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.

Real hypersurfaces in the complex projective plane satisfying an equality involving $\delta(2)$

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.

Almost $\eta$-Ricci solitons on Kenmotsu manifolds

Abstract: In this paper we characterize the Einstein metrics in such broader classes of metrics as almost $\eta$-Ricci solitons and $\eta$-Ricci solitons on Kenmotsu manifolds, and generalize some results of other authors. First, we prove that a Kenmotsu metric as an $\eta$-Ricci soliton is Einstein metric if either it is $\eta$-Einstein or the potential vector field $V$ is an infinitesimal contact transformation or $V$ is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost $\eta$-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of $\eta$-Ricci solitons and gradient $\eta$-Ricci solitons, which illustrate our results.

The AdS/CFT correspondence and generalized geometry

Abstract: The most general AdS$_5 imes Y$ solutions of type IIB string theory that are AdS/CFT dual to superconformal field theories in four dimensions can be fruitfully described in the language of generalized geometry, a powerful hybrid of complex and symplectic geometry. We show that the cone over the compact five-manifold $Y$ is generalized Calabi-Yau and carries a generalized holomorphic Killing vector field $xi$, dual to the R-symmetry. Remarkably, this cone always admits a symplectic structure, which descends to a contact structure on $Y$, with $xi$ as Reeb vector field. Moreover, the contact volumes of $Y$, which can be computed by localization, encode essential properties of the dual CFT, such as the central charge and the conformal dimensions of BPS operators corresponding to wrapped D3-branes. We then define a notion of ``generalized Sasakian geometry'', which can be characterized by a simple differential system of three symplectic forms on a four-dimensional transverse space. The correct Reeb vector field for an AdS$_5$ solution within a given family of generalized Sasakian manifolds can be determined---without the need of the explicit metric---by a variational procedure. The relevant functional to minimize is the type IIB supergravity action restricted to the space of generalized Sasakian manifolds, which turns out to be just the contact volume. We conjecture that this contact volume is equal to the inverse of the trial central charge whose maximization determines the R-symmetry of the dual superconformal field theory. The power of this volume minimization is illustrated by the calculation of the contact volumes for a new infinite family of solutions, in perfect agreement with the results of $a$-maximization in the dual mass-deformed generalized conifold theories.

Certain homogeneous paracontact three-dimensional Lorentzian metrics

Abstract: In this paper, we study three-dimensional homogeneous paracontact metric manifolds for which the Reeb vector field of the underlying paracontact structure satisfies a nullity condition. We give example of paraSasakian and non-paraSasakian (κ,μ)-manifolds. Finally, we exhibit explicit example of η-Einstein manifolds.

On some Properties of Riemannian Manifolds with Locally Conformal Almost Cosymplectic Structures

Abstract: Let M be a 2 m +1-dimensional Riemannian manifold and let ∇ be the Levi-Civita connection and ξ be the Reeb vector field, η the Reeb covector field and X be the structure vector field satisfying a certain property on M . In this paper the following properties are proved: ( i ) ξ and X define a 3-covariant vanishing structure; ( ii ) the Jacobi bracket corresponding to ξ vanishes; ( iii ) the harmonic operator acting on X b gives Δ X b = f ||X || 2 X b , which proves that X b is an eigenfunction of Δ, having f ||X|| 2 as eigenvalue; ( iv ) the 2-form Ω and the Reeb covector η define a Pfaffian transformation, i.e. L X Ω=0, L X η=0; ( v ) ∇ 2 X defines the Ricci tensor; ( vi ) one has ∇ X X = f X , f = scalar, which show that X is an affine geodesic vector field; ( vii ) the triple ( X, ξ,φ X ) is an involutive 3-distribution on M , in the sense of Cartan.