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.

Ricci solitons in almost $f$-cosymplectic manifolds

Abstract: In this article we study an almost $f$-cosymplectic manifold admitting a Ricci soliton. We first prove that there do not exist Ricci solitons on an almost cosymplectic $(\kappa,\mu)$-manifold. Further, we consider an almost $f$-cosymplectic manifold admitting a Ricci soliton whose potential vector field is the Reeb vector field and show that a three dimensional almost $f$-cosymplectic is a cosymplectic manifold. Finally we classify a three dimensional $\eta$-Einstein almost $f$-cosymplectic manifold admitting a Ricci soliton..

From a Reeb orbit trap to a Hamiltonian plug

Abstract: We present a simple construction of a plug for Hamiltonian flows on hypersurfaces of dimension at least five by doubling a trap for Reeb orbits.

The Space of Contact Forms Adapted to an Open Book

Abstract: We construct a complete set of two consecutive obstructions against homotopies of pointed families of adapted contact forms with parameters in the n-sphere. Using these obstructions, we show that there is a manifold with an open book decomposition together with both infinitely many adapted contact forms that all induce the same Liouville form on one page but such that the underlying contact manifolds are not contactomorphic, and infinitely many non-homotopic adapted contact forms that all induce the same Liouville form on one page and such that the underlying contact manifolds are contactomorphic. Following this, we use the neighbourhood theorem for the binding of an open book decomposition that we introduce in the construction of the obstructions to construct special generalised caps of contact manifolds. This leads us to a proof that, on closed manifolds, the Reeb vector field of every contact form defining a contact structure supported by a tower of open book decompositions has a contractible orbit provided the binding in the lowest level of this tower embeds into a subcritical Stein manifold as a hypersurface of restricted contact type or is supported by an open book decomposition with trivial monodromy. Moreover, we show that the strong Weinstein conjecture holds for contact manifolds supported by an open book whose binding is planar.

Ricci Almost Solitons on Three-Dimensional Quasi-Sasakian Manifolds

Abstract: In this paper it is shown that a three-dimensional non-cosymplectic quasi-Sasakian manifold admitting Ricci almost soliton is locally $$\phi $$-symmetric. It is proved that a Ricci almost soliton on a three-dimensional quasi-Sasakian manifold reduces to a Ricci soliton. It is also proved that if a three-dimensional non-cosymplectic quasi-Sasakian manifold admits gradient Ricci soliton, then the potential function is invariant in the orthogonal distribution of the Reeb vector field $$\xi$$. We also improve some previous results regarding gradient Ricci soliton on three-dimensional quasi-Sasakian manifolds. An illustrative example is given to support the obtained results.

Canonical Sasakian Metrics

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.