Deep learning-based 3D anomaly detection methods have demonstrated significant potential in industrial manufacturing. However, many approaches are specifically designed for anomaly detection tasks, which limits their generalizability. In contrast, self-supervised point cloud models aim for general-purpose representation learning, yet our investigation reveals that these classical models are suboptimal at anomaly detection under the unified fine-tuning paradigm. This motivates us to develop a more generalizable 3D model that can effectively detect anomalies without relying on task-specific designs. Interestingly, we find that using only the curvature of each point as its anomaly score already outperforms several classical self-supervised and dedicated anomaly detection models, highlighting the critical role of \textbf{curvature} in 3D anomaly detection. In this paper, we propose a Curvature-Augmented Self-supervised Learning (CASL) framework based on a reconstruction paradigm. Built upon the classical U-Net architecture, our approach introduces multi-scale curvature prompts to guide the decoder in predicting the spatial coordinates of each point. Without relying on any dedicated anomaly detection mechanisms, it achieves state-of-the-art performance through straightforward classification fine-tuning, improving the average O-AUROC by 5.6\% on the Real3D-AD dataset and 4.8\% on the Anomaly-ShapeNet dataset. Moreover, the learned representations generalize well to standard 3D understanding tasks such as point cloud classification and part segmentation.