Harmonic Riemannian submersions from 3-dimensional geometries

TL;DR

This paper classifies harmonic Riemannian submersions from various 3D geometries, including Thurston's geometries, BCV spaces, and Berger spheres, into surfaces, providing explicit examples and a comprehensive understanding.

Contribution

It offers a complete classification of harmonic Riemannian submersions from key 3D geometries into surfaces, expanding the understanding of their structure and explicit constructions.

Findings

Classified harmonic Riemannian submersions from Thurston's geometries.

Classified from BCV spaces and Berger spheres.

Provided explicit constructions of these submersions.

Abstract

In this paper, we study harmonic Riemannian submersions from 3-dimensional geometries using the ( generalized) integrability data associated to an orthonormal frame natural to a Riemannian submersion. We give complete classifications of harmonic Riemannian submersions from Thurston's 3-dimensional geometries, 3-dimensional BCV spaces and Berger sphere into a surface. We also give some explicit constructions of these harmonic Riemannian submersions.