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, a family of paracontact metric structures is presented, which is invariant under the condition that the metric structures are pseudo-Riemannian metrics of suitable signature on the unit tangent sphere bundle.
Abstract: Starting from $$g$$
-natural pseudo-Riemannian metrics of suitable signature on the unit tangent sphere bundle $$T_1 M$$
of a Riemannian manifold $$(M,\langle ,\rangle )$$
, we construct a family of paracontact metric structures. We prove that this class of paracontact metric structures is invariant under $$\mathcal {D}$$
-homothetic deformations, and classify paraSasakian and paracontact $$(\kappa ,\mu )$$
-spaces inside this class. We also present a way to build paracontact $$(\kappa ,\mu )$$
-spaces from corresponding contact metric structures on $$T_1 M$$
.
22 citations
TL;DR: In this article, a complete study of almost α -paracosymplectic manifolds with para-Kaehler leaves is presented and general curvature identities are proved.
22 citations
Cites background from "Riemannian Geometry of Contact and ..."
...This characterization is well known in the Riemannian case ([3, 23, 25])....
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01 Jan 2007
TL;DR: In this paper, the stability properties of nonminimal biharmonic anti-invariant submanifolds in Sasakian space forms were investigated and some classification results were obtained.
Abstract: We obtain some classification results and the stability condi- tions of nonminimal biharmonic anti-invariant submanifolds in Sasakian space forms.
22 citations
Cites background from "Riemannian Geometry of Contact and ..."
...Invariant submanifolds are minimal (see, [1]) and hence automatically critical points of the 2-energy functional, that is, biharmonic (2-harmonic) in the sense of Eells and Sampson [8]....
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...for all X, Y ∈ TN (see, for instance, [1])....
[...]
...The tangent planes in TpN 2n+1 which is invariant under φ are called φ-section (see, [1])....
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TL;DR: The Walczak formula is a very nice tool for understanding the geometry of a Riemannian manifold equipped with two orthogonal complementary distributions as discussed by the authors, and it simplifies to a Bochner-type formula when dealing with Kahler manifolds and holomorphic (integrable) distributions.
22 citations
Posted Content•
TL;DR: In this paper, the authors show that the partition function of Chern-Simons theory admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on a Seifert manifold.
Abstract: We reconsider Chern-Simons gauge theory on a Seifert manifold M (the total space of a nontrivial circle bundle over a Riemann surface). When M is a Seifert manifold, Lawrence and Rozansky have shown from the exact solution of Chern-Simons theory that the partition function has a remarkably simple structure and can be rewritten entirely as a sum of local contributions from the flat connections on M. We explain how this empirical fact follows from the technique of non-abelian localization as applied to the Chern-Simons path integral. In the process, we show that the partition function of Chern-Simons theory on M admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on M.
22 citations