Biharmonic Riemannian submersions

Abstract

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.

Harmonic maps and biharmonic Riemanian submersions

“…Remark 3.2. The bitension field τ 2 (π) for π has been obtained in a different way by Akyol and Ou [2] in which has referenced our paper.…”

Some recent work on biharmonic conformal maps

“…A biharmonic map is a map φ : (M m , g) → (N n , h) between Riemannian manifolds that is a critical point of the bienergy functional E 2 (φ, g) = 1 2 M |τ (φ)| 2 dv g , where τ (φ) is the tension field of φ. Biharmonic map equation is a system of the 4th order elliptic PDEs (see [23]):…”

“…Biharmonic Riemannian submersions were first studied in [28]. For some recent work on this subject, see [12], [26], [43], [21], [41], and [1].…”

Biharmonic Riemannian submersions from a 3-dimensional BCV space

“…By using the integrability data of a special orthonormal frame adapted to a Riemannian submersion, we proved in [25] that a Riemannian submersion from a 3-dimensional space form into a surface is biharmonic if and only if it is harmonic. In a recent paper [1], Akyol and Ou studied biharmonicity of a general Riemannian submersion and obtained biharmonic equations for Riemannian submersions with one-dimensional fibers and Riemannian submersions with basic mean curvature vector fields of fibers. In particular, in [1], biharmonic Riemannian submersions from (n+1)-dimensional spaces with one-dimensional fibers were studied using the integrability data.…”

“…In a recent paper [1], Akyol and Ou studied biharmonicity of a general Riemannian submersion and obtained biharmonic equations for Riemannian submersions with one-dimensional fibers and Riemannian submersions with basic mean curvature vector fields of fibers. In particular, in [1], biharmonic Riemannian submersions from (n+1)-dimensional spaces with one-dimensional fibers were studied using the integrability data. In a recent paper [11], the authors found many examples of biharmonic Riemannian submersions in their study of generalized harmonic morphisms which are maps between Riemannian manifolds that pull back local harmonic functions to local biharmonic functions.…”