Biharmonic Riemannian submersions

In this paper, we study biharmonic Riemannian submersions. We first derive bitension field of a general Riemannian submersion, and we then use it to obtain biharmonic equations for Riemannian submersions with one-dimensional fibers and Riemannian submersions with basic mean curvature vector fields of fibers. These are used to construct examples of proper biharmonic Riemannian submersions with one-dimensional fibers and to characterize warped products whose projections onto the first factor are biharmonic Riemannian submersions.