tailieunhanh - Harmonic functions and quadratic harmonic morphisms on Walker spaces

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. | Turk J Math (2016) 40: 1004 – 1019 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Harmonic functions and quadratic harmonic morphisms on Walker spaces ˘ Cornelia-Livia BEJAN∗, Simona-Luiza DRUT ¸ A-ROMANIUC Department of Math, “Gh. Asachi” Technical University of Ia¸si, Ia¸si, Romania Received: • Accepted/Published Online: • Final Version: 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 Garc´ıa-R´ıo et al. if and only if the almost Hermitian structure (J, q) is almost K¨ ahler. We classify all harmonic functions locally defined on (W, q, D) . We deal with the harmonicity of quadratic maps defined on R4 (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 . Key words: 4 -manifold, harmonic function, harmonic map, Walker manifold, almost complex structure 1. Introduction Walker manifolds are of special interest in an increasing number of works in mathematical physics [8, 11–13, 26], particularly in general relativity [10]. A Walker manifold is a semi-Riemannian n-dimensional manifold (. a manifold endowed with a nondegenerate symmetric (0, 2)-tensor field of arbitrary signature [28]) with an r -dimensional lightlike distribution (see [17]), which is parallel w. r. t. the Levi-Civita connection. Constituting the background of several specific semi-Riemannian structures, these manifolds are involved in many physical contexts and they are useful for furnishing