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Abstract.A new class of 3-dimensional contact metric manifolds is found. Moreover it is proved that there are no such manifolds in dimensions greater than 3.

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.

Given a function H ∈ C 1 (S 2 ), an H-surface Σ is a surface in the Euclidean space R 3 whose mean curvature H Σ satisfies H Σ = H • η, where η is the Gauss map of Σ. The purpose of this paper is to use a phase space analysis to give some classification results for helicoidal H-surfaces, when H is rotationally symmetric, that is,), where ν is the angle function of the surface. We prove a classification theorem for the case where h(t) is even and increasing for t ∈ [0, 1]. Finally, we provide examples of helicoidal H-surfaces in cases where h vanishes at some point.