For Riemannian manifolds and , admitting a submersion with compact fibres, we introduce the projection of a function via its decomposition intohorizontal and vertical components. By comparing the Laplacians on and , we determine conditions under which a harmonic function on projects down, via its horizontal component, to a harmonic function on .
"Harmonic Morphisms Projecting Harmonic Functions to Harmonic Functions." Abstr. Appl. Anal. 2012 (SI13) 1 - 8, 2012. https://doi.org/10.1155/2012/315757