2 1 2 D O C . 2 1 6 R I E M A N N I A N G E O M E T R Y & P A R A L L E L I S M § 2. Distant Parallelism and Rotational Invariance. The emplacement of the n-Bein orthogonal-frame field at the same time expresses the existence of a Riemannian metric and of distant parallelism. Let, for example, (A) and (B) be two vectors at the points P and Q, resp., which, with re- spect to the corresponding local n-Bein orthogonal frames, have the same respec- tive local coordinates (i.e., ) then they can be considered to be equal (ow- ing to (2)), and to be “parallel.”[8] If we regard only the metric and the distant parallelism as essential, i.e. as ob- jectively significant, then we can recognize that the n-Bein orthogonal frame is not completely determined by these emplacements. The metric and the parallelism in fact remain intact if one replaces the n-Bein orthogonal frames at all the points on the continuum by new frames that are derived from the original frames by means of a given rotation.[9] We call this replaceability of the n-Bein orthogonal-frame field a rotational invariance, and specify that only those mathematical relations that are rotationally invariant can lay claim to real meaning. With a fixed coordinate system, the for a given metric and a given parallel relation are thus still not completely determined a substitution of the is still possible, corresponding to the rotational invariance, i.e., to the equation ... , (6) where the are chosen to be orthogonal and independent of the coordinates. is an arbitrary vector referred to the original local system, and is a sim- ilar vector referred to the rotated local system. Using (1a), it follows from (6) that or , … (6a) where , … (6b) . … (6c) The postulate of rotational invariance then implies that among all the relations in which the quantity h occurs, only those can be regarded as reasonable that are trans- formed into similar relations in terms of the h*, provided that the h* are defined as in equations (6) etc. Or: n-Bein orthogonal-frame fields that differ only by uniform local rotation are equivalent. A a B a = h  a [p. 219] h  a A a d am A m = d am A a   Aa  h a A  d am h m A  = h a d am h m = d am d bm d ma d mb  ab = =  x d am 0 =