From Eqs. (2-46) and (2-47) it is inferred that its covariant derivatives vanish in

the directions tangent to the geodesic joining x and z, while Eq. (2-48) states that

it reduces to the ordinary metric (or Kronecker delta) in the coincidence limit.

Also, this bi-vector has symmetric reciprocity


 gab'(x, gba(z, x). (2-49)

 The role of the bi-vector gab' is to in !!',, i,,. ." the indices. For instance, a

local vector Ab' at the point z transforms into the local vector Aa at the point x

by parallel displacement. The application can also be extended to local tensors of

arbitrary order. In particular, one has


 gab'g cd'b'd' gac, (2-50)

 gab'g */, gb'd', (2-51)

 gab' ;b' -.;a, (2-52)

 gab ;a = --;b', (2-53)

 gab' cb' 6c, (2-54)

 gab' gad' = bd. (2-55)

 Tensor densities are also subjected to a geodesic parallel displacement by

means of the bi-vector gab'. One can introduce its determinant


 6= gJa (2-56)

This determinant is a bi-scalar density, having weight 1 at the point x and weight

-1 at the point z. It satisfies the equations


 9;agaba;b = 0, (2-57)

 ;a' a'bl' ;b = 0, (2-58)

 lim 6 1. (2-59)
 X~z