We　consider　the　following　nonlinear　Schrödinger　equation　on　exterior　domain.　(equation　presented)　where　1　＜　p　＜　n+2　n-2　,　⊂　Rn(n　≥　3)　is　an　exterior　domain　and　(Rn,　g)　is　a　complete　Riemannian　manifold.We　establish　Morawetz　estimates　for　the　system　(1)　without　dissipation　(a(x)　≡　0　in　(1))　and　meanwhile　prove　exponential　stability　of　the　system　(1)　with　a　dissipation　effective　on　a　neighborhood　of　the　infinity.　It　is　worth　mentioning　that　our　results　are　different　from　the　existing　studies.　First,　Morawetz　estimates　for　the　system　(1)　are　directly　derived　from　the　metric　g　and　are　independent　on　the　assumption　of　an　(asymptotically)　Euclidean　metric.　In　addition,　we　not　only　prove　exponential　stability　of　the　system　(1)　with　non-uniform　energy　decay　rate,　which　is　dependent　on　the　initial　data,　but　also　prove　exponential　stability　of　the　system　(1)　with　uniform　energy　decay　rate.　The　main　methods　are　the　development　of　Morawetz　multipliers　in　non　(asymptotically)　Euclidean　spaces　and　compactness-uniqueness　arguments.　©　2020　International　Press　of　Boston,　Inc..　All　rights　reserved.