Classification of biharmonic Riemannian submersions from manifolds with constant sectional curvature

Abstract

In 2011, Wang and Ou (Math. Z. {\bf 269}:917-925, 2011) showed that any biharmonic Riemannian submersion from a 3-dimensional Riemannian manifold with constant sectional curvature to a surface is harmonic. In this paper, we generalize the 3-dimensional setting to arbitrary dimensions. By constructing an adapted orthonormal frame, we simplify the biharmonic equation for Riemannian submersions and analyze the curvature properties of Riemannian manifolds with constant sectional curvature. As a result, we prove that a Riemannian submersion from an $(n+1)$-dimensional Riemannian manifold with constant sectional curvature to an $n$-dimensional Riemannian manifold is biharmonic if and only if it is harmonic. This result may also be viewed as an affirmative codimension-one Riemannian submersion analogue of Chen's conjecture, the generalized Chen's conjecture, and the BMO conjecture.