Graph Manifold: Rethinking Graph Learning Through Riemannian Geometry

Graph Neural Networks (GNNs) have achieved remarkable success in graph representation learning by modelling graph topology and the relationships among nodes embedded within it. However, existing graph learning typically regards graphs as discrete structures and optimises them based on local neighbourhood relationships, ignoring the intrinsic geometric properties and global evolution patterns. To address this, we rethink graph learning from the perspective of Riemannian geometry and introduce a novel **Graph Manifold**, which provides a continuous geometric space for graph representation and optimisation. Specifically, we reconstruct graph topology geometrically based on the Cholesky manifold, representing graphs as structured objects on a manifold space and enabling holistic geometric graph optimisation. Furthermore, we define a distributional structure on the Graph Manifold by modelling directional distributions in the tangent space, allowing the characterisation of intrinsic graph variations. Moreover, to extend the applicability of Graph Manifold to non-Euclidean computational spaces, we formulate Graph Manifold within a gyrovector algebraic framework and rigorously prove the intrinsic consistency of its geometric operations from a mathematical perspective. Finally, we apply Graph Manifold to the graph topology-oriented prompting task for pre-trained GNNs and conduct extensive experiments on four graph datasets under four pre-training strategies. Experimental results demonstrate that the proposed Graph Manifold consistently improves performance across multiple node classification tasks, validating its effectiveness as a novel Riemannian geometry for graph learning.