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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, it was shown that the Boothby-Wang fibration of the Iwasawa manifold is an unstable critical point for the energy of a distribution, and that it is the case for the case of any distribution.
Abstract: In this paper, we show that the Boothby-Wang fibration of the Iwasawa manifold \(H_{{\Bbb C}}/\Gamma\) is an unstable critical point for the energy of a distribution.
12 citations
Cites background or methods from "Riemannian Geometry of Contact and ..."
...In [14] (see also [ 2 ] p. 203), Korkmaz computed the covariant derivatives of G and H as...
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...In [1] and [ 2 ] the following are also listed for the complex Heisenberg group...
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TL;DR: In this paper, the geometric structure of reproducing kernel Hilbert space associated to the continuous wavelet transform generated by the irreducible representations of the group of Euclidean motion is studied.
Abstract: We study the geometric structure of the reproducing kernel Hilbert space associated to the continuous wavelet transform generated by the irreducible representations of the group of Euclidean motion...
12 citations
TL;DR: In this article, a Lancret invariant is obtained and the Legendre curves are analyzed as particular case of the Lancret Theorem of $3$-dimensional hyperbolic geometry.
Abstract: Slant curves are introduced in three-dimensional warped products with Euclidean factors; these curves are characterized through the scalar product between the normal at the curve and the
vertical vector field and an important feature is that the case of constant Frenet curvatures implies a proper mean curvature vector field. A Lancret invariant is obtained and the
Legendre curves are analyzed as particular case. An example of a slant curve is given for the exponential warping function; our example illustrates a proper (i.e. not reducible to the
two dimensions) case of the Lancret Theorem of $3$-dimensional hyperbolic geometry. We point out an eventually relationship with the geometry of relativistic models.
10.1017/S0004972712000809
12 citations
01 Jan 2009
TL;DR: In this article, it was shown that all non-trivial minimal-legendrian deformations of a certain 3-dimensional compact minimal Legendrian submanifolds embedded in the 7-dimensional standard Ein- stein Sasakian manifold (S 7 (1) and Stiefel manifold V2(R 5 ) with constant positive sectional curvature are given by the 7D family of minimal Legendrians, which is constructed by the group action of Sp(2,C).
Abstract: A minimal Legendrian submanifold in a Sasakian manifold is by definition a Leg- endrian submanifold in a Sasakian manifold which is a minimal submanifold in the sense of vanishing mean curvature vector field. The minimal Legendrian deformation means a smooth family of minimal Legendrian submanifolds. In this note we discuss minimal Legendrian deformations of certain 3-dimensional compact minimal Legendrian submanifolds embedded in the 7-dimensional standard Ein- stein Sasakian manifolds, 7-dimensional unit sphere S 7 (1) and Stiefel manifold V2(R 5 ). We prove that all non-trivial minimal Legendrian deformations of a certain non-totally geodesic minimal Legendrian orbit of SU(2) in S 7 (1) are given by the 7-dimensional family of minimal Legendrian submanifolds which is constructed by the group action of Sp(2,C). Moreover we show that a 3-dimensional compact minimal Legendrian submani- fold SO(3)/(Z2+Z2) in V2(R 5 ) with constant positive sectional curvature has no nontrivial minimal Legendrian deformation.
12 citations
TL;DR: In this paper, a generalized Sasakian space form with semi-symmetric non-metric connections is introduced. But the connection between warped products has not been considered.
Abstract: We introduce generalized Sasakian space forms with semi-symmetric non-metric connections. We show the existence of a generalized Sasakian space form with a semi-symmetric non-metric connection and give some examples by warped products endowed with semi-symmetric non-metric connections.
12 citations
Cites background from "Riemannian Geometry of Contact and ..."
...Such a manifold is said to be a contact metric manifold if dη = Φ, where Φ(X,Y ) = g(X,φY ) is called the fundamental 2-form of M (see [3])....
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...A normal contact metric manifold is called a Sasakian manifold ([3])....
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...Generalized Sasakian-space forms Let M be an n-dimensional almost contact metric manifold [3] with an almost contact metric structure (φ, ξ, η, g) consisting of a (1, 1) tensor field φ, a vector field ξ, a 1-form η and a Riemannian metric g on M satisfying φ(2)X = −X + η(X)ξ, η(ξ) = 1, φξ = 0, η ◦ φ = 0, (6) g(φX,φY ) = g(X,Y )− η(X)η(Y ), g(X, ξ) = η(X), (7)...
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...If N is a Kaehlerian manifold, it is well-known that M = N ×R with its usual product almost contact metric structure is a cosymplectic manifold [3]....
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