Turkish Journal of Mathematics
DOI
10.3906/mat-1504-87
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 $q$.
Keywords
$4$-manifold, harmonic function, harmonic map, Walker manifold, almost complex structure
First Page
1004
Last Page
1019
Recommended Citation
BEJAN, CORNELIA-LIVIA and DRUTA-ROMANIUC, SIMONA-LUIZA
(2016)
"Harmonic functions and quadratic harmonic morphisms on Walker spaces,"
Turkish Journal of Mathematics: Vol. 40:
No.
5, Article 7.
https://doi.org/10.3906/mat-1504-87
Available at:
https://journals.tubitak.gov.tr/math/vol40/iss5/7
