$n$-dimensional Riemann spaces are considered that are the direct product of the

$n-1$-dimensional Euclidean unit sphere and the half line and are equipped with a rotationally symmetric metric. Such spaces may be isometric to the Euclidean or to the hyperbolic space of dimension

$n$. Rotationally symmetric maps of one space to the other are considered and an energy integral is attached to each map. The Euler-Lagrange equation of this integral is a second order nonlinear scalar differential equation, and a map is called harmonic if its defining scalar function is a positive solution tending to zero at infinity. Conditions are given under which any bounded harmonic map is constant and other properties are studied, too.