Harmonic functions and quadratic harmonic morphisms on Walker spaces

Let (W, q,D) be a 4-dimensional Walker manifold. After providing a characterization and some examples for several special (1, 1)-tensor fields on (W, q,D) , we prove that the proper almost complex structure J , introduced by Matsushita, is harmonic in the sense of Garcia-Ŕio et al. if and only if the almost Hermitian structure (J, q) is almost Kahler. We classify all harmonic functions locally defined on (W, q,D) . We deal with the harmonicity of quadratic maps defined on R (endowed with a Walker metric q ) to the n -dimensional semi-Euclidean space of index r , and then between local charts of two 4-dimensional Walker manifolds. We obtain here the necessary and sufficient conditions under which these maps are harmonic, horizontally weakly conformal, or harmonic morphisms with respect to q .