Harmonic maps and biharmonic maps for double fibrations of compact Lie groups

This work is motivated by the works of W.Y. Hsiang and H.B. Lawson [7], Pages $12$ and $13$. In this paper, we deal with the following double fibration: \[ \xymatrix@R-0.5cm @C-0.5cm{ & (G,g) \ar[ld]_{\pi_1} \ar[rd]^{\pi_2} & \\ (G/H,h_1) && (K\backslash G,h_2) } \] We will show that every $K$-invariant minimal or biharmonic hypersurface $M$ in $(G/H,h_1)$ induces an $H$-invariant minimal or biharmonic hypersurface $\widetilde{M}$ in $(K\backslash G,h_2)$ by means of $\widetilde{M}:=\pi_2(\pi_1{}^{-1}(M))$ (cf. Theorems 3.2 and 4.1). We give a one to one correspondence between the class of all the $K$-invariant minimal or biharmonic hypersurfaces in $G/H$ and the one of all the $H$-invariant minimal or biharmonic hypersurfaces in $K\backslash G$ (cf. Theorem 4.2).