Rethinking Riemannian MeanFlow: Geometric Gaps and Geometry-Aligned Flow Maps

MeanFlow enables efficient one-step or few-step generation by learning average velocities over finite time intervals. Recent works have extended this principle to Riemannian manifolds for data with intrinsic non-Euclidean structure, such as rotations and directions. However, under curvature, the Euclidean relationships among average velocity, endpoint displacement, and generation error no longer directly hold. We revisit two formulations of Riemannian MeanFlow and identify two geometric gaps. First, parallel-transported path averages generally differ from endpoint log-displacements, potentially inducing one-step endpoint errors and flow-map composition defects. Second, even with endpoint-based representations, metric distortion through the exponential map causes tangent-space residuals and geodesic endpoint errors to weight directions differently. Motivated by these findings, we propose **Barycentric Semigroup Riemannian MeanFlow (RMF-BSC)**, which combines endpoint differential consistency with multi-split geodesic semigroup consistency to align training with endpoint geometry. We establish barycentric stability bounds and a variance-reduction result for multi-split averaging. Experiments validate the identified gaps and show that RMF-BSC achieves strong one-step performance across four diverse manifold benchmarks.