On the biharmonicity of product maps

where vg is the volume form on (M,g) and e( f )(x) := (1/2)‖df (x)‖T∗M⊗ f −1TN is the energy density of f at the point x ∈M. In local coordinates (x)i=1 on M and (y)a=1 on N , the energy density is given by e( f )(x)= (1/2)gi (x)hab( f (x))(∂ f a/∂xi)(∂ f b/∂x j). Critical points of the energy functional are called harmonic maps. The first variational formula of the energy gives the following characterization of harmonic maps: the map f is harmonic if and only if its tension field τ( f ) vanishes identically, where the tension field is given by