f-biharmonic and bi-f-harmonic Riemannian submersions

Abstract

In this paper, we study f-biharmonic Riemannian submersion, bi-f-harmonic Riemannian submersion and thus generalizing some results of [M. A. Akyol and Y.-L. Ou, Biharmonic Riemannian submersions, Ann. Math. Pura Appl. 198 (2019) 559–570; Z. Wang and Y.-L. Ou, Biharmonic Riemannian submersions from 3-manifolds, Math. Z. 269(3) (2011) 917–925]. Next, we obtain the conditions when the Riemannian submersion on its first factor is f-biharmonic as well as bi-f-harmonic. In the last section, we study f-biharmonic cylinders of a Riemannian submersion.