Defining the geometry of networks is typically associated with embedding in low-dimensional spaces such as manifolds. This approach has helped design efficient learning algorithms, unveil network symmetries and study dynamical network processes. However, the choice of embedding space is network-specific, and incompatible spaces can result in information loss. Here, we define a dynamic edge curvature for the study of arbitrary networks measuring the similarity between pairs of dynamical network processes seeded at nearby nodes. We show that the evolution of the curvature distribution exhibits gaps at characteristic timescales indicating bottleneck-edges that limit information spreading. Importantly, curvature gaps robustly encode communities until the phase transition of detectability, where spectral clustering methods fail. We use this insight to derive geometric modularity optimisation and demonstrate it on the European power grid and the C. elegans homeobox gene regulatory network finding previously unidentified communities on multiple scales. Our work suggests using network geometry for studying and controlling the structure of and information spreading on networks.