Biharmonic hypersurfaces in a product space Lᵐ× R

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In this paper, we study biharmonic hypersurfaces in a product of an Einstein space and a real line. We prove that a biharmonic hypersurface with constant mean curvature in such a product is either minimal or a vertical cylinder generalizing a result of OW and FOR. We derived the biharmonic equation for hypersurfaces in Sᵐ× R and Hᵐ× R in terms of the angle function of the hypersurface, and use it to obtain some classifications of biharmonic hypersurfaces in such spaces. These include classifications of biharmonic hypersurfaces which are totally umbilical or semi-parallel for m≥ 3, and some classifications of biharmonic surfaces in S²× R and H²× R which are constant angle or belong to certain classes of rotation surfaces.