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 volume of small sub-Riemannian balls in a contact 3-dimensional manifold has been shown to expand asymptotically with respect to geometric invariants.
Abstract: We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure
12 citations
Additional excerpts
...Theorem 12 (canonical connection, [20, 29, 22])....
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TL;DR: The characterizations of the curvatures of biharmonic Legendre curves in generalized Sasakian space forms with constant functions have been studied in this paper, where the curvature of the Legendre curve is characterized by a constant function.
Abstract: We find the characterizations of the curvatures of biharmonic Legendre curves in generalized Sasakian space forms with constant functions.
12 citations
Cites background from "Riemannian Geometry of Contact and ..."
...Such a manifold is said to be a contact metric manifold if dη = , where (X, Y ) = g(X, φY ) is called the fundamental 2-form of M [4]....
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...A normal contact metric manifold is called a Sasakian manifold [4]....
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...and an integral curve of the contact distribution is called a Legendre curve [4]....
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...A (2n + 1)-dimensional Riemannian manifold M is said to be an almost contact metric manifold [4], if there exist on M a (1, 1 ) tensor field φ, a vector field ξ , a 1-form η and a Riemannian metric g satisfying φ2 = −I + η ⊗ ξ, η(ξ) = 1, φξ = 0, η ◦ φ = 0 g(φX, φY ) = g(X, Y )− η(X)η(Y ), η(X) = g(X, ξ),...
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TL;DR: It is demonstrated that geometric computing in the Siegel–Klein disk allows one to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in theSiegel–Poincaré disk model, and to approximate fast and numerically the S Spiegel-Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.
Abstract: We study the Hilbert geometry induced by the Siegel disk domain, an open-bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel-Klein disk model to differentiate it from the classical Siegel upper plane and disk domains. In the Siegel-Klein disk, geodesics are by construction always unique and Euclidean straight, allowing one to design efficient geometric algorithms and data structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel-Poincare disk and in the Siegel-Klein disk: We demonstrate that geometric computing in the Siegel-Klein disk allows one (i) to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in the Siegel-Poincare disk model, and (ii) to approximate fast and numerically the Siegel-Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.
12 citations
Cites background from "Riemannian Geometry of Contact and ..."
...Presently, symplectic geometry is mainly understood as the study of symplectic manifolds [4] which are even-dimensional differentiable manifolds equipped with a closed and nondegenerate differential 2-form ω, called the symplectic form, studied in geometric mechanics....
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TL;DR: In this article, necessary and sufficient conditions for a slant curve in a Sasakian 3-manifold to have a C-parallel mean curvature vector were given.
Abstract: In this article, using the example of C. Camci((7)) we reconrm necessary suf- �cient condition for a slant curve. Next, wend some necessary and sufficient conditions for a slant curve in a Sasakian 3-manifold to have: (i) a C-parallel mean curvature vector
12 citations
Cites background from "Riemannian Geometry of Contact and ..."
...A 3-manifold M together with a contact form η is called a contact 3-manifold([4], [5])....
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31 Jul 2017
TL;DR: In this paper, the curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov’ev are discussed and a foliation on the cotangent bundle over a Lie group G whose leaves are tangent to invariant Hamiltonian vector fields is given.
Abstract: The author proved in the late 1980s that any homogeneous manifold with an intrinsic metric is isometric to some homogeneous quotient space of a connected Lie group by its compact subgroup with an invariant Finslerian or sub-Finslerian metric. In the case of a trivial compact subgroup, the invariant Riemannian or sub-Riemannian metrics are singled out from invariant Finslerian or sub-Finslerian metrics by being in one-to-one correspondence with special one-parameter Gaussian convolutions semigroups of absolutely continuous probability measures. Any such semigroup is generated by a second order hypoelliptic operator. In the present paper, in connection with this, the author discusses briefly the operator definition of lower bound for Ricci curvature by Baudoin–Garofalo. Earlier, Agrachev defined a notion of curvature for sub-Riemannian manifolds. As an alternative, the author discusses in some detail old definitions of curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov’ev. To calculate the Solov’ev sectional and Ricci curvatures for homogeneous sub-Riemannian manifolds, the author suggests to use in some cases special riggings of invariant bracket generating distributions on manifolds. As a justification, we find a foliation on the cotangent bundle over a Lie group G whose leaves are tangent to invariant Hamiltonian vector fields for the Pontryagin–Hamilton function. This function was applied in the Pontryagin maximum principle for the time-optimal problem. The foliation is entirely described by the coadjoint representation of the Lie group G. We also use the canonical symplectic form on and its values for the above mentioned invariant Hamiltonian vector fields. In particular, the above rigging method is applicable to contact sub-Riemannian manifolds, sub-Riemannian Carnot groups, and homogeneous sub-Riemannian manifolds possessing a submetry onto a Riemannian manifold. In Sects. 5 and 6, some examples are presented.
12 citations