Showing papers on "Reeb vector field published in 2006"
TL;DR: In this article, it was shown that the volume of a Sasaki-Einstein manifold is always an algebraic number, relative to that of the round sphere, and that it is a function on the space of Reeb vector fields.
Abstract: We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein-Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi-Yau cone X, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat-Heckman formula and also to a limit of a certain equivariant index on X that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a Sasaki-Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n=3 these results provide, via AdS/CFT, the geometric counterpart of a-maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kahler-Einstein metric.
46 citations
01 Jan 2006
TL;DR: In this paper, the authors discuss some of the more fundamental differential topological aspects of contact geometry, such as classification questions in dimension 3, dynamics of the Reeb vector field, various notions of symplectic fillability, transverse, and Legendrian knots and links.
Abstract: Publisher Summary There are many excellent surveys covering specific aspects of contact geometry—for example, classification questions in dimension 3, dynamics of the Reeb vector field, various notions of symplectic fillability, transverse, and Legendrian knots and links. The chapter discusses some of the more fundamental differential topological aspects of contact geometry.
39 citations
TL;DR: In this paper, the vanishing Lie derivative of the shape operator along the direction of the Reeb vector field was used to characterize real hypersurfaces of type A in a complex two-plane Grassmannian G2(C m+2 ) which are tubes over totally geodesic G 2(Cm+1 ) in G2m+2 ).
Abstract: In this paper we give a characterization of real hypersurfaces of type A in a complex two- plane Grassmannian G2(C m+2 ) which are tubes over totally geodesic G2(C m+1 ) in G2(C m+2 ) in terms of the vanishing Lie derivative of the shape operator A along the direction of the Reeb vector field �.
31 citations
TL;DR: In this paper, the authors study the functional given by the squared $L 2 -norm of the scalar curvature and prove that its critical points are Sasakian metrics that are transversally extremal.
Abstract: Let $M$ be a closed manifold of Sasaki type. A polarization of $M$ is defined by a Reeb vector field, and for one such, we consider the set of all Sasakian metrics compatible with it. On this space, we study the functional given by the squared $L^2$-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open.
6 citations
TL;DR: In this article, a volume-preserving fieldX on a 3-manifold is the one that satisfies LXΩ ≡ 0 for some volume Ω, while the Reeb vector field of a contact form is of volume preserving, but not conversely.
Abstract: Volume–preserving fieldX on a 3–manifold is the one that satisfies LXΩ ≡ 0 for some volume Ω. The Reeb vector field of a contact form is of volume–preserving, but not conversely. On the basis of Geiges–Gonzalo’s parallelization results, we obtain a volume–preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume–preserving fields. From many aspects, we discuss the distinction between volume–preserving fields and Reeb–like fields. We establish a duality between volume–preserving fields and h–closed 2–forms to understand such distinction. We also give two kinds of non–Reeb–like but volume–preserving vector fields to display such distinction.