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 $pm lul^p$ or $pm lul^{p-1}u$. 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.