  Since the conformal structure determines only an equivalence class of conformally equivalent symmetric linear connections \(K = [\Gamma]\), the affine connection in this type of geometry is not uniquely determined, and the parallel transport of vectors is not generally well defined. Moreover, the ratio of the lengths of two vectors located at different points is not determined even in a path-dependent way. According to Weyl, it is a fundamental principle of infinitesimal geometry that the metric structure on a manifold \(M\) determines a unique affine structure on \(M\). As was pointed out earlier, this principle is satisfied in Riemannian geometry where the metric determines a unique symmetric linear connection, namely, the metric connection according to (9). Evidently this fundamental principle of infinitesimal geometry is not satisfied for a structure which is merely a conformal structure, since the conformal structure only determines an equivalence class of conformally equivalent symmetric connections. Weyl showed that besides the conformal structure an additional structure is required in order to determine a unique symmetric linear connection from the equivalence class  \(K = [\Gamma]\) of conformally equivalent symmetric linear connections. Weyl showed that this additional structure is provided by the length connection or gauge field \(A_{j}\) that governs the congruent displacement of lengths. Weyl called this additional structure the “metric connection” on a manifold; however, we shall use the term “length connection” instead, in order to avoid confusion with the modern usage of the term “metric connection”, which today denotes the symmetric linear connection that is uniquely determined by a Riemannian metric tensor according to (9). 