# Harmonic and biharmonic Riemannain submersions from Sol space

## TL;DR

This paper classifies harmonic and biharmonic Riemannian submersions from Sol space to surfaces, proving their non-existence except when the base is a hyperbolic space form.

## Contribution

It provides a complete classification and non-existence results for harmonic and biharmonic submersions from Sol space, highlighting the special case of hyperbolic base spaces.

## Key findings

- No harmonic or biharmonic submersions from Sol space to any surface.
- Existence of submersions only when the base is a hyperbolic space form.
- Classification results for Riemannian 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.