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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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17 Jul 2003
TL;DR: In this article, an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol 2, is presented, along with a detailed exposition of the original proof of the Lutz-Martinet theorem.
Abstract: This is an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol 2 After discussing (and proving) some of the fundamental results of contact topology (neighbourhood theorems, isotopy extension theorems, approximation theorems), I move on to a detailed exposition of the original proof of the Lutz-Martinet theorem The text ends with a guide to the literature

106 citations

Journal ArticleDOI
TL;DR: This work introduces a general notion of contact Cauchy-Riemann (CR) lightlike submanifolds and study the geometry of leaves of their distributions and proves characterization theorems on the existence of contact SCR, screen real, invariant, and contact CR minimal light like sub manifolds.
Abstract: We first prove some results on invariant lightlike submanifolds of indefinite Sasakian manifolds. Then, we introduce a general notion of contact Cauchy-Riemann (CR) lightlike submanifolds and study the geometry of leaves of their distributions. We also study a class, namely, contact screen Cauchy-Riemann (SCR) lightlike submanifolds which include invariant and screen real subcases. Finally, we prove characterization theorems on the existence of contact SCR, screen real, invariant, and contact CR minimal lightlike submanifolds.

103 citations


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

  • ...Preliminaries An odd dimensional semi-Riemannian manifold (M̄, 1̄) is called a contact metric manifold [4] if there exists a (1, 1) tensor field φ, a vector fieldV, called the characteristic vector field, and its 1-form η satisfying 1̄(φX, φY) = 1̄(X,Y) − ε η(X)η(Y), 1̄(V,V) = ε, (1....

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  • ...We say thatM has a normal contact structure if Nφ + d η⊗ ξ = 0,whereNφ is the Nijenhuis tensor field of φ [4]....

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Journal ArticleDOI
TL;DR: The lecture notes for the course given at the "XXVII International Fall Workshop on Geometry and Physics" held in Seville (Spain) in September 2018 as mentioned in this paper, reviewed the geometric formulation of...
Abstract: These are the lecture notes for the course given at the “XXVII International Fall Workshop on Geometry and Physics” held in Seville (Spain) in September 2018. We review the geometric formulation of...

96 citations

Posted Content
TL;DR: In this paper, a Minkowski type formula relating the area, the mean curvature, and the volume is obtained for volume-preserving area-stationary surfaces enclosing a given region.
Abstract: We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H^1 endowed with its Carnot-Carath\'eodory distance. By analyzing the first variation of area, we characterize C^2 stationary surfaces for the area as those with mean curvature zero (or constant if a volume-preserving condition is assumed) and such that the characteristic curves meet orthogonally the singular curves. Moreover, a Minkowski type formula relating the area, the mean curvature, and the volume is obtained for volume-preserving area-stationary surfaces enclosing a given region. As a consequence of the characterization of area-stationary surfaces, we refine previous Bernstein type theorems in order to describe entire area-stationary graphs over the xy-plane in H^1. A calibration argument shows that these graphs are globally area-minimizing. Finally, by using the known description of the singular set, the characterization of area-stationary surfaces, and the ruling property of constant mean curvature surfaces, we prove our main results where we classify volume-preserving area-stationary surfaces in H^1 with non-empty singular set. In particular, we deduce the following counterpart to Alexandrov uniqueness theorem in Euclidean space: any compact, connected, C^2 surface in H^1 area-stationary under a volume constraint must be congruent with a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points. As a consequence, we solve the isoperimetric problem in H^1 assuming C^2 smoothness of the solutions.

95 citations

Journal ArticleDOI
TL;DR: In this paper, the role of gauge transformations and Legendre symmetries for metric contact manifolds and their significance in thermodynamics is investigated. And the authors use contact Hamiltonian dynamics to represent thermodynamic processes in a way that resembles the classical Hamiltonian formulation of conservative mechanics.

90 citations