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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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Journal ArticleDOI
TL;DR: Dai et al. as discussed by the authors proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in [Stable and unstable Einstein warped products, preprint (2015), arXiv:1507.01782v1].
Abstract: Riemannian manifolds with nonzero Killing spinors are Einstein manifolds. Kroncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in [Stable and unstable Einstein warped products, preprint (2015), arXiv:1507.01782v1]. In this paper, we obtain a new proof for this stability result by using a Bochner-type formula in [X. Dai, X. Wang and G. Wei, On the stability of Riemannian manifold with parallel spinors, Invent. Math. 161(1) (2005) 151–176; M. Wang, Preserving parallel spinors under metric deformations, Indiana Univ. Math. J. 40 (1991) 815–844]. Moreover, existence of real Killing spinors is closely related to the Sasaki–Einstein structure. A regular Sasaki–Einstein manifold is essentially the total space of a certain principal S1-bundle over a Kahler–Einstein manifold. We prove that if the base space is a product of two Kahler–Einstein manifolds then the regular Sasaki–Einstein manifold is unstable. This provides us many new examples of unstable manifolds with real Killing spinors.

9 citations

Posted Content
TL;DR: In this article, the generalized almost complex and almost Hermitian structures that are locally conformal to integrable and to generalized K\"ahler structures, respectively, are characterized.
Abstract: A conformal change of $TM\oplus T^*M$ is a morphism of the form $(X,\alpha)\mapsto(X,e^\tau\alpha)$ $(X\in TM,\alpha\in T^*M,\tau\in C^\infty(M))$ We characterize the generalized almost complex and almost Hermitian structures that are locally conformal to integrable and to generalized K\"ahler structures, respectively, and give examples of such structures

9 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that if a contact metric manifold admits a quasi-Yamabe soliton whose soliton field is the V-Ric vector field, then the Ricci operator Q commutes with the (1, 1) tensor.
Abstract: We prove that a contact metric manifold does not admit a proper quasi-Yamabe soliton ($$M,\, g,\,\xi ,\,\lambda ,\,\mu $$). Next we prove that if a contact metric manifold admits a quasi-Yamabe soliton ($$M,\, g,\, V,\, \lambda ,\, \mu $$) whose soliton field is pointwise collinear with the Reeb vector field, then the scalar curvature is constant, and the quasi-Yamabe soliton reduces to Yamabe soliton. Finally, it is shown that if a contact metric manifold admits a quasi-Yamabe soliton whose soliton field is the V-Ric vector field, then the Ricci operator Q commutes with the (1, 1) tensor $$\phi $$. As a consequence of the main result we obtain several corollaries.

9 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of conformally Anosov Reeb flows on a 3-manifold has been shown to be impossible and curvature conditions on a metric compatible with a contact structure have been given.
Abstract: We provide obstructions to the existence of conformally Anosov Reeb flows on a 3-manifold that partially generalize similar obstructions to Anosov Reeb flows. In particular, we show S 3 does not admit conformally Anosov Reeb flows. We also give a Riemannian geometric condition on a metric compatible with a contact structure implying that a Reeb field is Anosov. From this we can give curvature conditions on a metric compatible with a contact structure that implies universal tightness of the contact structure among other things.

9 citations

Journal ArticleDOI
TL;DR: In this paper, a generalization of classical coKahler geometry from the point of view of generalized contact metric geometry is proposed. But this generalization is restricted to manifolds that do not admit a classical co-kahler structure.

9 citations