Properties of α-harmonic morphisms

In this paper, we investigate [Formula: see text]-harmonic morphisms, obtaining their characterizations and studying their properties. Moreover, the fibers of the [Formula: see text]-harmonic morphisms are shown to be minimal submanifolds if the morphisms satisfy certain conditions. Furthermore, the precise mean curvature of a horizontal distribution associated to [Formula: see text]-harmonic morphism is determined through the analysis of the geometric nature of the underlying structure. The study is also extended to the exploration of polynomial [Formula: see text]-harmonic morphism and ends with potential applications of [Formula: see text]-harmonic maps.