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, the CR analogue of the three-circle theorem in a complete non-compact pseudohermitian manifold of vanishing torsion was shown to hold if its pseudhermitians sectional curvature is nonnegative.
Abstract: This paper mainly focuses on the CR analogue of the three-circle theorem in a complete noncompact pseudohermitian manifold of vanishing torsion being odd dimensional counterpart of K\"ahler geometry. In this paper, we show that the CR three-circle theorem holds if its pseudohermitian sectional curvature is nonnegative. As an application, we confirm the first CR Yau's uniformization conjecture and obtain the CR analogue of the sharp dimension estimate for CR holomorphic functions of polynomial growth and its rigidity when the pseudohermitian sectional curvature is nonnegative. This is also the first step toward second and third CR Yau's uniformization conjecture. Moreover, in the course of the proof of the CR three-circle theorem, we derive CR sub-Laplacian comparison theorem. Then Liouville theorem holds for positive pseudoharmonic functions in a complete noncompact pseudohermitian (2n+1)-manifold of vanishing torsion and nonnegative pseudohermitian Ricci curvature.
3 citations
Cites background from "Riemannian Geometry of Contact and ..."
...the rigidity part, we just claim that if (3.14) dimC O CR d (M) = dimC Od (H n) , then (M,J,θ) is CR-isomorphic to (2n+1)-dimensional Heisenberg group Hn. From Proposition 4.1in [T] (orTheorem 7.15 in[B]), it suffices toshow that M has constant J-holomorphic sectional curvature −3. From the equation, for any p ∈ M, Z ∈ (T 1,0M)p with |Z| = 1, Rθ Z,Z,Z,Z = R Z,Z,Z,Z +gθ Z ∧ Z Z,Z +2dθ Z,Z gθ JZ,Z = R Z,...
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TL;DR: In this paper, the notion of screen pseudo-slant light-like submanifolds of indefinite Kaehler manifolds was introduced, and the characterization theorem with some non-trivial examples of such submansifolds was given.
Abstract: In this paper, we introduce the notion of screen pseudo-slant lightlike submanifolds of indefinite Kaehler manifolds giving characterization theorem with some non-trivial examples of such submanifolds. Integrability conditions of distributions D1, D2 and RadTM on screen pseudo-slant lightlike submanifolds of an indefinite Kaehler manifold have been obtained. Further we obtain necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic.
3 citations
DOI•
15 Oct 2020
TL;DR: In this article, it was shown that for a -almost-Kenmotsu manifold with and, the tensor vanishes and every conformal vector field which leaves the curvature tensor invariant is Killing.
Abstract: First we consider almost Kenmotsu manifolds which satisfy Codazzi condition for and , and we prove that in such cases the tensor vanishes. Next, we prove that an almost Kenmotsu manifold having constant -sectional curvature which is locally symmetric is a Kenmotsu manifold of constant curvature . We also prove that, for a -almost Kenmotsu manifold of with , every conformal vector field is Killing. Finally, we prove that if is a -almost Kenmotsu manifold with and , then the vector field which leaves the curvature tensor invariant is Killing.
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Journal Article•
TL;DR: In this article, an anti-invariant, minimal, pseudoparallel and Riccigeneralized pseudop-arallel submanifold M of a Kenmotsu space form Mf(c) for which ξ is tangent to M is considered.
Abstract: We consider an anti-invariant, minimal, pseudoparallel and Riccigeneralized pseudoparallel submanifold M of a Kenmotsu space form Mf(c), for which ξ is tangent to M.
3 citations
Additional excerpts
...Submanifolds of Kenmotsu manifolds Let M̃ be a (2n + 1)-dimensional almost contact metric manifold with structure (φ, ξ, η, g), where φ is a tensor field of type (1, 1), ξ a vector field, η a 1-form and g the Riemannian metric on M̃ satisfying φ 2 = −I + η ⊗ ξ, φξ = 0, η(ξ) = 1, η ◦ φ = 0, g(φX, φY ) = g(X,Y ) − η(X)η(Y ), η(X) = g(X, ξ), g(φX,Y ) = −g(X,φY ), for all vector fields X, Y on M̃ [3]....
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Posted Content•
TL;DR: In this article, the authors characterize unimodular solvable Lie algebras with Vaisman structures in terms of coKahler flat Lie algesbras equipped with a suitable derivation.
Abstract: We characterize unimodular solvable Lie algebras with Vaisman structures in terms of Kahler flat Lie algebras equipped with a suitable derivation. Using this characterization we obtain algebraic restrictions for the existence of Vaisman structures and we establish some relations with other geometric notions, such as Sasakian, coKahler and left-symmetric algebra structures. Applying these results we construct families of Lie algebras and Lie groups admitting a Vaisman structure and we show the existence of lattices in some of these families, obtaining in this way many examples of new solvmanifolds equipped with invariant Vaisman structures.
3 citations