Biharmonic Riemannian Submersions from the Product Space \(M^2\times \mathbb {R}\)

In this paper, we study biharmonic Riemannian submersions \(\pi :M^2\times \mathbb {R}\rightarrow (N^2,h)\) $\pi :M^2\times \mathbb{R}\to (N^2,h)$ 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 \(\pi :M^2\times \mathbb {R}\rightarrow (N^2,h)\) $\pi :M^2\times \mathbb{R}\to (N^2,h)$ is locally a projection of a special twisted product, and when the target surface is non-flat, \(\pi \) $\pi$ 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 \(H^2\times \mathbb {R}\rightarrow \mathbb {R}^2\) $H^2\times \mathbb{R}\to {\mathbb{R}}^2$ given by the projection of a warped product onto the Euclidean plane.