We provide a simple and geometric proof of small data global existence of the shifted wave equation on hyperbolic space involving nonlinearities of the form $$extract_itex$$pm |u| ^p$$/extract_itex$$  or  $$extract_itex$$pm |u|^{p-1}u$$/extract_itex$$.  It is based on the weighted Strichartz estimates of Georgiev-Lindblad-Sogge(or Tataru) on Euclidean space.  We also prove a small data existence theorem for variably curved backgrounds which extends earlier ones for the constant curvature case of Anker-Pierfelice and Metcalfe Taylor.   It is based on the joint work with Yannick Sire and Chris Sogge.