Source localization, the inverse problem of information diffusion, shows fundamental importance for understanding social dynamics. While achieving notable progress, existing solutions are typically exposed to the risk of error accumulation, and require a large number of observations for effective inference. However, it is often impractical to obtain quantities of observations in real scenarios, highlighting the need for a transferable model with broad applicability. Recently, Riemannian geometry has demonstrated its effectiveness in information diffusion and offers guidance in knowledge transfer, but has yet to be explored in source localization. In light of the issues above, we propose to study transferable source localization from a fresh geometric perspective, and present a novel approach (Trace) on the Riemannian manifold. Concretely, we establish a structural Schrodinger bridge to directly model the map between source and final distributions, where a functional curvature, encapsulating the graph structure, is formulated to govern the Schrodinger bridge and facilitate domain adaptation. Furthermore, we design a simple yet effective learning algorithm for Riemannian Schrodinger bridges (geodesics bridge matching) in which we prove the optimal projection holds for Riemannian measure so that the expensive iterative procedure is avoided. Extensive experiments demonstrate the effectiveness and transferability of Trace on both synthetic and real datasets.