Contact metric manifolds whose characteristic vector field is a harmonic vector field

Taut contact circles on H-contact 3-manifolds

“…A contact metric manifold (M, η, g) is said to be a H-contact manifold if ξ is a harmonic vector field. The following characterization was proved in [12].…”

“…On the other hand, the first examples of harmonic vector fields, Hopf unit vector fields, are in fact the characteristic vector fields of the standard Sasakian structure on odd-dimensional spheres. In [12] the present author introduced the notion of H-contact manifold, namely, a contact metric manifold (M, η, g) for which the characteristic vector field ξ is a harmonic vector field, and proved that a contact metric manifold is H-contact if and only if ξ is an eigenvector of the Ricci operator. The class of H-contact manifolds is very large; it extends the class of Sasakian manifolds.…”

Instability of the geodesic flow for the energy functional

“…Contact metric manifolds which Reeb vector field is harmonic are called H -contact manifolds [Perrone 2004]. Recently, in [Perrone 2009a] we studied the stability of the Reeb vector field of a compact H -contact three manifold for the energy E z …”

Complete Riemannian manifolds admitting a pair of~Einstein-Weyl structures

“…⊲ The potential vector field V of the Ricci soliton is necessarily an infinitesimal harmonic transformation (see [18]). ⊲ The Reeb vector field ξ of a K-contact manifold is Killing, and hence it generates an infinitesimal harmonic transformation (see [16]). ⊲ Any vector field V on a contact metric manifold M 2n+1 (ϕ, ξ, η, g) that leaves the tensor ϕ invariant (i.e., £ V ϕ = 0) is necessarily an infinitesimal harmonic transformation (see [7]).…”