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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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Citations
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Journal ArticleDOI
TL;DR: In this paper, a fundamental exterior differential system of Riemannian geometry was discovered, which is an intrinsic and invariant global system of differential forms of degree n associated to any given oriented Riemmannian manifold of dimension n+1.
Abstract: We discover a fundamental exterior differential system of Riemannian geometry; indeed, an intrinsic and invariant global system of differential forms of degree $n$ associated to any given oriented Riemannian manifold $M$ of dimension $n+1$. The framework is that of the tangent sphere bundle of $M$. We generalise to a Riemannian setting some results from the theory of hypersurfaces in flat Euclidean space. We give new applications and examples of the associated Euler-Lagrange differential systems.

4 citations

Journal ArticleDOI
TL;DR: In this article, the Riemannian geometry of contact manifolds with respect to a fixed admissible metric was studied and the Reeb vector field was shown to be unitary and orthogonal to the contact distribution.
Abstract: We study the Riemannian geometry of contact manifolds with respect to a fixed admissible metric, making the Reeb vector field unitary and orthogonal to the contact distribution, under the assumption that the Levi–Tanaka form is parallel with respect to a canonical connection with torsion.

4 citations

Journal ArticleDOI
TL;DR: In this article, a new concept, namely, D-general warping, is introduced by extending some geometric notions defined by Blair and Tanno, and necessary and sufficient conditions for (M, g) to be Einstein, quasi-Einstein or −Einstein are provided.
Abstract: A new concept, namely, D-general warping $$(M=M_1\times M_2,g)$$ , is introduced by extending some geometric notions defined by Blair and Tanno. Corresponding to a result of Tanno in almost contact metric geometry, an outcome in almost Hermitian context is provided here. On $$T^*M$$ , the Riemann extension (introduced by Patterson and Walker) of the Levi–Civita connection on (M, g) is characterized. A Laplacian formula of g is obtained and the harmonicity of functions and forms on (M, g) is described. Some necessary and sufficient conditions for (M, g) to be Einstein, quasi-Einstein or $$\eta $$ -Einstein are provided. The cases when the scalar (resp. sectional) curvature is positive or negative are investigated and an example is constructed. Some properties of (M, g) for being a gradient Ricci soliton are considered. In addition, D-general warpings which are space forms (resp. of quasi-constant sectional curvature in the sense of Boju, Popescu) are studied.

4 citations

Journal ArticleDOI
TL;DR: In this article, the generalized Faddeev-Hopf model was used to construct pseudo horizontally weakly conformal maps, which extend both holomorphic and (semi)conformal maps into an almost Hermitian manifold.
Abstract: Pseudo horizontally weakly conformal maps [16] extend both holomorphic and (semi)conformal maps into an almost Hermitian manifold. We find critical points for the (generalized) Faddeev-Hopf model [28] in this larger class.

4 citations

OtherDOI
TL;DR: In this article, it was shown that for regular contact forms there exists a bijective correspondence between the $C^0$ limits of sequences of smooth strictly contact isotopies and the limits with respect to the contact distance of their corresponding Hamiltonians.
Abstract: We prove that for regular contact forms there exists a bijective correspondence between the $C^0$ limits of sequences of smooth strictly contact isotopies and the limits with respect to the contact distance of their corresponding Hamiltonians.

4 citations


Cites background from "Riemannian Geometry of Contact and ..."

  • ...ouple (g,J) of a riemannian metric gand a 1-1 tensor field Jsuch that (1) Jξ= 0 (2) J2X= −X+α(X)ξ (3) dα(X,Y) = g(X,JY) (4) g(X,Y) = g(JX,JY)+α(X)α(Y) Such contact metric structures exist in abundance [Bla02]. In the regular contact case, the connection form (contact form) αidentifies TxBwith Dx, where D = Ker αis the contact distribution. Hence (g,J) induces a split metric gM = π∗gB + α⊗ α. Denote by dM 0...

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