We　study　the　stability　of　the　defocusing　critical　semilinear　wave　equation　with　locally　distributed　damping　in　an　exterior　domain.　Under　some　assumptions　on　the　Riemannian　metric,　the　geometric　multiplier　technique　and　the　compactness　uniqueness　arguments　were　used　to　prove　the　unique　continuation,　the　observability　inequality　respectively.　Then　the　energy　decay　of　the　nonlinear　system　was　obtained.&　COPY;　2023　Elsevier　Inc.　All　rights　reserved.