scispace - formally typeset
Search or ask a question
Topic

Reeb vector field

About: Reeb vector field is a research topic. Over the lifetime, 254 publications have been published within this topic receiving 4118 citations.


Papers
More filters
01 Jan 2008
TL;DR: In this article, it was shown that the volume function of a Sasaki-Einstein manifold is a function on the space of Reeb vector fields, and that it can be computed in terms of topological fixed point data.
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 � � .

1 citations

Posted Content
TL;DR: In this paper, it was shown that every finite energy holomorphic map (J$-holomorphic map) converges to a periodic orbit of the Reeb vector field as $s\to \infty.
Abstract: We prove that every finite energy $J$-holomorphic map $u(s,t):\mathbb R\times S^1 \rightarrow {\mathbb R} \times \widetilde{M}$ exponentially converges to a periodic orbit of Reeb vector field of $\widetilde M,$ as $s\to \infty.$

1 citations

Posted Content
TL;DR: In this article, almost Ricci-like solitons on almost contact B-metric manifolds with torse-forming potential have been studied and necessary and sufficient conditions have been found for a number of properties of the curvature tensor and its Ricci tensor.
Abstract: Almost Ricci-like solitons on almost contact B-metric manifolds with torse-forming potential have been studied. The case in which this potential is further vertical is considered, i.e. the potential is pointwise collinear to the Reeb vector field. The conditions under which the investigated manifolds with almost Ricci-like solitons are equivalent to almost Einstein-like manifolds have been established. In this case, necessary and sufficient conditions have been found for a number of properties of the curvature tensor and its Ricci tensor. Then some results are obtained for a parallel symmetric second-order covariant tensor on the manifolds under study. Finally, an explicit example of an arbitrary dimension is given and some of the results are illustrated.

1 citations

Posted Content
TL;DR: In this paper, it was shown that there exist at least two geometrically distinct symmetric closed trajectories of the Reeb vector field on a convex Hamiltonian energy surface which is symmetric with respect to the origin.
Abstract: In this article, let $\Sigma\subset\R^{2n}$ be a compact convex Hamiltonian energy surface which is symmetric with respect to the origin. where $n\ge 2$. We prove that there exist at least two geometrically distinct symmetric closed trajectories of the Reeb vector field on $\Sg$.

1 citations

Posted Content
TL;DR: In this paper, the authors studied non-Hopf real hypersurfaces satisfying the equality case of the inequality under the condition that the mean curvature is constant along each integral curve of the Reeb vector field.
Abstract: It was proved in Chen's paper \cite{chen} that every real hypersurface in the complex projective plane of constant holomorphic sectional curvature $4$ satisfies $$ \delta(2)\leq \frac{9}{4}H^2+5,$$ where $H$ is the mean curvature and $\delta(2)$ is a $\delta$-invariant introduced by him. In this paper, we study non-Hopf real hypersurfaces satisfying the equality case of the inequality under the condition that the mean curvature is constant along each integral curve of the Reeb vector field. We describe how to obtain all such hypersurfaces.
Network Information
Related Topics (5)
Manifold
18.7K papers, 362.8K citations
89% related
Symplectic geometry
18.2K papers, 363K citations
85% related
Scalar curvature
12.7K papers, 296K citations
84% related
Differential geometry
10.9K papers, 305K citations
84% related
Holomorphic function
19.6K papers, 287.8K citations
84% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20221
202126
202028
201918
201813
201721