The rough Laplacian and harmonicity of Hopf vector fields

Let M be an orientable real hypersurface of a general Kaehler manifold N. The characteristic vector field ξ of the induced almost contact metric structure (ξ, η, g, ϕ) is also called the Hopf vector field of M. In this paper we compute the ‘rough’ Laplacian of ξ when N is a general Kaehler manifold, and as consequence we get some criteria for the harmonicity of ξ which include the criterium obtained by K.Tsukada and L. Vanhecke. Then we compute the ‘rough’ Laplacian of a Hopf vector field in terms of torsion L_ξ g as a natural generalization of the contact metric case, obtaining in this way other criteria of harmonicity for Hopf vector fields. Moreover, we consider an orientable real hypersurface (M, ξ, η, g, ϕ) of contact type, that is, dη = rg(•, ϕ•), where r is a nowhere zero function. In particular, if N is a complex space form, the Hopf vector field of an orientable real hypersurface of contact type is harmonic and determines a harmonic map. Moreover, when N is a Kaehler–Einstein manifold, we get some criteria for which an orientable real hypersurface of contact type admits an H-contact structure.