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 consider a 3D almost co-Kahler manifold where the Reeb vector field ξ is an eigenvector field of the Ricci operator Q, where ρ is a smooth function on M.
Abstract: Let (M3, g) be a three dimensional almost coKahler manifold such that the Reeb vector field ξ is an eigenvector field of the Ricci operator Q, i.e. Qξ = ρξ, where ρ is a smooth function on M. In th...
5 citations
Additional excerpts
...Concerning the operator the following identities, which were given in [1], are satisfied: { hξ = 0, φh = −hφ, ∇Xξ = −φhX, g(hX, Y ) = g(X,hY ), trace(h) = trace(φh) = 0, η ◦ h = 0....
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TL;DR: In this paper, the authors discuss the partial classification theorems of projective complex contact manifolds and discuss a distinguished contact cone structure, arising as the variety of minimal rational tangents.
Abstract: Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the first case we also discuss a distinguished contact cone structure, arising as the variety of minimal rational tangents. We discuss the repercussion of the aforementioned classification theorems for the geometry of quaternion-K\"ahler manifolds with positive scalar curvature and for the geometry of second-order PDEs imposed on hypersurfaces.
5 citations
5 citations
07 Jul 2011
TL;DR: In this paper, the authors studied geometric aspects of light-like hypersurfaces of indefinite Kenmotsu space forms, tangent to the structure vector field and whose shape operator is conformal to the shape operator of its screen distribution, showing that these hypersufaces are proper totally contact umbilical, semi-parallel and η-Einstein but not Ricci semi-symmetric.
Abstract: In this paper we deal with geometric aspects of lightlike hypersurfaces of indefinite Kenmotsu space forms, tangent to the structure vector field and whose shape operator is conformal to the shape operator of its screen distribution. We show that these hypersufaces are proper totally contact umbilical, semi-parallel and η-Einstein but not Ricci semi-symmetric. They are locally a product of lightlike curves and proper totally umbilical leaves of its screen distributions. Its mean curvature vectors have closed dual differential 1-forms. We also show that there exists an integrable distribution whose leave are space forms, proper totally umbilical, Einstein, locally symmetric and Ricci semi-symmetric. We finally characterize the relative nullity space in a screen conformal lightlike hypersurface of an indefinite Kenmotsu space form.
5 citations
Cites background from "Riemannian Geometry of Contact and ..."
...The contact geometry has significant use in differential equations, phase spaces of dynamical systems (see details in [15] and [24], for instance), but the literature about its lightlike case is very limited....
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Posted Content•
TL;DR: In this article, a frame is introduced on tangent bundle of a Finsler manifold in a manner that it makes some simplicity to study the properties of the natural foliations in tangent bundles.
Abstract: In this paper, a frame is introduced on tangent bundle of a Finsler manifold in a manner that it makes some simplicity to study the properties of the natural foliations in tangent bundle. Moreover, we show that the indicatrix bundle of a Finsler manifold with lifted sasaki metric and natural almost complex structure on tangent bundle cannot be a sasakian manifold.
5 citations
Cites background from "Riemannian Geometry of Contact and ..."
...5 Sasakian Structure and Indicatrix Bundle of a Finsler Manifold Now, let (M̄, φ̄, η̄, ξ̄, ḡ) be a contact metric manifold [6]....
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...1 The following argument was presented in Chapter 6 of [6]....
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...Finally in Section 5, it is proved that the indicatrix bundle with its contact structure given in [3] cannot be a Sasakian manifold [6]....
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