Temporal knowledge graphs (TKGs) are receiving increased attention due to their time-dependent properties and the evolving nature of knowledge over time. TKGs typically contain complex geometric structures, such as hierarchical, ring, and chain structures, which can often be mixed together. However, embedding TKGs into Euclidean space, as is typically done with TKG completion (TKGC) models, presents a challenge when dealing with high-dimensional nonlinear data and complex geometric structures. To address this issue, we propose a novel TKGC model called multicurvature adaptive embedding (MADE). MADE models TKGs in multicurvature spaces, including flat Euclidean space (zero curvature), hyperbolic space (negative curvature), and hyperspherical space (positive curvature), to handle multiple geometric structures. We assign different weights to different curvature spaces in a data-driven manner to strengthen the ideal curvature spaces for modeling and weaken the inappropriate ones. Additionally, we introduce the quadruplet distributor (QD) to assist the information interaction in each geometric space. Ultimately, we develop an innovative temporal regularization to enhance the smoothness of timestamp embeddings by strengthening the correlation of neighboring timestamps. Experimental results show that MADE outperforms the existing state-of-the-art TKGC models.