere, whose exact sub-Riemannian diameter is π, for all d ≥ 1.
11 citations
TL;DR: In this article, it was shown that a compact Lie group K admits left-invariant complex structures of maximal dimension with a transverse CR-action on a quasi-projective manifold X naturally associated to K. The authors also showed that X admits more general Abelian actions, also inducing complex or CR-structures on K which are generically not invariant.
Abstract: It was shown by Samelson [A class of complex-analytic manifolds. Portugaliae Math. 12, 129–132 (1953)] and Wang [Closed manifolds with homogeneous complex structure. Amer. J. Math. 76, 1–32 (1954)] that each compact Lie group K of even dimension admits left-invariant complex structures. When K has odd dimension it admits a left-invariant CR-structure of maximal dimension. This has been proved recently by Charbonnel and Khalgui [Classification des structures CR invariantes pour les groupes de Lie compactes. J. Lie theory 14, 165–198 (2004)] who have also given a complete algebraic description of these structures. In this article, we present an alternative and more geometric construction of this type of invariant structures on a compact Lie group K when it is semisimple. We prove that each left-invariant complex structure, or each CR-structure of maximal dimension with a transverse CR-action by \(\mathbb{R}\) , is induced by a holomorphic \(\mathbb{C}^l\) -action on a quasi-projective manifold X naturally associated to K. We then show that X admits more general Abelian actions, also inducing complex or CR-structures on K which are generically non-invariant.
11 citations
TL;DR: In this article, the stability of the Reeb vector field with respect to the energy functional with mean curvature correction was studied for H-contact manifolds in terms of the Webster scalar curvature.
Abstract: It is well known that a Hopf vector field on the unit sphere S
2n+1 is the Reeb vector field of a natural Sasakian structure on S
2n+1. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction.
11 citations
Cites background from "Riemannian Geometry of Contact and ..."
...(for n = 1 we obtain a left-invariant Sasakian structure on the classical three-dimensional Heisenberg group H [ 1 ], p.49)....
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...The 3-dimensional torus T 3 admits a natural flat contact metric structure ( ¯ η, ¯ g) (see [ 1 ], p. 68 and p. 100, for an explicit construction)....
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...We refer to [ 1 ] for more information about contact metric manifolds....
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...It is a unit field with respect to a contact Riemannian metric and plays a fundamental role in the study of the Riemannian geometry of a contact metric manifold (see [ 1 ])....
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TL;DR: In this paper, the authors considered energy-minimizing divergence-free eigenfields of the curl operator in dimension three from the perspective of contact topology and gave a negative answer to a question of Etnyre and the first author by constructing curl eigenfield which minimize L 2 energy on their co-adjoint orbit, yet are orthogonal to an overtwisted contact structure.
Abstract: This paper concerns topological and geometric properties of energy-minimizing solutions to the steady Euler equations for a fluid on a three-dimensional manifold. Specifically, we consider energy-minimizing divergence-free eigenfields of the curl operator in dimension three from the perspective of contact topology. We give a negative answer to a question of Etnyre and the first author by constructing curl eigenfields which minimize L2 energy on their co-adjoint orbit, yet are orthogonal to an overtwisted contact structure. We then show progress towards the conjecture that K-contact structures on Seifert fibred manifolds always define tight minimizers.
11 citations