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 existence and properties of uniform lattices in Lie groups were investigated and it was shown that in dimension 5, there are exactly seven connected and simply connected contact Lie groups with uniform lattice, all of which are solvable.
Abstract: We investigate the existence and properties of uniform lattices in Lie groups and use these results to prove that, in dimension 5, there are exactly seven connected and simply connected contact Lie groups with uniform lattices, all of which are solvable. Issues of symplectic boundaries are explored, as well. It is also shown that the special affine group has no uniform lattic e. 1
4 citations
TL;DR: In this article , a Ricci almost soliton whose associated vector field is projective is shown to have vanishing Cotton tensor, divergence-free Bach tensor and Ricci tensor as conformal Killing.
Abstract: Abstract A Ricci almost soliton whose associated vector field is projective is shown to have vanishing Cotton tensor, divergence-free Bach tensor and Ricci tensor as conformal Killing. For the compact case, a sharp inequality is obtained in terms of scalar curvature.We show that every complete gradient Ricci soliton is isometric to the Riemannian product of a Euclidean space and an Einstein space. A complete K-contact Ricci almost soliton whose associated vector field is projective is compact Einstein and Sasakian.
4 citations
27 Mar 2019
TL;DR: In this paper, the generalized Wintgen inequality for Legendrian submanifolds in almost Kenmotsu statistical manifolds was obtained, and the main interest of the present paper was to obtain the generalized generalized WINTgen inequality.
Abstract: Main interest of the present paper is to obtain the generalized Wintgen inequality for Legendrian submanifolds in almost Kenmotsu statistical manifolds.
4 citations
Cites background from "Riemannian Geometry of Contact and ..."
...The manifold M is said to be an almost contact manifold if it is endowed with an almost contact structure [6]....
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...If N (1) vanishes identically, then the almost contact manifold (structure) is said to be normal [6]....
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Posted Content•
TL;DR: In this paper, it was shown that any simply connected compact 3-Sasakian manifold, of dimension seven, is formal if and only if its second Betti number is $b_2 < 2.
Abstract: We prove that any simply connected compact 3-Sasakian manifold, of dimension seven, is formal if and only if its second Betti number is $b_2<2$. In the opposite, we show an example of a 7-dimensional Sasaki-Einstein manifold, with second Betti number $b_2\geq 2$, which is formal. Therefore, such an example does not admit any 3-Sasakian structure. Examples of 7-dimensional simply connected compact formal Sasakian manifolds, with $b_2\geq 2$, are also given.
4 citations
Cites methods from "Riemannian Geometry of Contact and ..."
...We recall the notion of 3-Sasakian manifolds following [4, 6, 5]....
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Journal Article•
TL;DR: In this paper, the mean curvature vector of almost contact curves has been analyzed on trans-Sasakian manifolds with gTWO connections and some properties of slant curves on the same manifolds have been established.
Abstract: The object of the present paper is to characterize three-dimensional trans-Sasakian generalized Sasakian space forms admitting biharmonic almost contact curves with respect to generalized Tanaka Webster Okumura (gTWO) connections and to give illustrative examples. The mean curvature vector of almost contact curves has been analyzed on trans-Sasakian manifolds with gTWO connections. Some properties of slant curves on the same manifolds have been established. Finally curvature and torsion, with respect to gTWO connections, of C-parallel and C-proper slant curves in three-dimensional almost contact metric manifolds have been deduced.
4 citations
Additional excerpts
...Let M be a connected almost contact metric manifold with an almost contact metric structure (φ, ξ, η, g), that is, φ is a (1, 1) tensor field, ξ is a vector field, η is a 1-form and g is a compatible Riemannian metric such that (see [6]) φ(2)X = −X + η(X)ξ, η(ξ) = 1, φξ = 0, ηφ = 0, g(φX, φY ) = g(X,Y )− η(X)η(Y ), g(X,φY ) = −g(φX, Y ), g(X, ξ) = η(X), for all X,Y ∈ T (M) [3]....
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