  We prove global-in-time Strichartz estimates for the shifted wave equations
on non-trapping asymptotically hyperbolic manifolds. The key tools are the
spectral measure estimates from \cite{CH2} and arguments borrowed from
\cite{HZ, Zhang}. As an application, we prove the small data global existence
for any power $p\in(1, 1+\frac{4}{n-1})$ for the shifted wave equation in this
setting, involving nonlinearities of the form $\pm|u|^p$ or $\pm|u|^{p-1}u$,
which answers partially an open question raised in \cite{SSW}.
