We　study　the　internal　stabilization　and　the　boundary　stabilization　of　the　critical　semilinear　Klein-Gordon　equation　in　compact　space.　The　observability　inequalities　are　proved　by　the　Morawetz　estimates　on　Riemannian　manifold,　and　then　the　compactness-uniqueness　arguments　are　used　to　prove　the　main　stabilization　results.