Graph-structured data, operating as an abstraction of data containing nodes and interactions between nodes, is pervasive in the real world. There are numerous ways dedicated to extract graph structure information explicitly or implicitly, but whether it has been adequately exploited remains an unanswered question. This work goes deeper by heuristically incorporating a geometric descriptor, the discrete Ricci curvature (DRC), in order to uncover more graph structure information. We present a curvature-based topology-aware graph transformer, termed Curvphormer. This work expands the expressiveness by using a more illuminating geometric descriptor to quantify the connections within graphs in modern models and to extract the desired structure information, such as the inherent community structure in graphs with homogeneous information. We conduct extensive experiments on a variety of scaled datasets, including PCQM4M-LSC, ZINC, and MolHIV, and obtain a remarkable performance gain on various graph-level tasks and fine-tuned tasks.