4 4 8 D O C U M E N T 5 2 1 U N I F I E D F I E L D T H E O R Y In a Riemannian continuum, the local coordinate system is determined only up to an arbitrary rotational transformation (a linear orthogonal transformation), so that the orientation of the local system can be freely chosen at every point. We now assume that the continuum can in addition be attributed with a certain structure. Let P and Q be points on the continuum. Among all those vectors (A) that originate at P and all those vectors (B) that originate at Q, there is presumed to be a relationship of mutual correspondence: to a particular vector (A), there corre- sponds a certain vector (B), and vice versa. Two such vectors are called “parallel” (although this word, in contrast to usual meaning, can also be thought to include a dimensional relation). Between the metric structure and the parallel structure of the continuum, there is a relation: Parallel vectors are metrically equivalent. From this, we can readily derive the proposition that the angle between two vectors originating at P is the same as the angle between the corresponding parallel vectors that originate at Q. From this, we furthermore conclude that if the orthogonal local system has been freely chosen in terms of its orientation at one point P, then the local systems at all other points Q are determinable and completely determined by that choice, so that correspondingly a unit vector would be parallel in all of them. We wish to make this choice definitively. Then the proposition holds that parallel vectors have the same local components. It is characteristic of the spatial structure chosen here that the local systems (lo- cal orthogonal “n-tuples”) can be subjected to an arbitrary common rotation. The spatial structure thus by no means defines n preferential directions at a spatial point P.[6] This double structure of the continuum that we have envisaged can thus evident- ly be completely described by specifying the components of the orthogonal n-tuples with respect to the Gaussian coordinate system as functions of the coordi- nates. In , there are 16 functions in there are . denotes the -com- ponent of the a-axis, referred to the general (Gaussian) coordinate system, of the local orthogonal n-tuple.[7] It follows immediately from this definition that the following relation holds be- tween the components with respect to the general coordinate system and the local coordinates of the same vector: … (2) Solving these equations for the local components*5 , if we let denote the normalized subdeterminants of the : , … (3) where the and the are related by the equations [p. 4] h a  R4 Rn n2 h a  A A a A h a A a = [p. 5] A a h a h a  A a h a A = h a h a 