  It is not advisable to bind the frame of reference in \(\Sigma_{p}\) to the coordinates \(x^{i}\) covering the neighborhood of \(p\) in \(M\). In this respect the old treatment of affinely connected manifolds is misleading. … [I]n the modern development of infinitesimal geometry in the large, where it combines with topology and the associated Klein spaces appear under the name of fibres, it has been found best to keep the répères, the frames of the fibre spaces, independent of the coordinates of the underlying manifold.