First start with a complex numbers on a plane
Represent the vector V as a complex number relative to a complex coordinate
axis: V=Vr+i Vi. The basis vectors are the unit real 1 represented as I
, and the unit imaginary number i. Rotate the I-i axis by some angle.
We can represent the rotation with a complex number 
of
unit length. The coordinates of the new (primed)axis relative to the old
are I'= I. Note
that the components of V transform oppositely from the basis vectors.
The components transform contravariantly to the way the basis vectors
transform. Unlike V , the rotation vector
is not a real object; it is derived from the coordinate transform. For
example the rotation vector effects another rotation on the primed coordinate
system is certainly not the
pictured above: that omega effects no rotation in the I'-i' system. This
is very different from the behavior of the vector V which is the same vector
regardless of the coordinate system. The transformation behavior of
is analogous to the Christoffel symbols of classical tensor analysis.
Next we draw the complex axis on a sphere surface. The axes may be thought of as geodesics. The displacements are shown here greatly exaggerated to demonstrate the effect of the surface curvature: One should imagine the displacements as infintesssemal.
A complex axis moves along the surface of a sphere of radius a.
The axis will rotate so that the imaginary axis t lines up with the coordinate
lines of 
;
the real axis lines up with 
. is
the geographic latitude ( equator=0 radian) and is
the longitude. The scaling differential form 
is:
The exterior derivative of
We now seek a one form
that solves:
is easily
found to be:
The curvature 2-form 
is
is a small complex
number proportional to the product 
representing
a parallel transversal about a little rectangle with sides .
We now scale the little rectangle in actual distance units:
The coefficient 1/a2 is indeed the scalar curvature.
Rev 5/7/0 Copyright ©2000 George Raetz <bestwork.1@pcisys.net>