This paper proposes a method called Geometric Prior Guided Feature Representation Learning for Long-Tailed Classification. The main idea is to use the geometric information of the feature distribution of the well-represented head class to guide the model in learning the underlying distribution of the tail class. The method first systematically defines the geometry of the feature distribution and the similarity measures between the geometries, and discovers four phenomena regarding the relationship between the geometries of different feature distributions. Based on these phenomena, feature uncertainty representation is proposed to perturb the tail features by utilizing the geometry of the head class feature distribution. The goal is to make the perturbed features cover the underlying distribution of the tail class as much as possible, thus improving the model's generalization performance in the test domain. A three-stage training scheme is designed to enable feature uncertainty modeling to be successfully applied. Experiments on CIFAR-10/100-LT, ImageNet-LT, and iNaturalist2018 show that the proposed approach outperforms other similar methods on most metrics. The code is available at https://github.com/mayanbiao1234/Geometric-metrics-for-perceptual-manifold.This paper proposes a method called Geometric Prior Guided Feature Representation Learning for Long-Tailed Classification. The main idea is to use the geometric information of the feature distribution of the well-represented head class to guide the model in learning the underlying distribution of the tail class. The method first systematically defines the geometry of the feature distribution and the similarity measures between the geometries, and discovers four phenomena regarding the relationship between the geometries of different feature distributions. Based on these phenomena, feature uncertainty representation is proposed to perturb the tail features by utilizing the geometry of the head class feature distribution. The goal is to make the perturbed features cover the underlying distribution of the tail class as much as possible, thus improving the model's generalization performance in the test domain. A three-stage training scheme is designed to enable feature uncertainty modeling to be successfully applied. Experiments on CIFAR-10/100-LT, ImageNet-LT, and iNaturalist2018 show that the proposed approach outperforms other similar methods on most metrics. The code is available at https://github.com/mayanbiao1234/Geometric-metrics-for-perceptual-manifold.