Biharmonic Riemannian submersions from $M^2\times R$

TL;DR

This paper investigates biharmonic Riemannian submersions from a product manifold to surfaces, providing local characterizations and explicit descriptions depending on the curvature of the target surface.

Contribution

It offers new local characterizations of biharmonic Riemannian submersions from product manifolds onto surfaces, including explicit forms for flat and non-flat targets.

Findings

Proper biharmonic submersions are locally projections of twisted products when the target is flat.

When the target surface is non-flat, submersions are locally maps between warped products with specific warping functions.

Uniqueness of a proper biharmonic submersion from hyperbolic space cross R to Euclidean space.

Abstract

In this paper, we study biharmonic Riemannian submersions $\pi:M^2\times\r\to (N^2,h)$ from a product manifold onto a surface and obtain some local characterizations of such biharmonic maps. Our results show that when the target surface is flat, a proper biharmonic Riemannian submersion $\pi:M^2\times\r\to (N^2,h)$ is locally a projection of a special twisted product, and when the target surface is non-flat, $\pi$ is locally a special map between two warped product spaces with a warping function that solves a single ODE. As a by-product, we also prove that there is a unique proper biharmonic Riemannian submersion $H^2\times \r\to \r^2$ given by the projection of a warped product.