In few-shot learning, utilizing local and global geometric priors to capture both subtle local class metrics and coarse global structures within the meta-task are important to obtain discriminative embeddings. However, existing graph-based and curvature-based few-shot approaches only focus on either one kind of geometric prior but neglect the other. To effectively utilize the pros of these two paradigms, we propose a novel Dual-Geometry Graph Network (DGGN) to adaptively integrate the local and global geometric priors via two key pathways. Specifically, the local-wise metric modeling pathway utilizes Ollivier-Ricci curvature to capture task-specific local class metrics among the instances, and the global-wise connectivity modeling pathway utilizes resistive embedding to capture global instance distributions and connectivity patterns of the entire meta-task. In addition, we introduce two new regularization loss functions to explicitly enhance the geometric representation ability of the local and global pathways respectively. We validate that DGGN's superior performance stems from its adaptively topological refinements by measuring the graph edit distance, demonstrating its ability to match the underlying data distribution. Extensive experiments show that DGGN sets a new state-of-the-art on standard, cross-domain, and semi-supervised few-shot benchmarks. Code is available in our Supplementary Material.