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-Rio 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-4 (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.