Biharmonic Legendre Curves on Totally η-Umbilical, Ruled, and Hopf Hypersurfaces in Four-Dimensional Complex Space Forms

The main purpose of the present paper is to investigate biharmonic Legendre curves on hypersurfaces in four-dimensional complex space forms. We examine the necessary conditions for the existence of such curves on totally η-umbilical, ruled, and Hopf hypersurfaces for which the shape operator AN satisfies a symmetry property. The obtained results are discussed in the complex projective space CP2. For any real hypersurfaces (M,g) of M˜(c), we have the relations ∇Xξ=ϕANX and (∇Xϕ)Y=η(Y)ANX−g(ANX,Y)ξ for any X,Y∈Γ(TM). Using these equations, we provide the main results and theorems for biharmonic curves on the mentioned hypersurfaces.