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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02 Apr 2018
TL;DR: In this paper, the notion of Golden Riemannian manifolds of type (r, s) was introduced, and starting from a GRS structure, some remarkable classes of the induced structures on RiemANNian manifold were constructed.
Abstract: We introduce the notion of Golden Riemannian manifolds of type (r, s) and starting from a Golden Riemannian structure, we construct some remarkable classes of the induced structures on Riemannian manifold. Concret examples are given.
7 citations
01 Jan 2008
TL;DR: In this article, Chen-Ricci inequalities for submanifolds tangent to the structure vector field in (κ, μ)-space forms and non-Sasakian manifolds are derived.
Abstract: A basic inequality for submanifolds of a Riemannian manifold, which involves Ricci curvature and squared mean curvature of the submanifold, is con- sidered and is named as Chen-Ricci inequality. The conditions under which the inequality becomes Chen-Ricci equality are discussed. The Chen-Ricci inequal- ity for C-totally real submanifolds in (κ, μ)-space forms and non-Sasakian (κ, μ)- manifolds are obtained. In particular, Chen-Ricci inequality for C-totally real sub- manifolds in Sasakian space forms is derived. Examples of C-totally real subman- ifolds of Sasakian space forms and non-Sasakian (κ, μ)-manifolds, which satisfy Chen-Ricci equality, are presented. The Chen-Ricci inequalities for submanifolds tangent to the structure vector field in (κ, μ)-space forms and non-Sasakian (κ, μ)- manifolds are obtained. It is shown that invariant submanifolds of non-Sasakian (κ, μ)-manifolds and non-Sasakian (κ, μ)-space forms always satisfy Chen-Ricci equality. An obstruction for an invariant submanifold of a non-Sasakian (κ, μ)- space form ˜ M (c) to be Einstein is obtained. In particular, it is deduced that invariant submanifolds of the tangent sphere bundle of a Riemannian manifold of constant curvature c = 1 and having constant ϕ-sectional curvature can not be Einstein. The Chen-Ricci inequality for anti-invariant submanifolds tangent to the structure vector field in non-Sasakian (κ, μ)-manifolds are also found. Con- trary to a known result that there is no anti-invariant submanifold of a Sasakian space form tangent to the structure vector field, which satisfies the correspond- ing Chen-Ricci equality; an example of anti-invariant submanifold tangent to the structure vector field in a non-Sasakian (κ, μ)-manifold with κ =0= μ is pre- sented, which satisfies Chen-Ricci equality. In last, basic inequalities involving scalar curvature and squared mean curvature for C-totally real submanifolds and submanifolds tangent to the structure vector field of (κ, μ)-space forms and non- Sasakian (κ, μ)-manifolds.
7 citations
Cites background from "Riemannian Geometry of Contact and ..."
...A submanifold M in a contact manifold is called an integral submanifold [6] if every tangent vector of M belongs to the contact distribution defined by η = 0....
[...]
...Furthermore, the internal contents of the theory of contact metric structures are very rich [6] and have close substantial interactions with other parts of geometry....
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...On the other hand, the roots of contact geometry lie in differential equations as in 1872 Sophus Lie introduced the notion of contact transformation as a geometric tool to study systems of differential equations (for more details see [1], [2], [6] and [19])....
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...For more details we refer to [6], [7] and [27]....
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...If M̃ is a contact metric manifold equipped with the contact metric structure (φ, ξ, η, g̃), then the structure vector field ξ becomes tangent to the invariant submanifold M , σ (X, ξ) = 0 and M is minimal [6]....
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TL;DR: In this article, the existence of a complete Sasakian sub-Riemannian 3-manifold of constant Webster scalar curvature with constant mean curvature is shown.
Abstract: Let M be a complete Sasakian sub-Riemannian 3-manifold of constant Webster scalar curvature $$\kappa $$
. For any point $$p\in M$$
and any number $$\lambda \in {\mathbb {R}}$$
with $$\lambda ^2+\kappa >0$$
, we show existence of a $$C^2$$
spherical surface $$\mathcal {S}_\lambda (p)$$
immersed in M with constant mean curvature $$\lambda $$
. Our construction recovers in particular the description of Pansu spheres in the first Heisenberg group (Pansu, Conference on differential geometry on homogeneous spaces (Turin, 1983), pp 159–174, 1984) and the sub-Riemannian 3-sphere (Hurtado and Rosales, Math Ann 340(3):675–708, 2008). Then, we study variational properties of $$\mathcal {S}_\lambda (p)$$
related to the area functional. First, we obtain uniqueness results for the spheres $$\mathcal {S}_\lambda (p)$$
as critical points of the area under a volume constraint, thus providing sub-Riemannian counterparts to the theorems of Hopf and Alexandrov for CMC surfaces in Riemannian 3-space forms. Second, we derive a second variation formula for admissible deformations possibly moving the singular set, and prove that $$\mathcal {S}_\lambda (p)$$
is a second order minimum of the area for those preserving volume. We finally give some applications of our results to the isoperimetric problem in sub-Riemannian 3-space forms.
7 citations
Posted Content•
TL;DR: In this article, it was shown that a normal nearly Kenmotsu manifold is a warped product of real line and nearly K\"ahler manifold, and that there exists no nearly K''ahler hypersurface of nearly K ''ahler manifolds.
Abstract: This is an expository paper, which provides a first approach to nearly Kenmotsu manifolds. The purpose of this paper is to focus on nearly Kenmotsu manifolds and get some new results from it. We prove that for a nearly Kenmotsu manifold is locally isometric to warped product of real line and nearly K\"ahler manifold. Finally, we prove that there exist no nearly Kenmotsu hypersurface of nearly K\"ahler manifold. It is shown that a normal nearly Kenmotsu manifold is Kenmotsu manifold.
7 citations
Cites background from "Riemannian Geometry of Contact and ..."
...and M is said to be an almost contact metric manifold if it is endowed with an almost contact metric structure [1], [16]....
[...]
19 May 2015
TL;DR: In this article, a correspondence between the Ganchev-Mihova-Gribachev classification of the studied manifolds and the explicit matrix representation of Lie groups is established.
Abstract: The object of investigation are Lie groups considered as almost contact B-metric manifolds of the lowest dimension three. It is established a correspondence of all basic-class-manifolds of the Ganchev-Mihova-Gribachev classification of the studied manifolds and the explicit matrix representation of Lie groups. Some known Lie groups are equipped with almost contact B-metric structure of different types.
7 citations