Abstract:In the present paper we give some properties of $f$-biharmonic hypersurfaces in real space forms. By using the $f$-biharmonic equation for a hypersurface of a Riemannian manifold, we characterize the $f$-biharmonicity of constant mean curvature and totally umbilical hypersurfaces in a Riemannian manifold and, in particular, in a real space form. As an example, we consider $f$-biharmonic vertical cylinders in $S^{2}\times \mathbb{R}$.
Keywords: $f$-biharmonic maps; $f$-biharmonic hypersurface
DOI: DOI 10.14712/1213-7243.2020.009
AMS Subject Classification: 58E20 53C25 53C43