This paper introduces METRIC FLOW MATCHING (MFM), a novel simulation-free framework for conditional flow matching that learns interpolants as approximate geodesics under a data-induced Riemannian metric. Unlike traditional methods that assume Euclidean geometry and produce straight interpolations, MFM constructs paths that stay close to the data manifold, improving the accuracy of interpolations in tasks such as trajectory inference, unpaired image translation, and modeling cellular dynamics. The framework minimizes the kinetic energy of a data-dependent Riemannian metric to approximate geodesics, leading to lower uncertainty and more meaningful interpolations. MFM is general and task-agnostic, allowing for the use of various metrics, including diagonal metrics like LAND and RBF, which are derived from the data. The paper demonstrates that MFM outperforms Euclidean baselines, particularly in single-cell trajectory prediction, and shows its effectiveness on a range of tasks including LiDAR navigation, image translation, and trajectory inference for single-cell data. The proposed method avoids the need for simulations by learning interpolants that approximate geodesics without explicitly parameterizing the lower-dimensional manifold. This approach enables more accurate and natural reconstructions of underlying dynamics, making MFM a promising tool for applications involving complex data structures.This paper introduces METRIC FLOW MATCHING (MFM), a novel simulation-free framework for conditional flow matching that learns interpolants as approximate geodesics under a data-induced Riemannian metric. Unlike traditional methods that assume Euclidean geometry and produce straight interpolations, MFM constructs paths that stay close to the data manifold, improving the accuracy of interpolations in tasks such as trajectory inference, unpaired image translation, and modeling cellular dynamics. The framework minimizes the kinetic energy of a data-dependent Riemannian metric to approximate geodesics, leading to lower uncertainty and more meaningful interpolations. MFM is general and task-agnostic, allowing for the use of various metrics, including diagonal metrics like LAND and RBF, which are derived from the data. The paper demonstrates that MFM outperforms Euclidean baselines, particularly in single-cell trajectory prediction, and shows its effectiveness on a range of tasks including LiDAR navigation, image translation, and trajectory inference for single-cell data. The proposed method avoids the need for simulations by learning interpolants that approximate geodesics without explicitly parameterizing the lower-dimensional manifold. This approach enables more accurate and natural reconstructions of underlying dynamics, making MFM a promising tool for applications involving complex data structures.