In　this　article,　we　consider　the　stabilization　of　a　wave　equation　with　variable　coefficients　and　internal　memory　in　an　open　bounded　domain,　by　the　Riemannian　geometry　approach.　For　the　wave　equation　with　a　locally　distributed　memory　with　a　kernel,　we　obtain　exponential　decay　of　the　energy　under　some　geometric　conditions.　In　addition,　for　the　wave　equation　with　nonlinear　internal　time-varying　delay　without　upper　bound,　we　obtain　uniform　decay　of　the　energy.