Harmonic Vector Fields

Our first examples of harmonic vector fields were Hopf vector fields on a sphere S ^{2 m +1} , that is Reeb vector fields underlying Sasakian structures on S ^{2 m +1} . It is therefore a natural problem to study harmonicity of Reeb vector fields on arbitrary contact Riemannian manifolds. As it turns out, looking at the harmonicity of the Reeb vector field of a contact metric manifold M is relevant for understanding the geometry of M itself. After a brief preparation of contact Riemannian geometry (cf. D.E. Blair, [42] ) one introduces one of the main object of study in this chapter, that is the notion of an H - contact manifold (a contact metric manifold whose Reeb vector is harmonic, cf. Definition 4.2 below). H -contact manifolds may be described ...