  We consider compact K\"ahler manifolds acted on by a connected compact Lie
group $K$ of isometries in Hamiltonian fashion. We prove that the squared
moment map $\|\mu\|^2$ is constant if and only if the manifold is
biholomorphically and $K$-equivariantly isometric to a product of a flag
manifold and a compact K\"ahler manifold which is acted on trivially by $K$.
The authors do not know whether the compactness of $M$ is essential in the main
theorem; more generally it would be interesting to have a similar result for
(compact) symplectic manifolds.
