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