  In this paper, we consider exterior problem of the critical semilinear wave
equation in three space dimensions with variable coefficients and prove global
existence of smooth solutions.
  Similar to the constant coefficients case, we show that the energy cannot
concentrate at any point $(t,x)\in(0,\infty)\times\Omega$. For that purpose,
following Ibrahim and Majdoub \cite{Ibrahim}, we use a geometric multiplier
close to the well-known Morawetz multiplier used in the constant coefficients
case. Then we use comparison theorem from Riemannian Geometry to estimate the
error terms. Finally, using Strichartz inequality as in Smith and Sogge
\cite{Sogge}, we get the global existence.
