Summary: We study a class of maps, called Pseudo Horizontally Weakly Conformal (PHWC), which includes horizontally weakly conformal mappings. We give geometrical conditions ensuring the harmonicity of a PHWC map, making it a pseudo harmonic morphism, a generalisation of harmonic morphism, for which we broaden the Baird-Eells Theorem.
Finally, considering pseudo horizontally homothetic maps, we extend a theorem of M. A. Aprodu, M. Aprodu and V. Brinzanescu [Int.J. Math. 11, No. 9, 1177–1191 (2000; Zbl 0978.58006)] to pseudo harmonic morphisms, and show that the dual stress-energy of such maps is horizontally covariant constant.