In this paper we discuss the role of curvature in the context of color spaces. Curvature is a differential geometric property of color spaces that has attracted less attention than other properties like the metric or geodesics. In this paper we argue that the curvature of a color space is important since curvature properties are essential in the construction of color coordinate systems. Only color spaces with negative or zero curvature everywhere allow the construction of Munsell-like coordinates with geodesics, shortest paths between two colors, that never intersect. In differential geometry such coordinate systems are known as Riemann coordinates and they are generalizations of the well-known polar coordinates.We investigate the properties of two measurement sets of just-noticeable-difference (jnd) ellipses and color coordinate systems constructed from them. We illustrate the role of curvature by investigating Riemann normal coordinates in CIELUV and CIELAB spaces. An algorithsm is also shown to build multipatch Riemann coordinates for spaces with the positive curvature.