  In this note we concern with the wave maps from the Lorentzian manifold with
the periodic in time metric into the Riemannian manifold, which belongs to the
one-parameter family of Riemannian manifolds. That family contains as a special
case the Poincare upper half-plane model. Our interest to such maps is
motivated with some particular type of the Robertson-Walker spacetime arising
in the cosmology. We show that small periodic in time perturbation of the
Minkowski metric generates parametric resonance phenomenon. We prove that, the
global in time solvability in the neighborhood of constant solutions is not a
stable property of the wave maps.
