Harmonic functions and quadratic harmonic morphisms on Walker spaces

Abstract

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 García-Río et al. if and only if the almost Hermitian structure (J, q) is almost Kähler. 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 .

Harmonic maps on the tangent bundle according to the ciconia metric

“…This crucial result on harmonic maps was obtained by Eells-Sampson in [16]. After that, harmonicity extended in a wide range of directions, such as harmonic morphisms, (see [5]), harmonic (semi-) Riemannian metrics (see [13]), harmonic sections [12,14,24,25], harmonic endomorphisms and (1, 1)-tensor fields (see [8,9,18]), harmonic connections [8,17], quasi-harmonic maps and morphisms (see [4,7]), etc.…”