scispace - formally typeset
Search or ask a question
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

Content maybe subject to copyright    Report

Citations
More filters
Journal ArticleDOI
TL;DR: In this paper, the energy functional on the space of sections of a sphere bundle over a Riemannian manifold equipped with the Sasaki metric is considered and the characterising condition for critical points is discussed.
Abstract: We consider the energy functional on the space of sections of a sphere bundle over a Riemannian manifold \((M,\langle \cdot, \cdot \rangle)\) equipped with the Sasaki metric and discuss the characterising condition for critical points. Furthermore, we provide a useful method for computing the tension field in some particular situations. Such a method is shown to be adequate for many tensor fields defined on manifolds M equipped with a G-structure compatible with \(\langle \cdot, \cdot \rangle\) . This leads to the construction of several new examples of differential forms which are harmonic sections or determine a harmonic map from \((M,\langle \cdot, \cdot \rangle)\) into its sphere bundle.

7 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that a complex (κ, µ)-space with κ < 1 is a locally homogeneous complex contact metric manifold and that such a complex space has either κ = 1 or is GH-local symmetric.
Abstract: We prove that a complex (κ, µ)-space with κ < 1 is a locally homogeneous complex contact metric manifold. Also, a complex (κ, µ)-space has either κ = 1 or is GH-locally symmetric.

7 citations


Cites background from "Riemannian Geometry of Contact and ..."

  • ...For a general reference see [3]....

    [...]

  • ..., Chapter 13 of [3] and the references therein)....

    [...]

  • ...For a general discussion of complex contact manifolds we refer to Chapter 12 of [3]....

    [...]

Journal ArticleDOI
TL;DR: Magnetic curves with respect to the almost cosymplectic structure of the space are determined and curvature properties of these curves are investigated in this paper, where the curvatures of the magnetic curves are also investigated.
Abstract: Magnetic curves with respect to the almost cosymplectic structure of the $$\mathrm {Sol}_3$$ space are determined and curvature properties of these curves are investigated

7 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied the problem of defining a Godbillon-Vey type invariant for a 3-dimensional manifold with a plane field under the assumption of convergence of certain integrals.
Abstract: Let M be a smooth three-dimensional manifold equipped with a vector field T transverse to a plane field $${\mathcal {D}}$$ —the kernel of a one form $$\omega $$ such that $$\omega (T)=1$$ . Recently, in Rovenski and Walczak (A Godbillon–Vey type invariant for a 3-dimensional manifold with a plane field, 2017. arXiv:1707.04847 ), we constructed a three-form analogous to that defining the Godbillon–Vey class of a foliation, showed how does this form depend on $$\omega $$ and T, and deduced Euler–Lagrange equations of the associated functional. In this paper, we continue our study when distributions/foliations and forms are defined outside a “singularity set” (a finite union of pairwise disjoint closed submanifolds of codimension $$\ge 2$$ ) under additional assumption of convergence of certain integrals. We characterize critical pairs $$(\omega ,T)$$ and foliations for different types of variations, find sufficient conditions for critical pairs when variations are among foliations, consider applications to transversely holomorphic flows, calculate the index form and present examples of critical foliations among Reeb foliations and twisted products.

7 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered a set E ⊂ Ω with prescribed mean curvature f ∈ C ( Ω ) and Euclidean Lipschitz boundary ∆ E = Σ inside a three-dimensional contact sub-Riemannian manifold M and proved that if Σ is locally a regular intrinsic graph, the characteristic curves are of class C 2.
Abstract: In this paper we consider a set E ⊂ Ω with prescribed mean curvature f ∈ C ( Ω ) and Euclidean Lipschitz boundary ∂ E = Σ inside a three-dimensional contact sub-Riemannian manifold M . We prove that if Σ is locally a regular intrinsic graph, the characteristic curves are of class C 2 . The result is shape and improves the ones contained in Capogna et al. (2009) and Galli and Ritore (2015).

7 citations