The geometry of perceived color space is widely recognized as non-Euclidean, with the Riemannian framework commonly adopted for its analysis. However, existing evidence, such as the principle of diminishing returns, suggests that the color space may be globally non-Riemannian. In this work, I investigate the local inhomogeneities of the perceived color space under the Riemannian setting. Specifically, the local agreement between the Riemannian model and the color-difference function are evaluated. In numerical experiments, the accuracy of the parallelogram law is assessed -- a necessary condition for the local validity of the metric tensor. Furthermore, I introduce several measures of local anisotropy to quantify directional variations in perceived color distances and compute these measures within the chromatic planes of the CIELAB color space. My findings describe the spatial variation of Riemannian inhomogeneities and distance anisotropy, which can be used to construct adaptive spatial meshes and improve the accuracy of computations in color space. While the proposed techniques are demonstrated on the CIELAB color model with the $\Delta E_{2000}$ metric, they are generalizable to the discretization of arbitrary non-Euclidean metric spaces.