| Summary: | <i>f</i>-biharmonic maps are generalizations of harmonic maps and biharmonic maps. In this paper, we give some descriptions of <i>f</i>-biharmonic curves in a space form. We also obtain a complete classification of proper <i>f</i>-biharmonic isometric immersions of a developable surface in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></semantics></math></inline-formula> by proving that a proper <i>f</i>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></semantics></math></inline-formula> exists only in the case when the surface is a cylinder. Riemannian submersions can be viewed as a dual notion of isometric immersions (i.e., submanifolds). We also study <i>f</i>-biharmonicity of Riemannian submersions from 3-manifolds by using the integrability data. Examples are given of proper <i>f</i>-biharmonic Riemannian submersions and <i>f</i>-biharmonic surfaces and curves.
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