  We introduce a class of rotationally invariant manifolds, which we call
\emph{admissible}, on which the wave flow satisfies smoothing and Strichartz
estimates. We deduce the global existence of equivariant wave maps from
admissible manifolds to general targets, for small initial data of critical
regularity $H^{\frac n2}$. The class of admissible manifolds includes in
particular asymptotically flat manifolds and perturbations of real hyperbolic
spaces $\mathbb{H}^{n}$ for $n\ge3$.
