Minimal and harmonic characteristic vector fields on three-dimensional contact metric manifolds

Abstract

We consider three-dimensional contact metric manifolds whose unit characteristic vector field is harmonic or minimal. (2000): 53C25. Mathematics Subject Classification

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

“…D. Perrone [2004] proved that M(η, ξ, φ, g) is H -contact metric manifold if and only if ξ is an eigenvector of the Ricci operator, generalizing the same result of J. C. González-Dávila and L. Vanhecke [2001] for n = 1. It is important to mention that the class of H -contact metric manifolds includes several interesting classes of contact metric manifolds such as Sasakian and η-Einstein manifolds, K -contact manifolds, strongly φ-symmetric spaces, (κ, µ)-contact metric manifolds, and generalized (κ, µ)-contact metric manifolds.…”

“…It is known, see [Perrone 2004, Theorem 3.1] and [González-Dávila and Vanhecke 2001], that a contact metric manifold is an H -contact metric manifold if and only if the characteristic vector field ξ is an eigenvector of the Ricci operator Q. So, by (3-2) we deduce that M is an H -contact metric manifold.…”

Harmonic sections of normal bundles for submanifolds and their stability

“…A smooth unit vector field is said to be a harmonic vector field if it is a stationary point of the energy functional of vector fields with unit length. We refer to [5], [6], [7], [8], [11], [12], [13], [15], for example. These formulations can be generalized to the case of the sphere bundle of a Riemannian vector bundle with metric connection.…”

Minimal and Harmonic Unit Vector Fields in and Its Dual Space

“…A series of examples of minimal and harmonic vector ®elds has been provided in [4], [5], [6], [7], [10], [11], [12], [16], [17]. In particular, in [17] we considered the unit Hopf vector ®elds on orientable real hypersurfaces in (non-¯at) complex space forms.…”