On the Expressive Power of Mixed-Curvature Representations in Product Manifolds

Mixed-curvature representations based on product manifolds of Euclidean, hyperbolic, and spherical spaces are widely used as latent geometries in representation learning and geometric deep learning. Their popularity stems from their analytical tractability and closed-form geodesic distances, which facilitate efficient backpropagation in deep learning. In the literature, they are often portrayed as richer representations than, for instance, hyperbolic embeddings. However, their expressive power is not well understood. In this work, we show that such mixed-curvature product manifolds impose rigid geometric constraints that fundamentally limit their expressive power. In particular, we prove that generic Riemannian manifolds cannot be locally represented isometrically by mixed-curvature product manifolds equipped with product metrics. The obstruction arises from curvature splitting: product metrics necessarily exhibit vanishing mixed sectional curvature (equivalently, flatness on mixed 2-planes), a rigidity that fails for a generic Riemannian metric.