  It is important to understand that there is no intrinsic notion of  infinitesimal parallel displacement on a differentiable manifold. A  notion of “parallelism” is not something that a manifold  would possess merely by virtue of being a smooth manifold; additional structure has to be introduced which resides on the manifold and  which permits the notion of infinitesimal parallelism. A manifold is  an “affine manifold” \((M, \Gamma)\) if in addition to its  manifold structure (differential topological structure) it is also  endowed with an affine structure \(\Gamma\) that assigns to each of its  points 64 coefficients \(\Gamma^{i}_{jk} (x^{i}(p))\) satisfying the symmetry condition \(\Gamma^{i}_{jk} (x^{i}(p)) = \Gamma^{i}_{kj} (x^{i}(p))\). 