On the biharmonicity of product maps (Q885579)

A harmonic (resp., biharmonic) map between Riemannian manifolds is a critical point of the energy (resp., bienergy) functional [\textit{J. Eells jun.} and \textit{J. H. Sampson}, Am. J. Math. 86, 109--160 (1964; Zbl 0122.40102)]. Let \(\phi_i :(M_i,g_i) \to (N_i,h_i)\), i=1,2, be two harmonic maps between Riemannian manifolds and let \(\alpha\), \(\beta\) be two positive functions on \(M=M_1 \times M_2\) and \(N=N_1 \times N_2\), respectively. Let \(G_{\alpha} = \pi_1*g + (\alpha \circ \pi_1)^2 \pi_2*g_2\) and similarly \(H_{\beta}\) be two warped product metrics on \(M\) and \(N\), respectively. Some necessary an sufficient conditions are given that the map \(\phi_1 \times \phi_2 : (M,G_{\alpha}) \to (N,H_{\beta})\) defined by \(\phi_1 \times \phi_2 (x,y) = (\phi_1 (x),\phi_2 (y))\) is harmonic. By [\textit{B. Chen, T. Nagano}, J. Math. Soc. Japan 36, 295--313 (1984; Zbl 0543.58015)], a Riemannian metric \(\sigma_2\) is harmonic (resp., biharmonic) w.r.t. a metric \(\sigma_1\) on a manifold \(S\) if the identity map \(i:(S,\sigma_1) \to (S,\sigma_2)\) is harmonic (resp., biharmonic). Some necessary and sufficient conditions are given here such that a warped product metric \(G_\alpha\) is harmonic (resp., nonharmonic biharmonic) w.r.t. a metric \(G\) on \(M\). Some examples are constructed.