$f$-Biharmonic submanifolds in space forms and $f$-biharmonic Riemannian submersions from 3-manifolds

TL;DR

This paper explores $f$-biharmonic maps, providing classifications of certain $f$-biharmonic submanifolds and Riemannian submersions in space forms, and presents examples illustrating these concepts.

Contribution

It offers a complete classification of proper $f$-biharmonic developable surfaces in $ ^3$ and studies $f$-biharmonic Riemannian submersions from 3-space forms, including examples.

Findings

Proper $f$-biharmonic developable surfaces are only cylinders.

Proper biharmonic conformal immersions of developable surfaces into $ ^3$ are only cylinders.

Examples of proper $f$-biharmonic Riemannian submersions and surfaces are provided.

Abstract

$f$-Biharmonic maps are generalizations of harmonic maps and biharmonic maps. In this paper, we obtain some descriptions of $f$-biharmonic curves in a space form. We also obtain a complete classification of proper $f$-biharmonic isometric immersions of a developable surface in $\r^3$ by proving that a proper $f$-biharmonic developable surface exists only in the case where the surface is a cylinder. Based on this, we show that a proper biharmonic conformal immersion of a developable surface into $\r^3$ exists only in the case when the surface is a cylinder. Riemannian submersions can be viewed as the dual notion of isometric immersions (i.e., submanifolds). We also study $f$-biharmonicity of Riemannian submersions from 3-space forms by using the integrability data. Examples are given of proper $f$-biharmonic Riemannian submersions and $f$-biharmonic surfaces and curves.