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 article, the authors studied almost cosymplectic statistical manifolds and obtained basic properties of such manifolds, including the properties of almost contact and almost contact statistical manifold.
Abstract: This paper is a study of almost contact statistical manifolds. Especially this study is focused on almost cosymplectic statistical manifolds. We obtained basic properties of such manifolds. A chara...
8 citations
Cites background from "Riemannian Geometry of Contact and ..."
...an almost contact manifold is called a contact metric manifold [3]....
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...The manifold M is said to be an almost contact manifold if it is endowed with an almost contact structure [3]....
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...If N (1) vanishes identically, then the almost contact manifold (structure) is said to be normal [3]....
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TL;DR: In this paper, the authors improved Tanno's result that a conformal vector field on a contact metric manifold M is a strictly infinitesimal contact transformation, and thus it is an automorphism of M by waiving the "strictness" in the hypothesis.
Abstract: First we improve a result of Tanno that says "If a conformal vector field on a contact metric manifold M is a strictly infinitesimal contact transformation, then it is an infinitesimal automorphism of M" by waiving the "strictness" in the hypothesis. Next, we prove that a (k, μ)-contact manifold admitting a non-Killing conformal vector field is either Sasakian or has k = –n – 1, μ = 1 in dimension > 3; and Sasakian or flat in dimension 3. In particular, we show that (i) among all compact simply connected (k, μ)-contact manifolds of dimension > 3, only the unit sphere S2n+1 admits a non-Killing conformal vector field, and (ii) a conformal vector field on the unit tangent bundle of a space-form of dimension > 2 is necessarily Killing.
8 citations
TL;DR: Oh and Wang as discussed by the authors introduced a family of canonical affine connections on the contact manifold, which are associated to each contact triad (Q, λ, J) where λ is a contact form and J: ξ → ξ is an endomorphism compatible to d λ.
Abstract: We introduce a family of canonical affine connections on the contact manifold (Q, ξ), which is associated to each contact triad (Q, λ, J) where λ is a contact form and J: ξ → ξ is an endomorphism with \(J^{2} = -id\) compatible to d λ. We call a particular one in this family the contact triad connection of (Q, λ, J) and prove its existence and uniqueness. The connection is canonical in that the pull-back connection ϕ ∗∇ of a triad connection ∇ becomes the triad connection of the pull-back triad \((Q,\phi ^{{\ast}}\lambda,\phi ^{{\ast}}J)\) for any diffeomorphism ϕ: Q → Q. It also preserves both the triad metric \(g:= d\lambda (\cdot,J\cdot ) +\lambda \otimes \lambda\) and J regarded as an endomorphism on \(TQ = \mathbb{R}\{X_{\lambda }\}\oplus \xi\), and is characterized by its torsion properties and the requirement that the contact form λ be holomorphic in the CR-sense. In particular, the connection restricts to a Hermitian connection ∇ π on the Hermitian vector bundle (ξ, J, g ξ ) with \(g_{\xi } = d\lambda (\cdot,J\cdot )\vert _{\xi }\), which we call the contact Hermitian connection of (ξ, J, g ξ ). These connections greatly simplify tensorial calculations in the sequels (Oh and Wang, The Analysis of Contact Cauchy-Riemann maps I: a priori Ck estimates and asymptotic convergence, preprint. arXiv:1212.5186, 2012; Oh and Wang, Analysis of contact instantons II: exponential convergence for the Morse-Bott case, preprint. arXiv:1311.6196, 2013) performed in the authors’ analytic study of the map w, called contact instantons, which satisfy the nonlinear elliptic system of equations
$$\displaystyle{\overline{\partial }^{\pi }w = 0,\,d(w^{{\ast}}\lambda \circ j) = 0}$$
of the contact triad (Q, λ, J).
8 citations
TL;DR: In this paper, the authors consider the Levi-Civita connection and the kth g-Tanaka-Webster connection on a real hypersurface M in a complex projective space.
Abstract: On a real hypersurface M in a complex projective space, we can consider the Levi-Civita connection and for any nonnull constant k the kth g-Tanaka–Webster connection Therefore, we can also consider their associated Lie derivatives We classify real hypersurfaces such that both the Lie derivatives associated with the Levi-Civita connection and the kth g-Tanaka–Webster connection either in the direction of the structure vector field ξ or in any direction of the maximal holomorphic distribution coincide when we apply them to the shape operator of M
8 citations