Multiple mobile robots play a significant role in various spatially distributed tasks.In unfamiliar and non-repetitive scenarios, reconstructing the global map is time-inefficient and sometimes unrealistic. Hence, research has focused on achieving real-time collaborative planning by utilizing sensor data from multiple robots located at different positions, all without relying on a global map.This paper introduces a Multi-Robot collaborative Path Planning method based on Geometric Graph Neural Network (MRPP-GeoGNN). We extract the features of each neighboring robot's sensory data and integrate the relative positions of neighboring robots into each interaction layer to incorporate obstacle information along with location details using geometric feature encoders. After that, a MLP layer is used to map the amalgamated local features to multiple forward directions for the robot's actual movement. We generated expert data in ROS to train the network and carried out both simulations and physical experiments to validate the effectiveness of the proposed method. Simulation results demonstrate an approximate 5% improvement in accuracy compared to the model based solely on CNN on expert datasets. The success rate is enhanced by about 4% compared to CNN, and the flowtime increase is reduced by approximately 18% in the ROS test, surpassing other GNN models. Besides, the proposed method is able to leverage neighbor's information and greatly improves path efficiency in real-world scenarios.