Abstract: Recently, several attempts have demonstrated that 3D deep learning models are as vulnerable to adversarial example attacks as 2D models. However, these methods are still far from stealthy and suffer from severe performance degradation in the physical world. Although 3D data is highly structured, it is difficult to bound the perturbations with simple metrics in the Euclidean space. In this paper, we propose a novel $\epsilon$-isometric ($\epsilon$-ISO) attack method to generate natural and robust 3D adversarial examples in the physical world by considering the geometric properties of 3D objects and the invariance to physical transformations. For naturalness, we constrain the adversarial example and the original one to be $\epsilon$-isometric by adopting the Gaussian curvature as the surrogate metric under a theoretical analysis. For robustness under physical transformations, we propose a maxima over transformation (MaxOT) method to actively search for the most difficult transformations rather than random ones to make the generated adversarial example more robust in the physical world. Extensive experiments on typical point cloud recognition models validate that our approach can improve the attack success rate and naturalness of the generated 3D adversarial examples than the state-of-the-art attack methods.