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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Journal Article•
TL;DR: In this article, the Ricci tensor tensor is used to classify a 3D almost co-Kahler manifold M3 under some additional conditions related to Ricci's tensor.
Abstract: Let M3 be a three-dimensional almost co-Kahler manifold whose Reeb vector field is harmonic. We obtain some local classification results of M3 under some additional conditions related to the Ricci tensor.
4 citations
Cites background from "Riemannian Geometry of Contact and ..."
...According to Blair [2], the normality of an almost contact structure is expressed by [φ, φ] = −2dη ⊗ ξ, where [φ, φ] denotes the Nijenhuis tensor of φ defined by [φ, φ](X,Y ) = φ2[X,Y ] + [φX, φY ]− φ[φX, Y ]− φ[X,φY ] for any vector fields X,Y on M2n+1....
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...Note that the (almost) co-Kähler manifolds in fact are the (almost) cosymplectic manifolds studied in [1, 2, 6, 7, 10]....
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...Almost co-Kähler manifolds were first introduced by Blair [1] and studied by Goldberg and Yano [7] and Olszak et al. [6, 10]....
[...]
...According to Blair [2], the normality of an almost contact structure is expressed by [φ, φ] = −2dη ⊗ ξ, where [φ, φ] denotes the Nijenhuis tensor of φ defined by [φ, φ](X,Y ) = φ2[X,Y ] + [φX, φY ]− φ[φX, Y ]− φ[X,φY ] for any vector fields X,Y on M2n+1....
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Posted Content•
TL;DR: In this article, the Ricci solitons and their analogs within the framework of contact geometry were studied and proved to be locally isometric to the product of a Euclidean space and a sphere of constant curvature.
Abstract: Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $(m,\rho)$-quasi-Einstein structure on a contact metric manifold. First, we prove that if a $K$-contact or Sasakian manifold $M^{2n+1}$ admits a closed $(m,\rho)$-quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $2n(2n+1)$, and for the particular case -- a non-Sasakian $(k,\mu)$-contact structure -- it is locally isometric to the product of a Euclidean space $\RR^{n+1}$ and a sphere $S^n$ of constant curvature $4$. Next, we prove that if a compact contact or $H$-contact metric manifold admits an $(m,\rho)$-quasi-Einstein structure, whose potential vector field $V$ is collinear to the Reeb vector field, then it is a $K$-contact $\eta$-Einstein manifold.
4 citations
TL;DR: In this article, it was shown that the stabilized manifolds X × S 1 and Y × S 2 are diffeomorphic and non-deformation equivalent in cosymplectic sense.
Abstract: The complex surface X obtained by 8 points blown up on ℂℙ2 and Barlow’s surface Y are homeomorphic, but not diffeomorphic. Using the Gromov-Witten invariant Ruan showed that the stabilized manifolds X × S 2 and Y × S 2 are not deformation equivalent. In this note, we show that the stabilized manifolds X × S 1 and Y × S 1 are diffeomorphic and non-deformation equivalent in cosymplectic sense.
4 citations
TL;DR: In this paper, the structure tensors of almost contact metric structures are explicitly calculated and the transformations of these tensors under conformal transformations are described, and the results obtained are used to study the behavior of the most interesting classes of most interesting almost contact structures.
Abstract: The so-called structure tensors of almost contact metric structures, which play a key role in the geometry of almost contact metric structures, are explicitly calculated. The transformations of these tensors under conformal transformations of almost contact metric structures are described. The results obtained are used to study the behavior of the most interesting classes of almost contact structures under conformal transformations.
4 citations
TL;DR: In this paper, the authors provide models that are as close as possible to being formal for a large class of compact manifolds that admit a transversely Kahler structure, including Vaisman and quasi-Sakian manifolds.
Abstract: We provide models that are as close as possible to being formal for a large class of compact manifolds that admit a transversely Kahler structure, including Vaisman and quasi-Sasakian manifolds. As an application we are able to classify the corresponding nilmanifolds.
4 citations