Here　we　show　that　the　Gross-Pitaevskii　equation　(GPE)　for　Bose-Einstein　condensates　(BECs)　admits　hydrodynamic　interpretation　in　a　general　Riemannian　metric,　and　show　that　in　this　metric　the　momentum　equation　has　a　new　term　that　is　associated　with　local　curvature　and　density　distribution　profile.　In　particular　conditions　of　steady　state　a　new　Einstein＇s　field　equation　is　determined　in　presence　of　negative　curvature.　Since　GPE　governs　BECs　defects　that　are　useful,　analogue　models　in　cosmology,　a　relativistic　form　of　GPE　is　also　considered　to　show　connection　with　models　of　analogue　gravity,　thus　providing　further　grounds　for　future　investigations　of　black　hole　dynamics　in　cosmology.