  Weyl defines what he means by an affine connection in the  following way: A point \(p\) on the manifold \(M\) is affinely  connected with its immediate neighborhood, if and only of for every  tangent vector \(v_{p}\) at \(p\), a tangent  vector \(v_{q}\) at \(q\) is determined to which the tangent vector \(v_{p}\) gives rise under  parallel displacement from \(p\) to the infinitesimally  neighboring point \(q\). This definition merely says that a  manifold is affinely connected if it admits the process of  infinitesimal parallel displacement of a vector. 