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 3 The law of the infinitesimal parallel displacement of a vector when it is shifted from a point (xν) to a neighboring point (xν + dxν) is evidently characterized by the relation ... , (7) that is, by the equation . Multiplying by and taking (5) into account converts this equation into where .[10] [11] (7a) This law of parallel displacement is rotationally invariant and is not symmetric with respect to the lower indices of the quantity Δν μ .[12] If the vector A is shifted according to this displacement law along a closed path, then it transforms into it- self this means that the Riemannian tensor formed from the displacement coeffi- cients Δν μ , [13] vanishes identically, owing to (7a), as can readily be verified. In addition to this parallel displacement law, there is also the (non-integrable) symmetric displacement law that belongs to the Riemannian metric according to (2) and (3). As is known, it is given by the equations .[14] (8) The Γν μ can be expressed using (3) in terms of the quantities h of the n-Bein orthogonal-frame field. Here, it must be noted that , … (9) since with this choice, owing to (4) and (5), the equations are fulfilled they define the g μν from the g μν . This displacement law, based entirely on the metric, is naturally also rotationally invariant in the above sense. dA a 0 = 0 d h a A     x h a A  dx  h a dA  + 0 = = = h  a dA    A  dx  – =    h a  x h a =      R k lm  i m x  kl i – l x  km i +  i l  km   m i  kl – + = [p. 220] dA     A  dx  – =    1 2 g   x g   x g   x g  – +       ·· . =        g  h a  h a  = g  g     =