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 doubly warped product CR submanifolds in locally conformal Kahler manifolds, and they found a B.Y. Chen's type inequality for the second fundamental form of these sub-mansifolds.
Abstract: In this paper we study doubly warped product CR submanifolds in locally conformal Kahler manifolds, and we found a B.Y. Chen's type inequality for the second fundamental form of these submanifolds
16 citations
TL;DR: In this paper, the authors developed new techniques for studying concentration of Laplace eigenfunctions as their frequency grows, by decomposing the set of geodesic tubes into those that are non self-looping for time $T$ and those that were.
Abstract: In this article we develop new techniques for studying concentration of Laplace eigenfunctions $\phi_\lambda$ as their frequency, $\lambda$, grows. The method consists of controlling $\phi_\lambda(x)$ by decomposing $\phi_\lambda$ into a superposition of geodesic beams that run through the point $x$. Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower than $\lambda^{-\frac{1}{2}}$. We control $\phi_\lambda(x)$ by the $L^2$-mass of $\phi_\lambda$ on each geodesic tube and derive a purely dynamical statement through which $\phi_\lambda(x)$ can be studied. In particular, we obtain estimates on $\phi_\lambda(x)$ by decomposing the set of geodesic tubes into those that are non self-looping for time $T$ and those that are. This approach allows for quantitative improvements, in terms of $T$, on the available bounds for $L^\infty$ norms, $L^p$ norms, pointwise Weyl laws, and averages over submanifolds.
16 citations
TL;DR: In this article, a new proof of the classification of contact real hypersurfaces with constant mean curvature in the complex hyperbolic quadric (Q^m}^* = SO{m,2}^o/SO_mSO_2, where $m\geq 3$ is given.
Abstract: We give a new proof of the classification of contact real hypersurfaces with constant mean curvature in the complex hyperbolic quadric ${Q^m}^* = SO_{m,2}^o/SO_mSO_2$, where $m\geq 3$. We show that a contact real hypersurface $M$ in ${Q^m}^*$ for $m\geq 3$ is locally congruent to a tube of radius $r{\in}{\mathbb R}^+$ around the complex hyperbolic quadric ${Q^{m-1}}^*$, or to a tube of radius $r\in\mathbb{R}^+$ around the $\mathfrak A$-principal $m$-dimensional real hyperbolic space ${\mathbb R}H^m$ in ${Q^m}^* = SO_{m,2}^o/SO_mSO_2$, or to a horosphere in ${Q^{m-1}}^*$ induced by a class of $\mathfrak A$-principal geodesics in ${Q^m}^*$.
16 citations
01 Mar 2015
TL;DR: In this paper, the authors consider the group of contact transformations of the manifold (contactomorphisms) and the subgroup of those transformations that precisely preserve the contact form (quantomorphisms), and obtain right-invariant metrics on both the contactomorphism and quantomorphism groups.
Abstract: Given an odd-dimensional compact manifold and a contact form, we consider the group of contact transformations of the manifold (contactomorphisms) and the subgroup of those transformations that precisely preserve the contact form (quantomorphisms). If the manifold also has a Riemannian metric, we can consider the \(L^2\) inner product of vector fields on it, which by restriction gives an inner product on the tangent space at the identity of each of the groups that we consider. We then obtain right-invariant metrics on both the contactomorphism and quantomorphism groups. We show that the contactomorphism group has geodesics at least for short time and that the quantomorphism group is a totally geodesic subgroup of it. Furthermore we show that the geodesics in this smaller group exist globally. Our methodology is to use the right invariance to derive an “Euler–Arnold” equation from the geodesic equation and to show using ODE methods that it has solutions which depend smoothly on the initial conditions. For global existence we then derive a “quasi-Lipschitz” estimate on the stream function, which leads to a Beale–Kato–Majda criterion which is automatically satisfied for quantomorphisms. Special cases of these Euler–Arnold equations are the Camassa–Holm equation (when the manifold is one-dimensional) and the quasi-geostrophic equation in geophysics.
16 citations
01 Apr 2009
TL;DR: In this paper, the authors introduce the notion of contact pair structure and corresponding associated metrics, in the same spirit of the geometry of almost contact structures, and prove that, with respect to these metrics, the integral curves of the Reeb vector field s are geodesics and that the leaves of the reeb action are totally geodesic.
Abstract: We introduce the notion of contact pair structure and the corresponding associated metrics, in the same spirit of the geometry of almost contact structures. We prove that, with respect to these metrics, the integral curves of the Reeb vector field s are geodesics and that the leaves of the Reeb action are totally geodesic. Moreover, we show that, in the case of a metric contact pair with decomposable endomorphism, the characteristic foliations are orthogonal and their leaves carry induced contact metric structures.
16 citations