D O C . 2 1 6 R I E M A N N I A N G E O M E T R Y & PA R A L L E L I S M 2 1 1 with respect to this local system (i.e., the n-Bein orthogonal frame). For the de- scription of a finite region, we also introduce the Gaussian coordinate system with length elements xν. Let Aν be the ν components of the vector (A) with respect to this latter coordinate system, and furthermore let hν a be the ν components of the unit vectors that form the n-Bein orthogonal frame then we have1 ... . (1) By inverting (1), and denoting the normalized subdeterminants of the hν a by h νa , we obtain ... . (1a) Then for the magnitude A of the vector A, owing to the Euclidean properties of the infinitesimal regions, the formula ... [4] (2) holds. The components of the metric tensor g μν can then be represented in the form , … (3) where of course a sum over a is implied. For fixed a, the are the components of a contravariant vector. Furthermore, the relations[5] ... and (4) , ... (5) hold, where δ = 1 or 0, depending on whether its two indices are the same or dif- ferent. The correctness of (4) and (5) follows from the above definition of the as the normalized subdeterminants of the .[6] The vector character of the is most readily seen from the fact that the left sides, i.e., also the right side of (1a), are invariant with respect to arbitrary coordinate transformations for any choice of the vector (A).[7] The n-Bein orthogonal frame is determined by the n2 functions , while the Rie- mannian metric is determined by only the quantities g μν . According to (3), the metric is determined by the n-Bein frame, but not conversely the last by the first. _____________________________________ 1 We denote the indices of the Gaussian coordinates by Greek letters, and those of the local frame by Roman letters.[3] A h a A a = [p. 218] A a h a A = A 2 A a 2 h a h a A A = = g h a h a = h a h a h a = h a h b ab = h a h a h a h a n n 1 + 2 --------------------