  We study the long time behavior of radial solutions to nonlinear
Schr\"{o}dinger equations on hyperbolic space. We show that the usual
distinction between short range and long range nonlinearity is modified: the
geometry of the hyperbolic space makes every power-like nonlinearity short
range. The proofs rely on weighted Strichartz estimates, which imply Strichartz
estimates for a broader family of admissible pairs, and on Morawetz type
inequalities. The latter are established without symmetry assumptions.
