Harmonic Morphisms Projecting Harmonic Functions to Harmonic Functions

Abstract

For Riemannian manifolds $M$ and $N$, admitting a submersion $\varphi$ with compact fibres, we introduce the projection of a function via its decomposition intohorizontal and vertical components. By comparing the Laplacians on $M$ and $N$, we determine conditions under which a harmonic function on $U={\varphi }^{-1}\left(V\right)\subset M$ projects down, via its horizontal component, to a harmonic function on $V\subset N$.