Knowledge distillation is a key approach for model compression, but existing methods face challenges due to dimensional mismatches between teacher and student models. This work introduces the first differential geometric framework that quantifies how Riemannian curvature differences limit knowledge transfer across differently parameterized models. Based on this, Geometry-Aware Knowledge Distillation (GeomKD) is proposed, incorporating two innovations: (1) geometric structure loss to address dimensional gaps, and (2) manifold alignment loss to align teacher and student feature spaces by Ricci curvature and Fisher information matching. These methods address the limitations of traditional approaches that rely solely on similarity-based optimization. Experiments on the CIFAR-100 dataset demonstrate GeomKD’s superiority in configurations with large capacity gaps. For example, in the ResNet50 \(\rightarrow \) MobileNetV2 setup (capacity ratio 10.00 \(\times \) ), GeomKD achieves 69.90% Top-1 accuracy, surpassing baseline KD (61.33%) by 8.57%. Grounded in differential geometry and optimal transport theory, this framework provides a novel theoretical foundation for knowledge distillation, offering a new geometric perspective to advance the field.