The paper introduces Metric Flow Matching (MFM), a novel simulation-free framework for conditional flow matching that learns interpolants to approximate geodesics on the data manifold. This approach addresses the limitation of traditional methods, which often produce straight interpolations that may lie outside the data manifold, leading to less meaningful reconstructions. MFM uses a data-dependent Riemannian metric to guide the learning of these interpolants, ensuring that the resulting probability paths stay close to the observed data. The framework is designed to be general and task-agnostic, with the ability to handle various types of data, including LiDAR navigation, unpaired image translation, and single-cell trajectory inference. The authors demonstrate that MFM outperforms Euclidean baselines, particularly in single-cell trajectory prediction, by providing more accurate and realistic reconstructions. The paper also discusses the theoretical foundations of MFM, including the construction of geodesics and the optimization of interpolants, and provides experimental results to validate its effectiveness.The paper introduces Metric Flow Matching (MFM), a novel simulation-free framework for conditional flow matching that learns interpolants to approximate geodesics on the data manifold. This approach addresses the limitation of traditional methods, which often produce straight interpolations that may lie outside the data manifold, leading to less meaningful reconstructions. MFM uses a data-dependent Riemannian metric to guide the learning of these interpolants, ensuring that the resulting probability paths stay close to the observed data. The framework is designed to be general and task-agnostic, with the ability to handle various types of data, including LiDAR navigation, unpaired image translation, and single-cell trajectory inference. The authors demonstrate that MFM outperforms Euclidean baselines, particularly in single-cell trajectory prediction, by providing more accurate and realistic reconstructions. The paper also discusses the theoretical foundations of MFM, including the construction of geodesics and the optimization of interpolants, and provides experimental results to validate its effectiveness.