Chapter Four - Harmonicity and Contact Metric Structures

Abstract

This chapter presents examples of harmonic vector fields; Hopf vector fields on a sphere, that is Reeb vector fields underlying Sasakian structures on. It is therefore a natural problem to study harmonicity of Reeb vector fields on arbitrary contact Riemannian manifolds. The harmonicity of the Reeb vector field of a contact metric manifold M is relevant for understanding the geometry of metric manifold itself. The main object of study in this chapter is the notion of an H-contact manifol. It reviews a few facts of contact Riemannian geometry and reports briefly recent results on the harmonicity of an almost contact structure.