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.

A new characterization of Finsler metrics with constant flag curvature 1

Abstract: The purpose of this article is to derive an integral inequality of Ricci curvature with respect to Reeb field in a Finsler space and give a new geometric characterization of Finsler metrics with constant flag curvature 1.

Ricci Curvature, Reeb Flows and Contact 3-Manifolds

Abstract: Given a contact 3-manifold, we consider the problem of when a given function can be realized as the Ricci curvature of a Reeb vector field for the contact structure. We will use topological tools to show that every admissible function can be realized as such Ricci curvature for a singular metric which is an honest compatible metric away from a measure zero set. However, we will see that resolving such singularities depends on contact topological data and is yet to be fully understood.

Reeb periodic orbits after a bypass attachment

Abstract: On a 3-dimensional contact manifold with boundary, a bypass attachment is an elementary change of the contact structure consisting in the attachment of a thickened half-disc with a prescribed contact structure along an arc on the boundary. We give a model bypass attachment in which we describe the periodic orbits of the Reeb vector field created by the bypass attachment in terms of Reeb chords of the attachment arc. As an application, we compute the contact homology of a product neighbourhood of a convex surface after a bypass attachment, and the contact homology of some contact structures on solid tori.

Three-dimensional almost co-Kähler manifolds with harmonic Reeb vector fields

Abstract: Let M3 be a three-dimensional almost co-Kahler manifold whose Reeb vector field is harmonic. We obtain some local classification results of M3 under some additional conditions related to the Ricci tensor.

On non-gradient $(m,\rho)$-quasi-Einstein contact metric manifolds.

Abstract: Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $(m,\rho)$-quasi-Einstein structure on a contact metric manifold. First, we prove that if a $K$-contact or Sasakian manifold $M^{2n+1}$ admits a closed $(m,\rho)$-quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $2n(2n+1)$, and for the particular case -- a non-Sasakian $(k,\mu)$-contact structure -- it is locally isometric to the product of a Euclidean space $\RR^{n+1}$ and a sphere $S^n$ of constant curvature $4$. Next, we prove that if a compact contact or $H$-contact metric manifold admits an $(m,\rho)$-quasi-Einstein structure, whose potential vector field $V$ is collinear to the Reeb vector field, then it is a $K$-contact $\eta$-Einstein manifold.