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

Themis Koufogiorgos, Michael Markellos and Vassilis J. Papantoniou

Vol. 234 (2008), No. 2, 325–344

DOI: 10.2140/pjm.2008.234.325

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.

Keywords

contact metric manifolds, harmonic characteristic vector fields, H-contact manifolds, (κ,μ,ν)-contact metric manifolds

Mathematical Subject Classification 2000

Primary: 53D10

Secondary: 53C25, 53C15

Milestones

Received: 18 April 2007

Revised: 10 October 2007

Accepted: 11 October 2007

Published: 1 February 2008