In Euclidean space the homogeneity of space is expressed by the invariance of the metric under parallel displacements (translations) of the Cartesian coordinate system. Each translation is determined by three parameters — the components of the displacement vector of the coordinate origin. All these transformations leave invariant the three independent differentials ("dx", "dy", "dz") from which the line element is constructed. In the general case of a non-Euclidean homogeneous space, the transformations of its group of motions again leave invariant three independent linear differential forms, which do not, however, reduce to total differentials of any coordinate functions. These forms are written as formula_53 where the Latin index ("a") labels three independent vectors (coordinate functions); these vectors are called a frame field or triad. The Greek letters label the three space-like curvilinear coordinates. A spatial metric invariant is constructed under the given group of motions with the use of the above forms: