REMARKS ON LEVI HARMONICITY OF CONTACT SEMI-RIEMANNIAN MANIFOLDS

In a recent paper [10] we introduced the notion of Levi harmonic map f from an almost contact semi-Riemannian manifold (M, phi, xi, eta, g) into a semi-Riemannian manifold M'. In particular, we computed the tension field T-H(f) for a CR map f between two almost contact semi-Riemannian manifolds satisfying the so-called phi-condition, where H = Ker(eta) is the Levi distribution. In the present paper we show that the condition (A) of Rawnsley [17] is related to the phi-condition. Then, we compute the tension field T-H (f) for a CR map between two arbitrary almost contact semi-Riemannian manifolds, and we study the concept of Levi pluriharmonicity. Moreover, we study the harmonicity on quasi-cosymplectic manifolds.