Graphs are ubiquitous, and learning on graphs has become a cornerstone in artificial intelligence and data mining communities. Unlike pixel grids in images or sequential structures in language, graphs exhibit a typical non-Euclidean structure with complex interactions among the objects. This paper argues that Riemannian geometry provides a principled and necessary foundation for graph representation learning, and that Riemannian graph learning should be viewed as a unifying paradigm rather than a collection of isolated techniques. While recent studies have explored the integration of graph learning and Riemannian geometry, most existing approaches are limited to a narrow class of manifolds, particularly hyperbolic spaces, and often adopt extrinsic manifold formulations. We contend that the central mission of Riemannian graph learning is to endow graph neural networks with intrinsic manifold structures, which remains underexplored. To advance this perspective, we identify key conceptual and methodological gaps in existing approaches and outline a structured research agenda along three dimensions: manifold type, neural architecture, and learning paradigm. We further discuss open challenges, theoretical foundations, and promising directions that are critical for unlocking the full potential of Riemannian graph learning. This paper aims to provide a coherent viewpoint and to stimulate broader exploration of Riemannian geometry as a foundational framework for future graph learning research.

翻译：图结构无处不在，图学习已成为人工智能和数据挖掘领域的基石。与图像中的像素网格或语言中的序列结构不同，图展现出典型的非欧几里得结构，其中对象之间存在复杂的相互作用。本文认为，黎曼几何为图表征学习提供了原则性且必要的基础，黎曼图学习应被视为一个统一范式，而非孤立技术的集合。尽管近期研究已探索图学习与黎曼几何的结合，但现有方法大多局限于狭窄的流形类别（特别是双曲空间），且常采用外在流形表述。我们主张黎曼图学习的核心使命是为图神经网络赋予内在流形结构，这一方向尚未得到充分探索。为推进这一观点，我们指出现有方法在概念与方法论上的关键缺口，并沿着三个维度——流形类型、神经架构与学习范式——勾勒出结构化研究议程。我们进一步讨论了开放挑战、理论基础以及关键的前沿方向，这些对于释放黎曼图学习的全部潜力至关重要。本文旨在提供一个连贯的视角，并激发更广泛的探索，将黎曼几何作为未来图学习研究的基础框架。