Describing the basic properties of road network systems, such as their robustness, vulnerability, and reliability, has been a very important research topic in the field of urban transportation. Current research mainly uses several statistical indicators of complex networks to analyze the road network systems. However, these methods are essentially node-based. These node-based methods are more concerned with the number of connections between nodes, and lack of consideration for interactions. So, this leads to the well-known node paradox problem, and their ability of characterizing the local and intrinsic properties of a network is weak. From the perspective of network intrinsic geometry, this paper proposes a method for measuring road network vulnerability using a discrete Ricci curvature, which can identify the key sections of a road network and indicate its fragile elements. The results show that our method performs better than complex network statistics on measuring the vulnerability of a road network. Additionally, it can characterize the evolution of the road network vulnerability among different periods of time in the same city through our method. Finally, we compare our method with the previous method of centrality and show the different between them. This article provides a new perspective on a geometry to analyze the vulnerability of a road network and describes the inherent nature of the vulnerability of a road system from a new perspective. It also contributes to enriching the analytical methods of complex road networks.