Pseudo Harmonic Morphisms on Riemannian Polyhedra

The aim of this paper is to extend the notion of pseudo harmonic morphism (introduced by Loubeau \cite {Lo}) to the case when the source manifold is an admissible Riemannian polyhedron. We define these maps to be harmonic in the sense of Eells-Fuglede \cite {EF} and pseudo-horizontally weakly conformal in our sense (see Section 3). We characterize them by means of germs of harmonic functions on the source polyhedron, in sense of Korevaar-Schoen \cite {KS}, and germs of holomorphic functions on the K\"ahler target manifold.