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.

Curvature restrictions for Levi-flat real hypersurfaces in complex projective planes

Abstract: We study curvature restrictions of Levi-flat real hypersurfaces in complex projective planes, whose existence is in question. We focus on its totally real Ricci curvature, the Ricci curvature of the real hypersurface in the direction of the Reeb vector field, and show that it cannot be greater than -4 along a Levi-flat real hypersurface. We rely on a finiteness theorem for the space of square integrable holomorphic 2-forms on the complement of the Levi-flat real hypersurface, where the curvature plays the role of the size of the infinitesimal holonomy of its Levi foliation.

The singular Weinstein conjecture

Abstract: In this article, we investigate Reeb dynamics on $b^m$-contact manifolds, previously introduced in [MiO], which are contact away from a hypersurface $Z$ but satisfy certain transversality conditions on $Z$. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the well-known Weinstein conjecture. Contrary to the initial expectations, examples of compact $b^m$-contact manifolds without periodic Reeb orbits outside $Z$ are provided. Furthermore, we prove that in dimension $3$, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the $b^m$-Reeb flow exist in any dimension. This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex type. In particular, we extend Hofer's arguments to open overtwisted contact manifolds that are $\mathbb R^+$-invariant in the open ends, obtaining as a corollary the existence of periodic $b^m$-Reeb orbits away from the critical set. The study of $b^m$-Reeb dynamics is motivated by well-known problems in fluid dynamics and celestial mechanics, where those geometric structures naturally appear. In particular, we prove that the dynamics on positive energy level-sets in the restricted planar circular three-body problem are described by the Reeb vector field of a $b^3$-contact form that admits an infinite number of periodic orbits at the critical set.

Ricci Solitons and Paracontact Geometry

Abstract: In this article, first we prove that if a metric of a para-Sasakian manifold is a Ricci soliton, then either it is an Einstein (and V Killing) or a $$\eta $$-Einstein (and V leaves $$\varphi $$ invariant) manifold. Next, we prove that if a K-paracontact metric g is a gradient Ricci soliton, then it becomes a expanding soliton which is Einstein with constant scalar curvature. Further, we study the Ricci soliton where the potential vector field V is point wise collinear with the Reeb vector field on paracontact manifold. Finally, we consider the gradient Ricci soliton on $$(\kappa ,\mu )$$-paracontact manifold.

Conformal great circle flows on the three-sphere

Abstract: We consider a closed orientable Riemannian 3-manifold $(M,g)$ and a vector field $X$ with unit norm whose integral curves are geodesics of $g$. Any such vector field determines naturally a 2-plane bundle contained in the kernel of the contact form of the geodesic flow of $g$. We study when this 2-plane bundle remains invariant under two natural almost complex structures. We also provide a geometric condition that ensures that $X$ is the Reeb vector field of the 1-form $\lambda$ obtained by contracting $g$ with $X$. We apply these results to the case of great circle flows on the 3-sphere with two objectives in mind: one is to recover the result in \cite{GG} that a volume preserving great circle flow must be Hopf and the other is to characterize in a similar fashion great circle flows that are conformal relative to the almost complex structure in the kernel of $\lambda$ given by rotation by $\pi/2$ according to the orientation of $M$.

D-homothetic warping

Abstract: The goal of this lecture will be to introduce the notion ofD-homothetic warping, give a few rudimentary properties and a couple of applications. As with the usual warped product it is hoped that this idea will prove useful for generating further results and examples of various structures. Details of the proofs will appear in [3]. For this purpose we must first review the geometry of contact metric and almost contact metric manifolds. By a contact manifold we mean a C manifold M2n+1 together with a 1-form η such that η ∧ (dη) 6= 0. It is well known that given η there exists a unique vector field ξ such that dη(ξ,X) = 0 and η(ξ) = 1. The vector field ξ is known as the characteristic vector field or Reeb vector field of the contact structure η. Denote by D the contact subbundle defined by {X ∈ TmM : η(X) = 0}. A Riemannian metric g is an associated metric for a contact form η if, first of all, η(X) = g(X, ξ) and secondly, there exists a field of endomorphisms, φ, such that φ2 = −I + η ⊗ ξ, dη(X,Y ) = g(X,φY ). We refer to (φ, ξ, η, g) as a contact metric structure and to M2n+1 with such a structure as a contact metric manifold. By an almost contact manifold we mean a C manifold M2n+1 together with a field of endomorphisms φ, a 1-form η and a vector field ξ such that