The harmonicity of the Reeb vector field on contact metric 3-manifolds

Abstract

A contact metric manifold whose characteristic vector field is a harmonic vector field is called an H-contact metric manifold. We introduce the notion of (κ, µ, ν)-contact metric manifolds in terms of a specific curvature condition. Then, we prove that a contact metric 3-manifold M is an H-contact metric manifold if and only if it is a (κ, µ, ν)-contact metric manifold on an everywhere open and dense subset of M. Also, we prove that, for dimensions greater than three, such manifolds are reduced to (κ, µ)-contact metric manifolds whereas, in three dimensions, (κ, µ, ν)-contact metric manifolds exist.

A Class of 3-dimensional Contact Metric Manifolds

“…Otherwise M is generalized (κ, µ) contact metric manifold. Since 15 years, this kind of manifolds especially was introduced and studied by Blair, Koufogiorgos, Papantoniou and Markellos in [1], [12].…”

“…In [12] the authors proved the existence of a new class of contact metric manifolds which is called (κ, µ, υ)-contact metric manifold. This means that curvature tensor R satisfies the condition…”

“…Furthermore, it is shown in [12] that if dimM > 3, then κ, µ are constants and υ is the zero function.…”

“…Koufogiorgos gave the examples satisfying (1.1) with κ, µ non constant smooth functions for dimension 3 and also proved in [10] that if dim M > 3 then κ,µ were necessarely constant. In [12] Koufogiorgos, Markellos and Papantoniou proved the existence of a new class of contact metric manifolds: the so called (κ, µ, v)-contact metric manifolds. Such a manifold M is defined through the condition for every X, Y ∈ χ(M ) and κ, µ, ν are smooth functions on M. Furthermore, it is shown in [12] that if dim M > 3, then κ, µ are constants and ν is the zero function.…”

“…In [12] Koufogiorgos, Markellos and Papantoniou proved the existence of a new class of contact metric manifolds: the so called (κ, µ, v)-contact metric manifolds. Such a manifold M is defined through the condition for every X, Y ∈ χ(M ) and κ, µ, ν are smooth functions on M. Furthermore, it is shown in [12] that if dim M > 3, then κ, µ are constants and ν is the zero function. In [13] locally classified the 3-dimensional generalized (κ, µ)-contact metric manifolds satisfied the condition gradκ =constant( = 0) .…”