Biharmonic product maps between doubly warped product manifolds

The biharmonicity of the product map Φ2=φ × ψ and the two generalized projections φ¯ and ψ¯ are analyzed. Some results are obtained, that is, Φ2 is a proper biharmonic map if and only if b is a non-constant solution of -1/f2 Jφ(dφ(grad(lnb)))+n/2 grad|dφ(grad(lnb))|2=0 and f is a non-constant solution of -1/b2 Jψ(dψ(grad(lnf)))+m/2 grad|dψ(grad(lnf))|2=0, and Φ2=φ × ψ is a proper biharmonic map if and only if φ¯ and ψ¯ are proper biharmonic maps. Copyright.