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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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TL;DR: In this article, it was shown that the Floer homology of the cotangent bundle of a closed manifold is isomorphic to the loop space homology relative to the constant loops.
Abstract: By a well-known theorem first proved by Viterbo, the Floer homology of the cotangent bundle of a closed manifold is isomorphic to the homology of its loop space. We prove that, when restricted to positive Floer homology resp. loop space homology relative to the constant loops, this isomorphism intertwines various constructions of secondary pair-of-pants coproducts with the loop homology coproduct. The proof uses compactified moduli spaces of punctured annuli. We extend this result to reduced Floer resp. loop homology (essentially homology relative to a point), and we show that on reduced loop homology the loop product and coproduct satisfy Sullivan's relation. Along the way, we show that the Abbondandolo-Schwarz quasi-isomorphism going from the Floer complex of quadratic Hamiltonians to the Morse complex of the energy functional can be turned into a filtered chain isomorphism by using linear Hamiltonians and the square root of the energy functional.
5 citations
03 Oct 2019
TL;DR: The Ricci tensor, phi-Ricci tensors, and the characteristic Jacobi operator on cosymplectic 3-manifolds were investigated in this paper.
Abstract: The Ricci tensor, phi-Ricci tensor and the characteristic Jacobi operator on cosymplectic 3-manifolds are investigated.
5 citations
Cites background from "Riemannian Geometry of Contact and ..."
...For general information on almost contact Riemannian geometry, we refer to [2]....
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TL;DR: In this paper, the authors studied the magnetic trajectories in the generalized Heisenberg group H ( n, 1 ) of dimension ( 2 n + 1 ) endowed with its quasi-Sasakian structure and proved that the trajectories are Frenet curves of maximum order 5.
Abstract: We study the magnetic trajectories in the generalized Heisenberg group H ( n , 1 ) of dimension ( 2 n + 1 ) endowed with its quasi-Sasakian structure. We prove that the trajectories are Frenet curves of maximum order 5 and we completely classify them.
5 citations
TL;DR: Among the studies on pseudo-Hermitian geometry of strictly pseudo-convex almost CR manifolds, this paper studied especially the two kinds of pseudo-Einstein structures and related problems.
5 citations
TL;DR: In this article, the authors show that the initial space-like slice of the generalized Robertson-Walker (GRW) space time is Einstein if and only if the electric part of the space-time Weyl conformal tensor vanishes.
Abstract: First we show that the initial space-like slice of the generalized Robertson–Walker (GRW) space–time is Einstein if and only if the electric part of the space–time Weyl conformal tensor vanishes. We also show that the purely spatial component of the space–time Weyl tensor vanishes if and only if the initial slice has constant curvature. Then, assuming that a spatially complete synchronous space–time solution of Einstein’s equations admits a conformal vector field whose normal component is non-constant and spatial component is conformal on each space-like slice, we show that each slice is conformal to (i) a Euclidean space, (ii) a sphere, or (iii) a hyperbolic space, or (iv) Riemannian product of an open interval and and a Riemannian space. Finally, under the preceding hypotheses, if a slice has a K -contact metric, then it is isometric to a unit sphere.
5 citations