Temporal　Knowledge　Graphs　(TKGs)　incorporate　a　temporal　dimension,　allowing　for　a　precise　capture　of　the　evolution　of　knowledge　and　reflecting　the　dynamic　nature　of　the　real　world.　Typically,　TKGs　contain　complex　geometric　structures,　with　various　geometric　structures　interwoven.　However,　existing　Temporal　Knowledge　Graph　Completion　(TKGC)　methods　either　model　TKGs　in　a　single　space　or　neglect　the　heterogeneity　of　different　curvature　spaces,　thus　constraining　their　capacity　to　capture　these　intricate　geometric　structures.　In　this　paper,　we　propose　a　novel　Integrating　Multi-curvature　shared　and　specific　Embedding　(IME)　model　for　TKGC　tasks.　Concretely,　IME　models　TKGs　into　multi-curvature　spaces,　including　hyperspherical,　hyperbolic,　and　Euclidean　spaces.　Subsequently,　IME　incorporates　two　key　properties,　namely　space-shared　property　and　space-specific　property.　The　space-shared　property　facilitates　the　learning　of　commonalities　across　different　curvature　spaces　and　alleviates　the　spatial　gap　caused　by　the　heterogeneous　nature　of　multi-curvature　spaces,　while　the　space-specific　property　captures　characteristic　features.　Meanwhile,　IME　proposes　an　Adjustable　Multi-curvature　Pooling　(AMP)　approach　to　effectively　retain　important　information.　Furthermore,　IME　innovatively　designs　similarity,　difference,　and　structure　loss　functions　to　attain　the　stated　objective.　Experimental　results　clearly　demonstrate　the　superior　performance　of　IME　over　existing　state-of-the-art　TKGC　models.　©　2024　ACM.