Distance functions are crucial in robotics for representing spatial relationships between a robot and its environment. They provide an implicit, continuous, and differentiable representation that integrates seamlessly with control, optimization, and learning. While standard distance fields rely on the Euclidean metric, many robotic tasks inherently involve non-Euclidean structures. To this end, we generalize Euclidean distance fields to more general metric spaces by solving the Riemannian eikonal equation, a first-order partial differential equation whose solution defines a distance field and its associated gradient flow on the manifold, enabling the computation of geodesics and globally length-minimizing paths. We demonstrate that geodesic distance fields, the classical Riemannian distance function represented as a global, continuous, and queryable field, are effective for a broad class of robotic problems where Riemannian geometry naturally arises. To realize this, we present a neural Riemannian eikonal solver (NES) that solves the equation as a mesh-free implicit representation without grid discretization, scaling to high-dimensional robot manipulators. Training leverages a physics-informed neural network (PINN) objective that constrains spatial derivatives via the PDE residual and boundary and metric conditions, so the model is supervised by the governing equation and requires no labeled distances or geodesics. We propose two NES variants, conditioned on boundary data and on spatially varying Riemannian metrics, underscoring the flexibility of the neural parameterization. We validate the effectiveness of our approach through extensive examples, yielding minimal-length geodesics across diverse robot tasks involving Riemannian geometry.