On η-biharmonic hypersurfaces in pseudo-Riemannian space forms

Abstract

Abstract In this paper, η-biharmonic hypersurfaces with constant scalar curvature in 5-dimensional pseudo-Riemannian space forms are studied. We prove that such hypersurfaces with diagonalizable shape operator have constant mean curvature, which gives an affirmative partial answer to the conjecture in [Arvanitoyeorgos, A.—Kaimakamis, F. G.: Hypersurfaces of type $\begin{array}{} \displaystyle M^3_2 \end{array}$ in $\begin{array}{} \displaystyle \mathbb{E}^4_2 \end{array}$ with proper mean curvature vector, J. Geom. Phys. 63 (2013), 99–106]. As a result, we give several partial classification results.