Biharmonic Riemannian submersions from a 3-dimensional BCV space

TL;DR

This paper classifies proper biharmonic Riemannian submersions from 3D BCV spaces, showing they only exist in specific cases and constructing infinite families in those cases, extending previous classifications.

Contribution

It provides a complete classification of proper biharmonic Riemannian submersions from BCV spaces and constructs infinite examples, extending prior work on space forms.

Findings

Proper biharmonic submersions only in specific BCV cases

Existence of infinitely many proper biharmonic submersions in these cases

Extension of previous classifications to BCV spaces

Abstract

BCV spaces are a family of 3-dimensional Riemannian manifolds which include six of Thurston's eight geometries. In this paper, we give a complete classification of proper biharmonic Riemannian submersions from a 3-dimensional BCV space by proving that such biharmonic maps exist only in the cases of $H^2\times\mathbb{R}\to \mathbb{R}^2$ or $\widetilde{SL}(2,\mathbb{R})\to \mathbb{R}^2$. In each of these two cases, we are able to construct a family of infinitely many proper biharmonic Riemannian submersions. Our results on one hand, extend a previous result of the authors which gave a complete classification of proper biharmonic Riemannian submersions from a 3-dimensional space form, and on the other hand, can be viewed as the dual study of biharmonic surfaces (i.e., biharmonic isometric immersions) in a BCV space studied in some recent literature.