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Learning a latent embedding to understand the underlying nature of data distribution is often formulated in Euclidean spaces with zero curvature. However, the success of the geometry constraints, posed in the embedding space, indicates that curved spaces might encode more structural information, leading to better discriminative power and hence richer representations. In this work, we investigate benefits of the curved space for analyzing anomalies or out-of-distribution objects in data. This is achieved by considering embeddings via three geometry constraints, namely, spherical geometry (with positive curvature), hyperbolic geometry (with negative curvature) or mixed geometry (with both positive and negative curvatures). Three geometric constraints can be chosen interchangeably in a unified design given the task at hand. Tailored for the embeddings in the curved space, we also formulate functions to compute the anomaly score. Two types of geometric modules (i.e., Geometric-in-One and Geometric-in-Two models) are proposed to plug in the original Euclidean classifier, and anomaly scores are computed from the curved embeddings. We evaluate the resulting designs under a diverse set of visual recognition scenarios, including image detection (multi-class OOD detection and one-class anomaly detection) and segmentation (multi-class anomaly segmentation and one-class anomaly segmentation). The empirical results show the effectiveness of our proposal through the consistent improvement over various scenarios.