  Manifolds discovered by machine learning models provide a compact
representation of the underlying data. Geodesics on these manifolds define
locally length-minimising curves and provide a notion of distance, which are
key for reduced-order modelling, statistical inference, and interpolation. In
this work, we propose a model-based parameterisation for distance fields and
geodesic flows on manifolds, exploiting solutions of a manifold-augmented
Eikonal equation. We demonstrate how the geometry of the manifold impacts the
distance field, and exploit the geodesic flow to obtain globally
length-minimising curves directly. This work opens opportunities for statistics
and reduced-order modelling on differentiable manifolds.
