  Weyl’s next definition characterizes the essential nature of infinitesimal parallel displacement. The definition says that at any  arbitrary point of the manifold there exists a geodesic  coordinate system such that the components of any vector at that point are not altered by an infinitesimal parallel displacement with  respect to it. This is a geometrical way of expressing Einstein’s  requirement that the gravitational field can always be made to vanish locally. According to Weyl (1923b, 115), it characterizes the  nature of an affine connection on the manifold. A manifold  which is an affine manifold is homogeneous in this sense. Moreover,  manifolds do not exist whose affine structure is of a different  nature. 