Temporal Knowledge Graphs (TKGs) represent a crucial source of structured temporal information and exhibit significant utility in various real-world applications. However, TKGs are susceptible to incompleteness, necessitating Temporal Knowledge Graph Completion (TKGC) to predict missing facts. Existing models have encountered limitations in effectively capturing the intricate temporal dynamics and hierarchical relations within TKGs. To address these challenges, HyGNet is proposed, leveraging hyperbolic geometry to effectively model temporal knowledge graphs. The model comprises two components: the Hyperbolic Gated Graph Neural Network (HGGNN) and the Hyperbolic Convolutional Neural Network (HCNN). HGGNN aggregates neighborhood information in hyperbolic space, effectively capturing the contextual information and dependencies between entities. HCNN interacts with embeddings in hyperbolic space, effectively modeling the complex interactions between entities, relations, and timestamps. Additionally, a consistency loss is introduced to ensure smooth transitions in temporal embeddings. The extensive experimental results conducted on four benchmark datasets for TKGC highlight the effectiveness of HyGNet. It achieves state-of-the-art performance in comparison to previous models, showcasing its potential for real-world applications that involve temporal reasoning and knowledge prediction.