Generalized warped product manifolds and biharmonic maps (Q2913255)

A biharmonic map between two Riemannian manifolds, \(\varphi \:(M,g) \rightarrow (N,h)\), is a critical point of the bi-energy functional \(E_2(\varphi)= \frac {1}{2} \int _M | \tau (\varphi)| ^2 v_g\). The authors consider a generalized warped product \((M \times _f N, G_f)\) of two Riemannian manifolds, \(f\) being a smooth positive function defined on \(M\times N\), and they give an explicit expression of the bitension field \(\tau _2(\phi)\) in different situations, in particular for the inclusion \(\phi \: (N,h)\rightarrow (M\times _f N ,G_f)\), with \(\phi (y)=(x_0,y)\) and \(x_0\) a fixed point in \(M\), and for a map \(\phi :(M\times _f N ,G_f)\rightarrow (P,k)\). Moreover, given a conformal map \(\varphi \:(M,g)\rightarrow (P,l)\) with dilation \(\lambda \), they state a formula for the bitension of the map \(\phi \:(M\times _f N, G_f)\rightarrow (P,l)\) defined by \(\phi (x,y)=\varphi (x)\). In particular, if \(f \in C^{\infty }(M)\) out \({\operatorname {dim}} M ={\operatorname {dim}} P =m \geq 3\), then the biharmonicity of the map \(\phi \) is characterized by an equation involving \({\operatorname {ln}} \lambda , {\operatorname {ln}} f\), through the operators \(\operatorname {grad}\), \(\nabla\), and \(\Delta \) evaluated on \(M\), and \({\operatorname {Ricci}}^M\).