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 intrinsic geometry of hypersurfaces in Calabi-Yau manifolds of real dimension 6 and, more generally, SU(2)-structures on 5-manifolds defined by a generalized Killing spinor was studied.
Abstract: We study the intrinsic geometry of hypersurfaces in Calabi-Yau manifolds of real dimension 6 and, more generally, SU(2)-structures on 5-manifolds defined by a generalized Killing spinor. We prove that in the real analytic case, such a 5-manifold can be isometrically embedded as a hypersurface in a Calabi-Yau manifold in a natural way. We classify nilmanifolds carrying invariant structures of this type, and present examples of the associated metrics with holonomy SU(3).
3 citations
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TL;DR: In this paper, the helicity of a vector field preserving a regular contact form on a closed three-dimensional manifold is computed and shown to be invariant under conjugation by volume preserving homeomorphisms.
Abstract: We compute the helicity of a vector field preserving a regular contact form on a closed three-dimensional manifold, and improve results by J.-M. Gambaudo and \'E. Ghys [GG97] relating the helicity of the suspension of a surface isotopy to the Calabi invariant of the latter. Based on these results, we provide positive answers to two questions posed by V. I. Arnold [Arn86]. In the presence of a regular contact form that is also preserved, the helicity extends to an invariant of an isotopy of volume preserving homeomorphisms, and is invariant under conjugation by volume preserving homeomorphisms. A similar statement also holds for suspensions of surface isotopies and surface diffeomorphisms. This requires the techniques of topological Hamiltonian and contact dynamics developed in [MO07, M\"ul08b, Vit06, BS11b, BS11a, MS11]. Moreover, we generalize an example of H. Furstenberg [Fur61] of topologically conjugate but not C^1-conjugate area preserving diffeomorphisms of the two-torus to trivial T^2-bundles, and construct examples of Hamiltonian and contact vector fields that are topologically conjugate but not C^1-conjugate. Higher-dimensional helicities are considered briefly at the end of the paper.
3 citations
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TL;DR: In this paper, the authors characterize general pseudo-harmonic morphisms from a Riemannian manifold to a Hermitian manifold as pseudo horizontally weakly conformal maps with an additional property.
Abstract: We characterize general pseudo-harmonic morphisms from a Riemannian manifold to a Hermitian manifold as pseudo horizontally weakly conformal maps with an additional property. We study to what extent we can (locally) describe these submersive pseudo-harmonic morphisms via the foliation given by the kernel of the associated f-structure. In a second part, we point out that, in the case of pseudo-harmonic morphisms with one and two-dimensional fibers, the induced f-structure gives rise to an almost contact, respectively almost complex structure on the domain. We give criteria for normality and integrability of these structures and we show how these two particular cases are interrelated.
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TL;DR: In this paper, the Sasaki metric has been studied on the unit tangent sphere bundle SM of any oriented Riemannian 4-manifold M and the equations of calibration and cocalibration for W3 pure type and nearly-parallel type were derived.
Abstract: We study natural variations of the G2 structure 0 2 3 existing on the unit tangent sphere bundle SM of any oriented Riemannian 4-manifold M. We nd a circle of structures for which the induced metric is the usual one, the so-called Sasaki metric, and prove how the original structure has a preferred role in the theory. We deduce the equations of calibration and cocalibration, as well as those of W3 pure type and nearly-parallel type.
3 citations
TL;DR: In this article, the authors studied the geometry of a G$-structure inside the oriented orthonormal frame bundle over an oriented Riemannian manifold and showed that the minimality of the structure is equivalent to the harmonicity of an induced section of the homogeneous bundle.
Abstract: We study the geometry of a $G$-structure $P$ inside the oriented orthonormal frame bundle ${\rm SO}(M)$ over an oriented Riemannian manifold $M$. We assume that $G$ is connected and closed, so the quotient ${\rm SO}(n)/G$, where $n=\dim M$, is a normal homogeneous space and we equip ${\rm SO}(M)$ with the natural Riemannian structure induced from the structure on $M$ and the Killing form of ${\rm SO}(n)$. We show, in particular, that minimality of $P$ is equivalent to harmonicity of an induced section of the homogeneous bundle ${\rm SO}(M)\times_{{\rm SO}(n)}{\rm SO}(n)/G$, with a Riemannian metric on $M$ obtained as the pull-back with respect to this section of the Riemannian metric on the considered associated bundle, and to the minimality of the image of this section. We apply obtained results to the case of almost product structures, i.e., structures induced by plane fields.
3 citations