Turkish Journal of Mathematics
Abstract
Let $(W,q, \mathcal{D})$ be a 4-dimensional Walker manifold. After providing a characterization and some examples for several special $(1,1)$-tensor fields on $(W,q, \mathcal{D})$, 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 $(W,q, \mathcal{D})$. We deal with the harmonicity of quadratic maps defined on $\mathbb{R}^4$ (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 $q$.
DOI
10.3906/mat-1504-87
Keywords
$4$-manifold, harmonic function, harmonic map, Walker manifold, almost complex structure
First Page
1004
Last Page
1019
Recommended Citation
BEJAN, C, & DRUTA-ROMANIUC, S (2016). Harmonic functions and quadratic harmonic morphisms on Walker spaces. Turkish Journal of Mathematics 40 (5): 1004-1019. https://doi.org/10.3906/mat-1504-87
