Harmonic functions and quadratic harmonic morphisms on Walker spaces

Authors : Cornelia-livia BEJAN, Simona-luiza DRUTA-ROMANIUC

Pages : 1004-1019

View : 11 | Download : 4

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Let $insert ignore into journalissuearticles values(W,q, \mathcal{D});$ be a 4-dimensional Walker manifold. After providing a characterization and some examples for several special $insert ignore into journalissuearticles values(1,1);$-tensor fields on $insert ignore into journalissuearticles values(W,q, \mathcal{D});$, 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 $insert ignore into journalissuearticles values(J,q);$ is almost Kahler. We classify all harmonic functions locally defined on $insert ignore into journalissuearticles values(W,q, \mathcal{D});$. We deal with the harmonicity of quadratic maps defined on $\mathbb{R}^4$ insert ignore into journalissuearticles values(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