Abstract: Graph-structured data plays a crucial role in diverse fields such as social networks, drug discovery, and logistics transportation. However, the scaling laws of graph learning paradigm remains unclear, and the task generalization capability is limited. Therefore, the key challenge is how lies in understanding the semantics of graph complex structures, to enable graph foundation models to break through from connection generalization to isomorphic generalization. Graph geometry (Riemannian geometry) provides a unified and elegant mathematical tool for topological representation, and has potential and power as a new theoretical underpinning for graph foundation models. Firstly, we introduce the mathematical foundation of graph geometry and geometric deep learning. Secondly, we attempt to re-interpret the emergence phenomenon of graph-based models from a geometric perspective. Thirdly, we summarize the research status of graph foundation models based on Riemannian geometry. Finally, we discuss the typical applications and future prospects of Riemannian graph foundation models. This article aims to systematically review the research taxonomy of graph-based models from a geometric perspective, and deeply discuss their mathematical principles, key technical frameworks, core research progress and future development directions.