Harmonic and biharmonic Riemannain submersions from Sol space

Abstract

In this paper, we give a complete classification of harmonic and biharmonic Riemannian submersions $\pi:(R^3,g_{Sol})\to (N^2,h)$ from Sol space into a surface by proving that there is neither harmonic nor biharmonic Riemannian submersion $\pi:(R^3,g_{Sol})\to (N^2,h)$ from Sol space no matter what the base space $(N^2,h)$ is. We also prove that a Riemannian submersion $\pi:(R,g_{Sol})\to (N^2,h)$ from Sol space exists only when the base space is a hyperbolic space form.