  In this paper we prove an abstract result of almost global existence for
small and smooth solutions of some semilinear PDEs on Riemannian manifolds with
globally integrable geodesic flow. Some examples of such manifolds are Lie
groups (including flat tori), homogeneous spaces and rotational invariant
surfaces. As applications of the abstract result we prove almost global
existence for a nonlinear Schr\"odinger equation with a convolution potential
and for a nonlinear beam equation. We also prove $H^s$ stability of the ground
state in NLS equation. The proof is based on a normal form procedure.
